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"# 2023-04-21 Stiff equations\n",
"\n",
"## Last time\n",
"\n",
"* Notes on integration\n",
"* Ordinary differential equations (ODE)\n",
"* Explicit methods for solving ODE\n",
"* Stability\n",
"\n",
"## Today\n",
"\n",
"* Implicit and explicit methods\n",
"* Stiff equations\n",
"* PDE examples"
]
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"ode_rk_explicit (generic function with 1 method)"
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"source": [
"using LinearAlgebra\n",
"using Plots\n",
"default(linewidth=4, legendfontsize=12)\n",
"\n",
"struct RKTable\n",
" A::Matrix\n",
" b::Vector\n",
" c::Vector\n",
" function RKTable(A, b)\n",
" s = length(b)\n",
" A = reshape(A, s, s)\n",
" c = vec(sum(A, dims=2))\n",
" new(A, b, c)\n",
" end\n",
"end\n",
"\n",
"function rk_stability(z, rk)\n",
" s = length(rk.b)\n",
" 1 + z * rk.b' * ((I - z*rk.A) \\ ones(s))\n",
"end\n",
"\n",
"rk4 = RKTable([0 0 0 0; .5 0 0 0; 0 .5 0 0; 0 0 1 0], [1, 2, 2, 1] / 6)\n",
"heun = RKTable([0 0; 1 0], [.5, .5])\n",
"Rz_theta(z, theta) = (1 + (1 - theta)*z) / (1 - theta*z)\n",
"\n",
"function ode_rk_explicit(f, u0; tfinal=1, h=0.1, table=rk4)\n",
" u = copy(u0)\n",
" t = 0.\n",
" n, s = length(u), length(table.c)\n",
" fY = zeros(n, s)\n",
" thist = [t]\n",
" uhist = [u0]\n",
" while t < tfinal\n",
" tnext = min(t+h, tfinal)\n",
" h = tnext - t\n",
" for i in 1:s\n",
" ti = t + h * table.c[i]\n",
" Yi = u + h * sum(fY[:,1:i-1] * table.A[i,1:i-1], dims=2)\n",
" fY[:,i] = f(ti, Yi)\n",
" end\n",
" u += h * fY * table.b\n",
" t = tnext\n",
" push!(thist, t)\n",
" push!(uhist, u)\n",
" end\n",
" thist, hcat(uhist...)\n",
"end"
]
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"id": "1bcd691a",
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"source": [
"# Ordinary Differential Equations\n",
"\n",
"Given initial condition $y_0 = y(t=0)$, find $y(t)$ for $t > 0$ that satisfies\n",
"\n",
"$$ y' \\equiv \\dot y \\equiv \\frac{\\partial y}{\\partial t} = f(t, y) $$"
]
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"source": [
"| Application | $y$ | $f$ |\n",
"| --- | --- | --- |\n",
"| Orbital dynamics | position, momentum | conservation of momentum|\n",
"| Chemical reactions | concentration | conservation of atoms |\n",
"| Epidemiology | infected/recovered population | transmission and recovery |"
]
},
{
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"id": "e8089acc",
"metadata": {
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"source": [
"* $y$ can be a scalar or a vector"
]
},
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"id": "1858f407",
"metadata": {
"cell_style": "center",
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"source": [
"# Solving differential equations\n"
]
},
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"id": "87c81fd0",
"metadata": {
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"source": [
"## Linear equations\n",
"\n",
"$$ y' = A(t) y + \\text{source}(t)$$\n",
"\n",
"* Autonomous if $A(t) = A$ and source independent of $t$\n",
"\n",
"* Suppose $y$ and $a = A$ are scalars: $y(t) = e^{at} y_0$"
]
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"source": [
"## Can do the same for systems\n",
"\n",
"$$ y(t) = e^{A t} y_0 $$\n",
"\n",
"### What does it mean to exponentiate a matrix?\n",
"\n",
"Taylor series!\n",
"\n",
"$$ e^A = 1 + A + \\frac{A^2}{2} + \\frac{A^3}{3!} + \\dotsb $$\n",
"and there are many [practical ways to compute it](https://www.cs.cornell.edu/cv/files/2021/10/19ways.pdf).\n"
]
},
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"source": [
"## Question\n",
"Suppose that the diagonalization $A = X \\Lambda X^{-1}$ exists and derive a finite expression for the matrix exponential using the scalar `exp` function."
]
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"source": [
"# Forward Euler Method\n",
"The simplest method for solving $y'(t) = f(t,y)$ is\n",
"to use numerical differentiation to write\n",
"$$ y' \\approx \\frac{y(h) - y(0)}{h} $$\n",
"which yields the solution estimate\n",
"$$ \\tilde y(h) = y(0) + h f(0, y(0)) $$\n",
"where $h$ is the step size.\n",
"Let's try this on a scalar problem\n",
"$$ y' = -k (y - \\cos t) $$\n",
"where $k$ is a parameter controlling the rate at which the solution $u(t)$ is pulled toward the curve $\\cos t$."
]
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"ode_euler (generic function with 1 method)"
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"source": [
"function ode_euler(f, y0; tfinal=10., h=0.1)\n",
" y = copy(y0)\n",
" t = 0.\n",
" thist = [t]\n",
" yhist = [y0]\n",
" while t < tfinal\n",
" tnext = min(t+h, tfinal)\n",
" h = tnext - t\n",
" y += h * f(t, y)\n",
" t = tnext\n",
" push!(thist, t)\n",
" push!(yhist, y)\n",
" end\n",
" thist, hcat(yhist...)\n",
"end"
]
},
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},
"execution_count": 117,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f1(t, y; k=10) = -k * (y .- cos(t))\n",
"\n",
"thist, yhist = ode_euler(f1, [1.], tfinal=20, h=.2)\n",
"plot(thist, yhist[1,:], marker=:circle)\n",
"plot!(cos)"
]
},
{
"cell_type": "markdown",
"id": "eedad8e0",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# Forward Euler on a linear system\n",
"\n",
"$$ \\begin{bmatrix} y_1 \\\\ y_2 \\end{bmatrix}' = \\begin{bmatrix} 0 & 1 \\\\ -1 & 0 \\end{bmatrix} \\begin{bmatrix} y_1 \\\\ y_2 \\end{bmatrix}$$"
]
},
{
"cell_type": "code",
"execution_count": 118,
"id": "d9a8f791",
"metadata": {
"cell_style": "split"
},
"outputs": [
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" 550.328,728.117 551.392,727.862 552.457,727.628 553.522,727.417 554.587,727.228 555.651,727.063 556.716,726.92 557.781,726.801 558.845,726.706 559.91,726.635 \n",
" 560.975,726.587 562.04,726.564 563.104,726.565 564.169,726.59 565.234,726.639 566.298,726.711 567.363,726.808 568.428,726.929 569.493,727.073 570.557,727.24 \n",
" 571.622,727.43 572.687,727.643 573.752,727.878 575.881,728.413 578.01,729.032 579.075,729.372 580.14,729.731 581.205,730.108 582.269,730.503 584.233,731.277 \n",
" 586.196,732.106 588.159,732.983 590.122,733.906 592.085,734.867 594.048,735.861 596.011,736.883 597.975,737.926 601.901,740.052 605.827,742.19 607.79,743.248 \n",
" 609.754,744.289 611.717,745.309 613.68,746.301 615.643,747.259 617.606,748.178 619.569,749.052 621.533,749.876 623.496,750.646 625.459,751.356 627.422,752.003 \n",
" 629.385,752.583 630.367,752.846 631.348,753.092 632.33,753.319 633.312,753.528 634.293,753.717 635.275,753.887 636.256,754.038 637.238,754.169 638.219,754.279 \n",
" 639.201,754.37 640.183,754.441 641.164,754.491 642.146,754.52 643.127,754.529 644.109,754.518 645.091,754.486 646.143,754.429 647.195,754.349 648.248,754.245 \n",
" 649.3,754.119 650.352,753.969 651.405,753.797 652.457,753.603 653.509,753.386 654.562,753.148 655.614,752.889 656.667,752.609 657.719,752.309 658.771,751.988 \n",
" 659.824,751.649 660.876,751.291 661.928,750.914 664.033,750.11 666.138,749.241 668.242,748.314 670.347,747.334 672.452,746.308 674.557,745.244 676.661,744.148 \n",
" 678.766,743.028 682.975,740.745 687.185,738.456 689.29,737.329 691.394,736.224 693.499,735.148 695.604,734.108 696.656,733.604 697.709,733.112 698.761,732.632 \n",
" 699.813,732.166 701.918,731.276 704.023,730.448 705.075,730.06 706.127,729.689 707.18,729.336 708.232,729.002 709.284,728.688 710.337,728.394 711.389,728.12 \n",
" 712.442,727.867 713.661,727.601 714.88,727.364 716.099,727.157 717.319,726.98 718.538,726.834 719.757,726.719 720.976,726.635 722.196,726.582 723.415,726.562 \n",
" 724.634,726.572 725.853,726.615 727.073,726.689 728.292,726.794 729.511,726.93 730.73,727.097 731.95,727.295 733.169,727.522 734.388,727.779 735.607,728.065 \n",
" 736.827,728.379 738.046,728.721 739.265,729.089 740.485,729.483 741.704,729.902 742.923,730.346 744.142,730.812 745.362,731.3 746.581,731.81 749.019,732.886 \n",
" 751.458,734.032 753.896,735.237 756.335,736.49 758.773,737.779 761.212,739.094 766.089,741.75 770.966,744.363 773.405,745.623 775.843,746.838 778.282,747.996 \n",
" 780.72,749.086 781.939,749.603 783.159,750.099 784.378,750.574 785.597,751.026 786.816,751.454 788.036,751.858 789.255,752.236 790.474,752.587 791.558,752.877 \n",
" 792.641,753.145 793.725,753.39 794.808,753.612 795.892,753.811 796.975,753.986 798.059,754.137 799.143,754.264 800.226,754.366 801.31,754.444 802.393,754.497 \n",
" 803.477,754.525 804.56,754.527 805.644,754.505 806.727,754.458 807.811,754.387 808.894,754.29 809.978,754.169 811.061,754.024 812.145,753.854 813.228,753.661 \n",
" 814.312,753.444 815.396,753.204 816.479,752.942 818.646,752.351 820.813,751.676 821.897,751.309 822.98,750.922 824.064,750.516 825.147,750.093 827.314,749.196 \n",
" 829.482,748.238 831.649,747.224 833.816,746.163 835.983,745.062 838.15,743.928 840.317,742.771 842.484,741.597 846.818,739.236 851.152,736.912 853.319,735.784 \n",
" 855.486,734.691 857.653,733.64 859.821,732.638 861.721,731.805 863.621,731.021 865.521,730.289 867.422,729.613 869.322,728.997 871.222,728.445 873.123,727.959 \n",
" 875.023,727.542 875.973,727.36 876.923,727.197 877.874,727.051 878.824,726.925 879.774,726.817 880.724,726.728 881.674,726.657 882.624,726.606 883.575,726.574 \n",
" 884.525,726.561 885.475,726.568 886.425,726.594 887.375,726.638 888.325,726.702 889.276,726.785 890.226,726.887 891.176,727.008 892.126,727.147 893.076,727.304 \n",
" 894.026,727.48 895.927,727.885 897.827,728.36 899.727,728.901 901.628,729.507 903.528,730.173 905.428,730.896 907.329,731.672 909.229,732.497 911.129,733.366 \n",
" 913.03,734.275 914.93,735.218 916.83,736.19 918.731,737.186 920.631,738.2 924.775,740.45 928.918,742.701 930.99,743.81 933.062,744.897 935.134,745.956 \n",
" 937.206,746.979 938.242,747.475 939.278,747.96 940.314,748.433 941.35,748.893 943.421,749.772 945.493,750.59 946.529,750.975 947.565,751.343 948.601,751.693 \n",
" 949.637,752.025 950.673,752.338 951.709,752.632 952.745,752.906 953.781,753.16 954.817,753.394 955.853,753.606 956.889,753.798 957.924,753.967 958.96,754.115 \n",
" 959.996,754.24 961.032,754.343 962.068,754.424 963.104,754.482 964.14,754.517 965.176,754.529 966.212,754.519 967.248,754.486 968.284,754.429 969.32,754.351 \n",
" 970.356,754.249 971.392,754.126 972.428,753.98 973.463,753.812 974.499,753.623 975.535,753.412 976.571,753.18 977.607,752.928 978.643,752.655 979.679,752.363 \n",
" 980.715,752.051 981.751,751.721 982.787,751.372 984.859,750.622 986.931,749.806 989.015,748.925 991.099,747.988 993.183,747.002 995.267,745.973 999.436,743.815 \n",
" 1003.6,741.571 1007.77,739.3 1011.94,737.061 1014.02,735.973 1016.11,734.915 1018.19,733.894 1020.28,732.917 1022.36,731.99 1024.45,731.12 1026.53,730.312 \n",
" 1028.61,729.572 1029.66,729.229 1030.7,728.904 1031.74,728.599 1032.78,728.314 1033.82,728.048 1034.87,727.804 1035.91,727.58 1036.95,727.378 1037.99,727.197 \n",
" 1039.04,727.039 1040.08,726.903 1041.12,726.789 1042.16,726.698 1043.2,726.63 1044.25,726.585 1045.29,726.564 1046.33,726.565 1047.37,726.589 1048.41,726.636 \n",
" 1049.46,726.707 1050.5,726.8 1051.54,726.916 1052.58,727.054 1053.62,727.215 1054.78,727.418 1055.93,727.648 1057.08,727.904 1058.23,728.185 1059.39,728.492 \n",
" 1060.54,728.823 1061.69,729.177 1062.84,729.554 1063.99,729.954 1065.15,730.375 1066.3,730.816 1067.45,731.277 1069.75,732.255 1072.06,733.299 1074.36,734.402 \n",
" 1076.67,735.555 1078.97,736.748 1081.28,737.971 1085.88,740.47 1090.49,742.972 1092.8,744.198 1095.1,745.395 1097.41,746.553 1099.71,747.662 1102.02,748.714 \n",
" 1104.32,749.7 1105.47,750.166 1106.62,750.612 1107.78,751.038 1108.93,751.443 1110.08,751.825 1111.23,752.185 1112.38,752.522 1113.54,752.834 1114.69,753.122 \n",
" 1115.84,753.384 1116.99,753.62 1118.15,753.829 1119.3,754.012 1120.45,754.168 1121.6,754.296 1122.75,754.396 1123.91,754.469 1125.06,754.513 1126.21,754.529 \n",
" 1127.36,754.517 1128.46,754.479 1129.56,754.416 1130.65,754.327 1131.75,754.214 1132.84,754.075 1133.94,753.911 1135.04,753.723 1136.13,753.511 1137.23,753.276 \n",
" 1138.33,753.016 1139.42,752.735 1140.52,752.43 1141.61,752.105 1142.71,751.758 1143.81,751.39 1144.9,751.003 1147.1,750.171 1149.29,749.27 1151.48,748.304 \n",
" 1153.67,747.282 1155.87,746.211 1158.06,745.098 1160.25,743.952 1162.44,742.781 1166.83,740.398 1171.21,738.02 1173.41,736.855 1175.6,735.716 1177.79,734.613 \n",
" 1179.98,733.553 1182.18,732.544 1184.37,731.593 1185.46,731.142 1186.56,730.708 1187.66,730.292 1188.75,729.895 1189.85,729.517 1190.95,729.16 1192.04,728.823 \n",
" 1193.14,728.508 1194.23,728.215 1195.33,727.944 1196.43,727.696 1197.52,727.472 1198.66,727.264 1199.8,727.083 1200.94,726.928 1202.08,726.8 1203.22,726.699 \n",
" 1204.36,726.626 1205.5,726.58 1206.64,726.562 1207.78,726.571 1208.92,726.608 1210.06,726.672 1211.2,726.764 1212.34,726.883 1213.48,727.03 1214.62,727.203 \n",
" 1215.76,727.402 1216.9,727.627 1218.04,727.878 1219.18,728.154 1220.32,728.454 1221.46,728.778 1222.6,729.126 1223.74,729.496 1224.88,729.888 1227.16,730.734 \n",
" 1229.44,731.658 1231.72,732.651 1234,733.708 1236.28,734.818 1238.56,735.974 1240.84,737.166 1243.12,738.384 1247.68,740.862 1252.24,743.331 1254.52,744.537 \n",
" 1256.8,745.711 1259.07,746.845 1261.35,747.929 1263.63,748.954 1265.91,749.913 1267.05,750.366 1268.19,750.798 1269.33,751.211 1270.47,751.602 1272.39,752.212 \n",
" 1274.31,752.756 1275.27,753.002 1276.23,753.231 1277.19,753.443 1278.15,753.636 1279.11,753.811 1280.07,753.967 1281.03,754.104 1281.99,754.223 1282.95,754.322 \n",
" 1283.91,754.402 1284.87,754.463 1285.83,754.504 1286.79,754.526 1287.74,754.528 1288.7,754.51 1289.66,754.473 1290.62,754.416 1291.58,754.34 1292.54,754.245 \n",
" 1293.5,754.13 1295.42,753.844 1297.34,753.484 1298.3,753.276 1299.26,753.051 1300.22,752.808 1301.18,752.548 1303.1,751.977 1305.02,751.343 1306.93,750.648 \n",
" 1308.85,749.897 1310.77,749.093 1312.69,748.241 1314.61,747.346 1316.53,746.413 1318.45,745.448 1320.37,744.455 1322.29,743.439 1324.21,742.408 1328.04,740.32 \n",
" 1331.88,738.237 1334.23,736.988 1336.57,735.768 1338.91,734.589 1341.25,733.459 1343.6,732.388 1345.94,731.386 1347.11,730.913 1348.28,730.46 1349.45,730.028 \n",
" 1350.63,729.618 1351.8,729.23 1352.97,728.867 1354.14,728.528 1355.31,728.214 1356.48,727.925 1357.65,727.663 1358.83,727.428 1360,727.22 1361.17,727.04 \n",
" 1362.34,726.889 1363.51,726.766 1364.68,726.671 1365.85,726.606 1367.03,726.569 1368.2,726.562 1369.37,726.584 1370.54,726.635 1371.71,726.716 1372.88,726.825 \n",
" 1374.05,726.962 1375.23,727.128 1376.4,727.323 1377.57,727.544 1378.74,727.793 1379.91,728.069 1381.08,728.37 1382.25,728.697 1383.43,729.049 1384.6,729.424 \n",
" 1385.77,729.823 1386.94,730.245 1388.11,730.687 1390.45,731.634 1392.8,732.654 1395.14,733.741 1397.48,734.884 1399.83,736.075 1402.17,737.303 1404.51,738.558 \n",
" 1406.86,739.83 1411.15,742.167 1415.44,744.459 1417.59,745.568 1419.73,746.642 1421.88,747.673 1424.03,748.654 1426.17,749.579 1428.32,750.44 1429.39,750.845 \n",
" 1430.46,751.231 1431.54,751.6 1432.61,751.948 1433.68,752.277 1434.76,752.585 1435.83,752.872 1436.9,753.138 1437.98,753.381 1439.05,753.602 1440.12,753.8 \n",
" 1441.2,753.975 1442.27,754.127 1443.34,754.254 1444.41,754.358 1445.49,754.437 1446.56,754.492 1447.63,754.522 1448.71,754.529 1449.78,754.51 1450.85,754.467 \n",
" 1451.93,754.4 1453,754.308 1454.07,754.193 1455.15,754.053 1456.22,753.89 1457.29,753.703 1458.37,753.493 1459.44,753.261 1460.51,753.006 1461.58,752.73 \n",
" 1462.66,752.432 1463.73,752.113 1464.8,751.774 1465.88,751.416 1466.95,751.038 1469.1,750.228 1471.24,749.351 1473.39,748.411 1475.54,747.417 1477.52,746.456 \n",
" 1479.5,745.459 1481.48,744.433 1483.46,743.384 1487.43,741.241 1491.39,739.081 1495.36,736.956 1499.32,734.917 1501.3,733.945 1503.29,733.013 1505.27,732.125 \n",
" 1507.25,731.288 1509.23,730.506 1511.22,729.784 1513.2,729.126 1515.18,728.537 1516.17,728.269 1517.16,728.019 1518.15,727.788 1519.14,727.576 1520.14,727.384 \n",
" 1521.13,727.211 1522.12,727.058 1523.11,726.925 1524.1,726.813 1525.09,726.721 1526.08,726.65 1527.07,726.6 1528.06,726.57 1529.06,726.561 1530.05,726.574 \n",
" 1531.04,726.607 1532.03,726.661 1533.02,726.735 1534.01,726.831 1535,726.946 1535.99,727.082 1536.98,727.239 1537.98,727.415 1538.97,727.611 1539.97,727.829 \n",
" 1540.97,728.066 1541.98,728.323 1542.98,728.598 1544.99,729.204 1547,729.878 1549.01,730.619 1551.01,731.42 1553.02,732.277 1555.03,733.185 1557.04,734.138 \n",
" 1559.05,735.13 1561.05,736.155 1563.06,737.208 1567.08,739.367 1571.09,741.555 1575.11,743.719 1579.13,745.805 1581.13,746.803 1583.14,747.762 1585.15,748.677 \n",
" 1587.16,749.542 1589.17,750.353 1591.17,751.102 1593.18,751.788 1595.19,752.404 1596.19,752.685 1597.2,752.947 1598.2,753.191 1599.2,753.415 1600.21,753.619 \n",
" 1601.21,753.804 1602.22,753.967 1603.22,754.111 1604.35,754.247 1605.47,754.356 1606.6,754.439 1607.72,754.495 1608.85,754.525 1609.97,754.527 1611.1,754.502 \n",
" 1612.22,754.451 1613.35,754.373 1614.47,754.268 1615.6,754.137 1616.72,753.979 1617.85,753.796 1618.97,753.587 1620.1,753.353 1621.22,753.094 1622.35,752.812 \n",
" 1623.47,752.505 1624.6,752.176 1625.72,751.824 1626.85,751.45 1627.97,751.055 1629.1,750.641 1630.22,750.206 1632.47,749.283 1634.73,748.292 1635.85,747.774 \n",
" 1636.98,747.241 1638.1,746.696 1639.23,746.139 1641.48,744.994 1643.73,743.815 1645.98,742.61 1648.23,741.39 1652.73,738.939 1657.23,736.537 1659.48,735.378 \n",
" 1661.73,734.259 1663.98,733.188 1666.23,732.174 1667.36,731.69 1668.48,731.224 1669.61,730.775 1670.73,730.346 1671.86,729.936 1672.98,729.546 1674.11,729.178 \n",
" 1675.23,728.832 1676.25,728.537 1677.27,728.262 1678.29,728.006 1679.31,727.769 1680.33,727.553 1681.35,727.358 1682.37,727.183 1683.39,727.03 1684.41,726.898 \n",
" 1685.43,726.787 1686.45,726.699 1687.47,726.632 1688.49,726.587 1689.51,726.564 1690.53,726.564 1691.54,726.585 1692.56,726.629 1693.58,726.694 1694.6,726.782 \n",
" 1695.62,726.891 1696.64,727.022 1697.66,727.174 1698.68,727.347 1699.7,727.542 1700.72,727.757 1701.74,727.992 1702.76,728.246 1703.78,728.521 1704.8,728.814 \n",
" 1705.82,729.126 1706.84,729.456 1707.86,729.804 1709.9,730.549 1711.94,731.358 1713.98,732.225 1716.01,733.144 1718.05,734.11 1720.09,735.117 1722.13,736.158 \n",
" 1724.17,737.227 1728.25,739.421 1732.33,741.644 1736.41,743.838 1740.48,745.95 1742.82,747.1 1745.15,748.196 1746.32,748.721 1747.49,749.229 1748.66,749.719 \n",
" 1749.82,750.19 1750.99,750.641 1752.16,751.071 1753.33,751.479 1754.49,751.865 1755.66,752.227 1756.83,752.565 1757.99,752.878 1759.16,753.165 1760.33,753.426 \n",
" 1761.5,753.661 1762.66,753.868 1763.83,754.047 1765,754.199 1766.17,754.323 1767.33,754.417 1768.5,754.483 1769.67,754.52 1770.84,754.529 1772,754.508 \n",
" 1773.17,754.458 1774.34,754.379 1775.51,754.272 1776.67,754.136 1777.84,753.972 1779.01,753.781 1780.17,753.561 1781.34,753.315 1782.51,753.043 1783.68,752.744 \n",
" 1784.84,752.42 1786.01,752.072 1787.18,751.699 1788.35,751.304 1789.51,750.886 1790.68,750.447 1791.85,749.987 1793.02,749.507 1794.18,749.01 1795.35,748.494 \n",
" 1796.52,747.962 1798.85,746.853 1801.19,745.692 1803.52,744.488 1805.86,743.251 1810.53,740.722 1815.2,738.186 1817.36,737.033 1819.52,735.906 1821.68,734.811 \n",
" 1823.84,733.757 1826,732.751 1828.17,731.801 1830.33,730.913 1832.49,730.094 1833.57,729.711 1834.65,729.348 1835.73,729.005 1836.81,728.682 1837.89,728.381 \n",
" 1838.97,728.101 1840.06,727.843 1841.14,727.608 1842.22,727.396 1843.3,727.207 1844.38,727.042 1845.46,726.9 1846.54,726.783 1847.62,726.691 1848.7,726.623 \n",
" 1849.78,726.58 1850.86,726.562 1851.94,726.569 1853.03,726.6 1854.11,726.656 1855.19,726.737 1856.27,726.843 1857.35,726.972 1858.43,727.126 1859.51,727.304 \n",
" 1860.59,727.505 1861.67,727.73 1862.75,727.977 1863.83,728.247 1864.91,728.538 1866,728.851 1867.08,729.185 1868.16,729.539 1869.24,729.912 1870.32,730.304 \n",
" 1871.4,730.715 1873.56,731.587 1875.72,732.524 1877.89,733.517 1880.05,734.56 1882.21,735.646 1884.37,736.766 1888.63,739.047 1892.9,741.369 1897.16,743.668 \n",
" 1901.42,745.881 1903.55,746.936 1905.68,747.946 1907.81,748.906 1909.94,749.808 1912.08,750.646 1914.21,751.414 1915.27,751.77 1916.34,752.107 1917.4,752.424 \n",
" 1918.47,752.721 1919.54,752.996 1920.6,753.25 1921.67,753.482 1922.73,753.691 1923.8,753.878 1924.86,754.042 1925.93,754.182 1926.99,754.299 1928.06,754.392 \n",
" 1929.13,754.461 1930.19,754.507 1931.26,754.528 1932.32,754.525 1933.39,754.497 1934.45,754.446 1935.52,754.371 1936.59,754.271 1937.65,754.148 1938.72,754.002 \n",
" 1939.78,753.832 1940.85,753.64 1941.91,753.424 1942.98,753.187 1944.04,752.927 1945.11,752.647 1946.18,752.345 1947.24,752.023 1948.31,751.681 1949.37,751.32 \n",
" 1950.44,750.94 1951.5,750.542 1952.57,750.127 1954.64,749.275 1956.71,748.366 1958.78,747.405 1960.85,746.401 1962.92,745.358 1964.98,744.283 1967.05,743.185 \n",
" 1969.12,742.069 1973.26,739.815 1977.4,737.579 1979.47,736.487 1981.54,735.421 1983.61,734.389 1985.68,733.397 1987.75,732.451 1989.82,731.558 1991.88,730.723 \n",
" 1993.95,729.953 1994.99,729.593 1996.02,729.251 1997.06,728.927 1998.09,728.623 1999.13,728.338 2000.16,728.072 2001.2,727.827 2002.23,727.603 2003.27,727.4 \n",
" 2004.3,727.218 2005.33,727.058 2006.37,726.92 2007.4,726.804 2008.44,726.711 2009.47,726.64 2010.51,726.591 2011.54,726.566 2012.58,726.563 2013.61,726.583 \n",
" 2014.65,726.625 2015.68,726.691 2016.72,726.779 2017.75,726.889 2018.78,727.021 2019.85,727.18 2020.91,727.362 2021.97,727.566 2023.03,727.792 2024.09,728.041 \n",
" 2025.15,728.31 2026.21,728.601 2027.27,728.912 2028.33,729.243 2029.39,729.593 2030.45,729.962 2031.51,730.349 2033.63,731.175 2035.75,732.065 2037.88,733.013 \n",
" 2040,734.012 2042.12,735.057 2044.24,736.139 2048.48,738.385 2052.72,740.691 2056.97,742.992 2061.21,745.227 2063.33,746.3 2065.45,747.334 2067.57,748.321 \n",
" 2069.69,749.255 2071.82,750.13 2073.94,750.938 2075,751.316 2076.06,751.676 2077.12,752.017 2078.18,752.338 2079.24,752.638 2080.3,752.919 2081.36,753.177 \n",
" 2082.42,753.415 2083.48,753.63 2084.54,753.823 2085.6,753.993 2086.66,754.14 2087.83,754.275 2088.99,754.381 2090.15,754.459 2091.31,754.508 2092.47,754.529 \n",
" 2093.64,754.521 2094.8,754.484 2095.96,754.418 2097.12,754.324 2098.28,754.202 2099.45,754.052 2100.61,753.873 2101.77,753.668 2102.93,753.436 2104.09,753.177 \n",
" 2105.26,752.892 2106.42,752.582 2107.58,752.247 2108.74,751.888 2109.9,751.505 2111.07,751.101 2112.23,750.674 2113.39,750.227 2114.55,749.76 2116.88,748.769 \n",
" 2119.2,747.712 2121.52,746.595 2123.85,745.429 2128.5,742.986 2133.14,740.463 2137.79,737.943 2142.44,735.508 2144.76,734.347 2147.09,733.238 2149.41,732.189 \n",
" 2151.74,731.208 2152.9,730.746 2154.06,730.304 2155.22,729.883 2156.39,729.484 2157.55,729.107 2158.71,728.755 2159.87,728.426 2161.03,728.122 2162.72,727.727 \n",
" 2164.41,727.387 2166.1,727.104 2167.79,726.879 2169.47,726.714 2171.16,726.608 2172.85,726.563 2174.54,726.579 2176.23,726.655 2177.91,726.791 2179.6,726.987 \n",
" 2181.29,727.241 2182.98,727.554 2184.67,727.922 2186.35,728.345 2188.04,728.822 2189.73,729.349 2191.42,729.924 2193.11,730.546 2194.79,731.211 2196.48,731.916 \n",
" 2198.17,732.659 2199.86,733.436 2201.55,734.244 2204.92,735.937 2208.3,737.711 2211.67,739.534 2215.05,741.374 2218.43,743.2 2221.8,744.98 2223.49,745.843 \n",
" 2225.18,746.683 2226.87,747.496 2228.55,748.28 2230.24,749.03 2231.93,749.743 2233.62,750.416 2235.31,751.046 2236.99,751.631 2238.68,752.168 2240.37,752.655 \n",
" 2242.06,753.089 2243.75,753.468 2245.43,753.792 2247.12,754.058 2248.81,754.266 2250.5,754.414 2252.19,754.502 2253.88,754.529 2255.56,754.496 2257.25,754.403 \n",
" 2258.94,754.249 2260.63,754.036 2262.32,753.765 2264,753.436 2265.69,753.051 2267.38,752.612 2269.07,752.121 2270.45,751.681 2271.84,751.208 2273.22,750.703 \n",
" 2274.61,750.169 2275.99,749.607 2277.38,749.019 2278.76,748.406 2280.15,747.77 2282.92,746.436 2285.69,745.034 2288.46,743.579 2291.23,742.089 \n",
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" 552.457,745.903 554.587,744.813 556.716,743.693 560.975,741.397 565.234,739.077 567.363,737.928 569.493,736.797 571.622,735.692 573.752,734.621 575.881,733.59 \n",
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" 594.048,727.369 595.03,727.199 596.011,727.049 596.993,726.919 597.975,726.809 598.956,726.719 599.938,726.649 600.919,726.599 601.901,726.57 602.883,726.561 \n",
" 603.864,726.573 604.846,726.606 605.827,726.658 606.809,726.731 607.79,726.825 608.772,726.938 609.754,727.072 610.735,727.225 611.717,727.398 612.698,727.59 \n",
" 613.68,727.801 614.662,728.03 615.643,728.278 616.625,728.544 617.606,728.828 619.569,729.446 621.533,730.13 623.496,730.874 625.459,731.675 626.441,732.096 \n",
" 627.422,732.528 628.404,732.973 629.385,733.428 631.348,734.37 633.312,735.348 635.275,736.357 637.238,737.39 641.164,739.505 645.091,741.645 647.195,742.784 \n",
" 649.3,743.909 651.405,745.011 653.509,746.082 655.614,747.117 657.719,748.107 659.824,749.046 661.928,749.928 664.033,750.747 666.138,751.497 667.19,751.845 \n",
" 668.242,752.173 669.295,752.482 670.347,752.771 671.4,753.039 672.452,753.287 673.504,753.512 674.557,753.716 675.609,753.898 676.661,754.057 677.714,754.194 \n",
" 678.766,754.307 679.818,754.397 680.871,754.464 681.923,754.508 682.975,754.528 684.028,754.524 685.08,754.497 686.133,754.447 687.185,754.372 688.237,754.275 \n",
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" 699.813,751.741 700.866,751.387 701.918,751.016 702.97,750.626 704.023,750.22 706.127,749.359 708.232,748.439 709.284,747.959 710.337,747.466 711.389,746.962 \n",
" 712.442,746.446 714.88,745.216 717.319,743.943 719.757,742.639 722.196,741.316 727.073,738.662 731.95,736.075 734.388,734.837 736.827,733.65 738.046,733.08 \n",
" 739.265,732.526 740.485,731.99 741.704,731.474 742.923,730.978 744.142,730.504 745.362,730.053 746.581,729.625 747.8,729.222 749.019,728.845 750.239,728.494 \n",
" 751.458,728.17 752.677,727.875 753.896,727.608 755.116,727.37 756.335,727.162 757.554,726.984 758.773,726.837 759.993,726.721 761.212,726.637 762.431,726.583 \n",
" 763.65,726.562 764.87,726.572 766.089,726.613 767.308,726.686 768.528,726.791 769.747,726.926 770.966,727.092 772.185,727.289 773.405,727.516 774.624,727.772 \n",
" 775.843,728.057 777.062,728.37 778.282,728.711 779.501,729.079 780.72,729.472 781.939,729.891 783.159,730.334 784.378,730.799 785.597,731.287 788.036,732.324 \n",
" 790.474,733.436 792.641,734.478 794.808,735.564 796.975,736.685 799.143,737.833 803.477,740.18 807.811,742.538 809.978,743.699 812.145,744.838 814.312,745.947 \n",
" 816.479,747.017 818.646,748.04 820.813,749.01 822.98,749.92 825.147,750.762 826.231,751.157 827.314,751.532 828.398,751.888 829.482,752.223 830.565,752.538 \n",
" 831.649,752.831 832.732,753.102 833.816,753.351 834.899,753.577 835.983,753.78 837.066,753.959 838.15,754.114 839.233,754.245 840.317,754.351 841.4,754.433 \n",
" 842.484,754.49 843.567,754.522 844.651,754.529 845.735,754.511 846.818,754.468 847.902,754.4 848.985,754.308 850.069,754.19 851.152,754.049 852.236,753.883 \n",
" 853.319,753.694 854.403,753.481 855.486,753.245 856.57,752.986 857.653,752.705 858.737,752.403 859.821,752.079 861.721,751.462 863.621,750.784 865.521,750.051 \n",
" 867.422,749.265 869.322,748.432 871.222,747.555 873.123,746.639 875.023,745.69 878.824,743.713 882.624,741.666 886.425,739.595 890.226,737.545 892.126,736.541 \n",
" 894.026,735.56 895.927,734.606 897.827,733.685 899.727,732.801 901.628,731.96 903.528,731.166 905.428,730.424 907.329,729.737 909.229,729.109 911.129,728.545 \n",
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" 922.703,726.615 923.739,726.577 924.775,726.562 925.811,726.569 926.847,726.599 927.882,726.653 928.918,726.728 929.954,726.827 930.99,726.948 932.026,727.091 \n",
" 933.062,727.256 934.098,727.442 935.134,727.65 936.17,727.879 937.206,728.129 938.242,728.399 939.278,728.689 940.314,728.998 941.35,729.326 943.421,730.037 \n",
" 945.493,730.816 947.565,731.659 949.637,732.56 951.709,733.512 953.781,734.511 955.853,735.549 957.924,736.62 959.996,737.716 962.068,738.831 966.212,741.087 \n",
" 970.356,743.329 972.428,744.426 974.499,745.498 976.571,746.538 978.643,747.538 980.715,748.493 982.787,749.396 984.859,750.241 986.931,751.023 987.973,751.391 \n",
" 989.015,751.741 990.057,752.072 991.099,752.384 992.141,752.677 993.183,752.95 994.225,753.202 995.267,753.433 996.309,753.643 997.352,753.831 998.394,753.998 \n",
" 999.436,754.142 1000.48,754.263 1001.52,754.362 1002.56,754.438 1003.6,754.492 1004.65,754.522 1005.69,754.529 1006.73,754.513 1007.77,754.474 1008.81,754.412 \n",
" 1009.86,754.327 1010.9,754.219 1011.94,754.088 1012.98,753.936 1014.02,753.761 1015.07,753.564 1016.11,753.346 1017.15,753.106 1018.19,752.846 1019.24,752.566 \n",
" 1020.28,752.265 1021.32,751.946 1022.36,751.607 1023.4,751.25 1024.45,750.876 1026.53,750.076 1028.61,749.214 1030.7,748.294 1032.78,747.323 1034.87,746.307 \n",
" 1036.95,745.254 1041.12,743.06 1045.29,740.8 1049.46,738.533 1053.62,736.319 1055.93,735.14 1058.23,734.004 1060.54,732.921 1062.84,731.899 1063.99,731.414 \n",
" 1065.15,730.948 1066.3,730.5 1067.45,730.073 1068.6,729.668 1069.75,729.284 1070.91,728.923 1072.06,728.585 1073.21,728.271 1074.36,727.983 1075.52,727.719 \n",
" 1076.67,727.482 1077.82,727.271 1078.97,727.087 1080.12,726.93 1081.28,726.8 1082.43,726.698 1083.58,726.625 1084.73,726.579 1085.88,726.561 1087.04,726.572 \n",
" 1088.19,726.611 1089.34,726.678 1090.49,726.773 1091.65,726.896 1092.8,727.047 1093.95,727.225 1095.1,727.429 1096.25,727.66 1097.41,727.918 1098.56,728.2 \n",
" 1099.71,728.508 1100.86,728.84 1102.02,729.195 1103.17,729.574 1104.32,729.974 1105.47,730.396 1106.62,730.839 1107.78,731.301 1108.93,731.782 1111.23,732.795 \n",
" 1113.54,733.872 1115.84,735.002 1118.15,736.177 1122.75,738.622 1127.36,741.13 1131.75,743.5 1136.13,745.783 1138.33,746.872 1140.52,747.914 1142.71,748.902 \n",
" 1144.9,749.829 1146,750.268 1147.1,750.689 1148.19,751.091 1149.29,751.474 1151.48,752.179 1153.67,752.8 1154.77,753.076 1155.87,753.33 1156.96,753.561 \n",
" 1158.06,753.767 1159.15,753.95 1160.25,754.108 1161.35,754.241 1162.44,754.349 1163.54,754.432 1164.64,754.49 1165.73,754.522 1166.83,754.529 1167.92,754.509 \n",
" 1169.02,754.465 1170.12,754.395 1171.21,754.299 1172.31,754.179 1173.41,754.034 1174.5,753.863 1175.6,753.669 1176.69,753.451 1177.79,753.208 1178.89,752.943 \n",
" 1179.98,752.655 1181.08,752.345 1182.18,752.014 1183.27,751.661 1184.37,751.288 1185.46,750.896 1186.56,750.484 1187.66,750.055 1188.75,749.608 1190.95,748.665 \n",
" 1193.14,747.663 1195.33,746.608 1197.52,745.51 1202.08,743.119 1206.64,740.646 1211.2,738.171 1215.76,735.77 1218.04,734.622 1220.32,733.52 1222.6,732.474 \n",
" 1224.88,731.492 1227.16,730.581 1229.44,729.749 1230.58,729.365 1231.72,729.002 1232.86,728.663 1234,728.347 1235.14,728.055 1236.28,727.788 1237.42,727.546 \n",
" 1238.56,727.33 1239.7,727.139 1240.84,726.976 1241.98,726.839 1243.12,726.729 1244.26,726.647 1245.4,726.592 1246.54,726.565 1247.68,726.565 1248.82,726.593 \n",
" 1249.96,726.648 1251.1,726.731 1252.24,726.841 1253.38,726.979 1254.52,727.143 1255.66,727.334 1256.8,727.55 1257.94,727.793 1259.07,728.061 1260.21,728.353 \n",
" 1261.35,728.669 1262.49,729.009 1263.63,729.372 1264.77,729.757 1265.91,730.163 1267.05,730.59 1268.19,731.036 1269.33,731.501 1270.47,731.984 1272.39,732.835 \n",
" 1274.31,733.729 1276.23,734.662 1278.15,735.627 1281.99,737.634 1285.83,739.707 1289.66,741.799 1293.5,743.862 1295.42,744.869 1297.34,745.851 1299.26,746.804 \n",
" 1301.18,747.722 1303.1,748.599 1305.02,749.432 1306.93,750.214 1308.85,750.943 1310.77,751.613 1312.69,752.221 1314.61,752.764 1316.53,753.238 1317.49,753.449 \n",
" 1318.45,753.642 1319.41,753.816 1320.37,753.972 1321.33,754.109 1322.29,754.227 1323.25,754.325 1324.21,754.405 1325.17,754.465 1326.13,754.505 1327.08,754.526 \n",
" 1328.04,754.528 1329,754.509 1329.96,754.472 1330.92,754.414 1331.88,754.338 1333.05,754.218 1334.23,754.069 1335.4,753.893 1336.57,753.688 1337.74,753.456 \n",
" 1338.91,753.197 1340.08,752.912 1341.25,752.601 1342.43,752.265 1343.6,751.904 1344.77,751.519 1345.94,751.112 1347.11,750.683 1348.28,750.232 1349.45,749.761 \n",
" 1350.63,749.271 1352.97,748.237 1355.31,747.139 1357.65,745.986 1360,744.788 1364.68,742.295 1369.37,739.744 1371.71,738.473 1374.05,737.22 1376.4,735.994 \n",
" 1378.74,734.806 1381.08,733.666 1383.43,732.584 1385.77,731.568 1388.11,730.627 1389.28,730.187 1390.45,729.769 1391.63,729.373 1392.8,729 1393.97,728.652 \n",
" 1395.14,728.328 1396.31,728.03 1397.48,727.758 1398.66,727.513 1399.83,727.295 1401,727.105 1402.17,726.942 1403.34,726.808 1404.51,726.703 1405.68,726.627 \n",
" 1406.86,726.58 1407.93,726.562 1409,726.569 1410.07,726.6 1411.15,726.656 1412.22,726.736 1413.29,726.84 1414.37,726.968 1415.44,727.12 1416.51,727.296 \n",
" 1417.59,727.495 1418.66,727.716 1419.73,727.96 1420.81,728.226 1421.88,728.514 1422.95,728.823 1424.03,729.152 1425.1,729.502 1426.17,729.87 1427.24,730.258 \n",
" 1428.32,730.663 1430.46,731.525 1432.61,732.451 1434.76,733.432 1436.9,734.464 1439.05,735.539 1441.2,736.648 1445.49,738.941 1449.78,741.278 1454.07,743.596 \n",
" 1458.37,745.827 1460.51,746.892 1462.66,747.911 1464.8,748.88 1466.95,749.79 1468.02,750.221 1469.1,750.635 1470.17,751.031 1471.24,751.409 1472.32,751.768 \n",
" 1473.39,752.107 1474.46,752.426 1475.54,752.725 1476.53,752.981 1477.52,753.219 1478.51,753.438 1479.5,753.638 1480.49,753.818 1481.48,753.978 1482.47,754.118 \n",
" 1483.46,754.238 1484.46,754.338 1485.45,754.417 1486.44,754.475 1487.43,754.512 1488.42,754.528 1489.41,754.524 1490.4,754.499 1491.39,754.452 1492.38,754.385 \n",
" 1493.38,754.298 1494.37,754.19 1495.36,754.061 1496.35,753.912 1497.34,753.743 1498.33,753.555 1499.32,753.347 1500.31,753.12 1501.3,752.874 1502.3,752.61 \n",
" 1503.29,752.327 1505.27,751.71 1507.25,751.026 1509.23,750.28 1511.22,749.475 1513.2,748.617 1515.18,747.711 1517.16,746.762 1519.14,745.776 1521.13,744.758 \n",
" 1523.11,743.716 1527.07,741.58 1531.04,739.42 1535,737.286 1538.97,735.23 1540.97,734.235 1542.98,733.277 1544.99,732.365 1547,731.503 1549.01,730.696 \n",
" 1551.01,729.949 1553.02,729.268 1555.03,728.655 1556.03,728.376 1557.04,728.116 1558.04,727.874 1559.05,727.652 1560.05,727.45 1561.05,727.268 1562.06,727.106 \n",
" 1563.06,726.965 1564.07,726.845 1565.07,726.746 1566.07,726.668 1567.08,726.611 1568.08,726.575 1569.09,726.561 1570.09,726.569 1571.09,726.598 1572.1,726.648 \n",
" 1573.1,726.72 1574.11,726.812 1575.11,726.926 1576.11,727.061 1577.12,727.216 1578.12,727.392 1579.13,727.588 1580.13,727.804 1581.13,728.039 1582.14,728.294 \n",
" 1583.14,728.567 1585.15,729.169 1587.16,729.84 1589.17,730.577 1591.17,731.375 1593.18,732.229 1595.19,733.134 1597.2,734.085 1599.2,735.075 1601.21,736.099 \n",
" 1603.22,737.15 1607.72,739.571 1612.22,742.023 1614.47,743.236 1616.72,744.429 1618.97,745.592 1621.22,746.716 1622.35,747.26 1623.47,747.792 1624.6,748.31 \n",
" 1625.72,748.813 1627.97,749.77 1630.22,750.656 1631.35,751.07 1632.47,751.464 1633.6,751.836 1634.73,752.188 1635.85,752.516 1636.98,752.822 1638.1,753.104 \n",
" 1639.23,753.362 1640.35,753.595 1641.48,753.803 1642.6,753.985 1643.73,754.142 1644.85,754.272 1645.98,754.376 1647.1,754.453 1648.23,754.504 1649.35,754.527 \n",
" 1650.48,754.524 1651.6,754.494 1652.73,754.437 1653.85,754.353 1654.98,754.242 1656.1,754.106 1657.23,753.943 1658.35,753.754 1659.48,753.54 1660.6,753.3 \n",
" 1661.73,753.037 1662.85,752.749 1663.98,752.437 1665.1,752.103 1666.23,751.747 1667.36,751.368 1668.48,750.969 1669.61,750.55 1670.73,750.112 1672.98,749.181 \n",
" 1675.23,748.184 1677.27,747.229 1679.31,746.231 1681.35,745.198 1683.39,744.136 1687.47,741.949 1691.54,739.727 1695.62,737.526 1699.7,735.401 1701.74,734.384 \n",
" 1703.78,733.406 1705.82,732.473 1707.86,731.591 1709.9,730.766 1711.94,730.003 1712.96,729.646 1713.98,729.306 1714.99,728.984 1716.01,728.68 1717.03,728.396 \n",
" 1718.05,728.13 1719.07,727.884 1720.09,727.658 1721.11,727.452 1722.13,727.267 1723.15,727.103 1724.17,726.961 1725.19,726.839 1726.21,726.74 1727.23,726.662 \n",
" 1728.25,726.606 1729.27,726.573 1730.29,726.561 1731.31,726.572 1732.33,726.604 1733.35,726.659 1734.37,726.736 1735.39,726.834 1736.41,726.954 1737.43,727.096 \n",
" 1738.45,727.259 1739.46,727.443 1740.48,727.648 1741.65,727.907 1742.82,728.193 1743.99,728.504 1745.15,728.84 1746.32,729.2 1747.49,729.584 1748.66,729.991 \n",
" 1749.82,730.42 1752.16,731.339 1754.49,732.334 1755.66,732.858 1756.83,733.398 1757.99,733.952 1759.16,734.52 1761.5,735.693 1763.83,736.906 1766.17,738.149 \n",
" 1768.5,739.412 1773.17,741.956 1777.84,744.452 1780.17,745.657 1782.51,746.82 1784.84,747.931 1787.18,748.98 1788.35,749.479 1789.51,749.959 1790.68,750.42 \n",
" 1791.85,750.861 1793.02,751.28 1794.18,751.677 1795.35,752.051 1796.52,752.4 1797.68,752.726 1798.85,753.026 1800.02,753.3 1801.19,753.548 1802.35,753.769 \n",
" 1803.52,753.962 1804.69,754.128 1805.86,754.265 1807.02,754.374 1808.19,754.454 1809.36,754.506 1810.53,754.528 1811.69,754.522 1812.86,754.486 1814.03,754.422 \n",
" 1815.2,754.329 1816.28,754.217 1817.36,754.081 1818.44,753.921 1819.52,753.737 1820.6,753.53 1821.68,753.3 1822.76,753.047 1823.84,752.771 1824.92,752.474 \n",
" 1826,752.156 1827.08,751.817 1828.17,751.458 1829.25,751.08 1830.33,750.683 1831.41,750.268 1832.49,749.836 1834.65,748.923 1836.81,747.95 1838.97,746.924 \n",
" 1841.14,745.853 1845.46,743.606 1849.78,741.273 1854.11,738.918 1858.43,736.61 1860.59,735.494 1862.75,734.413 1864.91,733.377 1867.08,732.391 1869.24,731.463 \n",
" 1871.4,730.6 1872.48,730.194 1873.56,729.807 1874.64,729.439 1875.72,729.091 1876.8,728.763 1877.89,728.456 1878.97,728.17 1880.05,727.907 1881.13,727.666 \n",
" 1882.21,727.448 1883.29,727.253 1884.37,727.081 1885.44,726.936 1886.5,726.814 1887.57,726.716 1888.63,726.642 1889.7,726.591 1890.76,726.565 1891.83,726.563 \n",
" 1892.9,726.585 1893.96,726.632 1895.03,726.702 1896.09,726.796 1897.16,726.914 1898.22,727.056 1899.29,727.221 1900.35,727.408 1901.42,727.619 1902.49,727.852 \n",
" 1903.55,728.107 1904.62,728.383 1905.68,728.68 1906.75,728.998 1907.81,729.336 1908.88,729.693 1909.94,730.069 1912.08,730.874 1914.21,731.746 1916.34,732.679 \n",
" 1918.47,733.667 1920.6,734.701 1922.73,735.777 1924.86,736.885 1926.99,738.018 1931.26,740.329 1935.52,742.646 1937.65,743.787 1939.78,744.906 1941.91,745.994 \n",
" 1944.04,747.044 1946.18,748.05 1948.31,749.004 1950.44,749.9 1952.57,750.73 1953.6,751.109 1954.64,751.47 1955.67,751.813 1956.71,752.138 1957.74,752.444 \n",
" 1958.78,752.731 1959.81,752.998 1960.85,753.244 1961.88,753.47 1962.92,753.675 1963.95,753.859 1964.98,754.02 1966.02,754.16 1967.05,754.278 1968.09,754.373 \n",
" 1969.12,754.446 1970.16,754.496 1971.19,754.524 1972.23,754.528 1973.26,754.51 1974.3,754.469 1975.33,754.406 1976.37,754.32 1977.4,754.211 1978.43,754.08 \n",
" 1979.47,753.928 1980.5,753.753 1981.54,753.557 1982.57,753.339 1983.61,753.101 1984.64,752.843 1985.68,752.564 1986.71,752.266 1987.75,751.948 1988.78,751.613 \n",
" 1989.82,751.259 1991.88,750.499 1993.95,749.675 1996.02,748.791 1998.09,747.853 2000.16,746.868 2002.23,745.842 2006.37,743.693 2010.51,741.462 2014.65,739.207 \n",
" 2018.78,736.987 2020.91,735.882 2023.03,734.808 2025.15,733.774 2027.27,732.785 2029.39,731.85 2031.51,730.975 2032.57,730.561 2033.63,730.165 2034.69,729.786 \n",
" 2035.75,729.426 2036.81,729.085 2037.88,728.763 2038.94,728.462 2040,728.181 2041.06,727.921 2042.12,727.683 2043.18,727.467 2044.24,727.274 2045.3,727.103 \n",
" 2046.36,726.955 2047.42,726.83 2048.48,726.729 2049.54,726.652 2050.6,726.598 2051.66,726.568 2052.72,726.562 2053.78,726.58 2054.85,726.622 2055.91,726.688 \n",
" 2056.97,726.777 2058.03,726.89 2059.09,727.026 2060.15,727.186 2061.21,727.368 2062.27,727.573 2063.33,727.8 2064.39,728.049 2065.45,728.319 2066.51,728.611 \n",
" 2067.57,728.922 2068.63,729.254 2069.69,729.605 2071.82,730.362 2073.94,731.189 2076.06,732.08 2078.18,733.029 2080.3,734.029 2082.42,735.074 2084.54,736.157 \n",
" 2086.66,737.269 2091.31,739.774 2095.96,742.305 2098.28,743.554 2100.61,744.778 2102.93,745.967 2105.26,747.112 2107.58,748.202 2109.9,749.23 2112.23,750.187 \n",
" 2114.55,751.064 2115.71,751.471 2116.88,751.855 2118.04,752.216 2119.2,752.553 2120.36,752.866 2121.52,753.153 2122.69,753.414 2123.85,753.649 2125.01,753.857 \n",
" 2126.17,754.037 2127.33,754.19 2128.5,754.315 2129.66,754.411 2130.82,754.479 2131.98,754.518 2133.14,754.529 2134.31,754.511 2135.47,754.464 2136.63,754.389 \n",
" 2137.79,754.285 2138.95,754.153 2140.12,753.993 2141.28,753.805 2142.44,753.59 2143.6,753.349 2144.76,753.081 2145.93,752.787 2147.09,752.468 2148.25,752.125 \n",
" 2149.41,751.758 2150.58,751.368 2151.74,750.955 2154.06,750.067 2156.39,749.101 2157.55,748.591 2158.71,748.064 2159.87,747.522 2161.03,746.966 2164.41,745.279 \n",
" 2167.79,743.51 2171.16,741.69 2174.54,739.85 2177.91,738.022 2181.29,736.237 2182.98,735.371 2184.67,734.528 2186.35,733.71 2188.04,732.922 2189.73,732.167 \n",
" 2191.42,731.449 2193.11,730.77 2194.79,730.133 2196.48,729.541 2198.17,728.997 2199.86,728.503 2201.55,728.062 2203.23,727.674 2204.92,727.342 2206.61,727.068 \n",
" 2208.3,726.851 2209.99,726.695 2211.67,726.598 2213.36,726.562 2215.05,726.586 2216.74,726.671 2218.43,726.816 2220.11,727.02 2221.8,727.283 2223.49,727.604 \n",
" 2225.18,727.98 2226.87,728.411 2228.55,728.895 2230.24,729.429 2231.93,730.011 2233.62,730.64 2235.31,731.31 2236.99,732.021 2238.68,732.769 2240.37,733.551 \n",
" 2242.06,734.363 2245.43,736.064 2248.81,737.842 2252.19,739.667 2255.56,741.507 2258.94,743.331 2262.32,745.106 2264,745.966 2265.69,746.803 2267.38,747.612 \n",
" 2269.07,748.391 2270.45,749.004 2271.84,749.593 2273.22,750.156 2274.61,750.691 2275.99,751.196 2277.38,751.67 2278.76,752.111 2280.15,752.519 2281.53,752.891 \n",
" 2282.92,753.228 2284.3,753.528 2285.69,753.789 2287.08,754.013 2288.46,754.196 2289.85,754.34 2291.23,754.444 \n",
" \"/>\n",
"<path clip-path=\"url(#clip660)\" d=\"\n",
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]
},
"execution_count": 118,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f2(t, y) = [0 1; -1 0] * y\n",
"\n",
"thist, yhist = ode_euler(f2, [0., 1], h=.1, tfinal=80)\n",
"scatter(thist, yhist')\n",
"plot!([cos, sin])"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "95eb6628",
"metadata": {
"cell_style": "split"
},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{ComplexF64}:\n",
" 0.0 - 1.0im\n",
" 0.0 + 1.0im"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"eigvals([0 1; -1 0])"
]
},
{
"cell_type": "markdown",
"id": "53e73e58",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# Runge-Kutta 4"
]
},
{
"cell_type": "code",
"execution_count": 127,
"id": "747257b9",
"metadata": {
"cell_style": "split"
},
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" 470.115,1108.43 473.278,1107.24 476.441,1105.91 479.605,1104.45 482.768,1102.87 485.931,1101.16 489.095,1099.33 492.258,1097.39 495.421,1095.34 498.584,1093.19 \n",
" 501.748,1090.95 504.911,1088.63 508.074,1086.22 514.401,1081.21 520.727,1075.99 527.054,1070.61 533.38,1065.15 539.707,1059.67 546.033,1054.26 552.36,1048.97 \n",
" 558.686,1043.87 564.035,1039.76 569.384,1035.87 572.059,1034.03 574.733,1032.25 577.408,1030.54 580.082,1028.91 582.757,1027.36 585.431,1025.9 588.106,1024.52 \n",
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" 928.143,1031.09 931.47,1029.04 934.797,1027.12 938.124,1025.33 941.451,1023.67 944.778,1022.16 948.105,1020.8 951.432,1019.59 954.759,1018.53 958.086,1017.64 \n",
" 961.413,1016.91 964.74,1016.35 968.067,1015.96 971.394,1015.74 974.72,1015.69 978.047,1015.81 981.374,1016.11 984.701,1016.57 988.028,1017.2 991.355,1018 \n",
" 994.682,1018.96 998.009,1020.07 1001.34,1021.35 1004.66,1022.78 1007.99,1024.35 1011.32,1026.06 1014.64,1027.91 1017.97,1029.89 1021.3,1031.98 1024.25,1033.94 \n",
" 1027.21,1035.98 1030.17,1038.11 1033.12,1040.3 1036.08,1042.56 1039.04,1044.89 1041.99,1047.26 1044.95,1049.68 1056.78,1059.71 1068.6,1069.93 1074.52,1074.98 \n",
" 1080.43,1079.9 1086.34,1084.65 1092.26,1089.17 1095.21,1091.33 1098.17,1093.41 1101.13,1095.41 1104.08,1097.32 1107.04,1099.15 1110,1100.87 1112.95,1102.49 \n",
" 1115.91,1104 1118.5,1105.24 1121.09,1106.39 1123.69,1107.45 1126.28,1108.41 1128.87,1109.28 1131.46,1110.05 1134.06,1110.73 1136.65,1111.3 1139.24,1111.77 \n",
" 1141.83,1112.15 1144.43,1112.41 1147.02,1112.58 1149.61,1112.64 1152.21,1112.6 1154.8,1112.45 1157.39,1112.2 1159.98,1111.84 1162.58,1111.39 1165.17,1110.83 \n",
" 1167.76,1110.17 1170.35,1109.41 1172.95,1108.56 1175.54,1107.61 1178.13,1106.57 1180.72,1105.44 1183.32,1104.21 1185.91,1102.91 1188.5,1101.52 1191.1,1100.04 \n",
" 1193.69,1098.5 1196.28,1096.87 1198.87,1095.18 1201.7,1093.26 1204.53,1091.27 1207.35,1089.2 1210.18,1087.08 1215.83,1082.65 1221.49,1078.04 1232.79,1068.43 \n",
" 1244.1,1058.66 1249.75,1053.83 1255.41,1049.1 1261.06,1044.53 1266.71,1040.16 1269.54,1038.07 1272.37,1036.04 1275.19,1034.08 1278.02,1032.2 1280.85,1030.4 \n",
" 1283.67,1028.69 1286.5,1027.07 1289.33,1025.54 1292.17,1024.1 1295.01,1022.77 1297.86,1021.54 1300.7,1020.42 1303.55,1019.42 1306.39,1018.53 1309.23,1017.76 \n",
" 1312.08,1017.1 1314.92,1016.57 1317.76,1016.17 1320.61,1015.88 1323.45,1015.73 1326.29,1015.69 1329.14,1015.78 1331.98,1016 1334.82,1016.34 1337.67,1016.81 \n",
" 1340.51,1017.39 1343.35,1018.1 1346.2,1018.93 1349.04,1019.87 1351.88,1020.93 1354.73,1022.1 1357.57,1023.37 1360.42,1024.76 1363.26,1026.24 1366.1,1027.82 \n",
" 1368.95,1029.5 1371.79,1031.27 1374.63,1033.12 1377.48,1035.05 1380.32,1037.05 1386.61,1041.73 1392.89,1046.69 1399.18,1051.87 1405.47,1057.2 1418.05,1068.07 \n",
" 1430.62,1078.74 1433.76,1081.31 1436.91,1083.83 1440.05,1086.29 1443.2,1088.67 1446.34,1090.98 1449.48,1093.21 1452.63,1095.34 1455.77,1097.38 1458.92,1099.31 \n",
" 1462.06,1101.13 1465.2,1102.83 1468.35,1104.41 1471.49,1105.86 1474.63,1107.19 1477.78,1108.37 1480.92,1109.42 1483.91,1110.28 1486.91,1111.01 1489.9,1111.61 \n",
" 1492.89,1112.07 1495.88,1112.4 1498.87,1112.59 1501.86,1112.64 1504.85,1112.55 1507.84,1112.32 1510.84,1111.96 1513.83,1111.46 1516.82,1110.82 1519.81,1110.05 \n",
" 1522.8,1109.15 1525.79,1108.12 1528.78,1106.96 1531.77,1105.69 1534.77,1104.29 1537.76,1102.78 1540.75,1101.16 1543.74,1099.43 1546.73,1097.61 1549.72,1095.68 \n",
" 1552.71,1093.67 1555.7,1091.58 1558.7,1089.4 1561.69,1087.15 1564.68,1084.84 1570.66,1080.04 1576.64,1075.07 1589.08,1064.38 1601.53,1053.69 1607.75,1048.5 \n",
" 1613.97,1043.51 1617.08,1041.11 1620.19,1038.77 1623.3,1036.52 1626.41,1034.35 1629.52,1032.27 1632.63,1030.29 1635.74,1028.42 1638.85,1026.66 1641.96,1025.01 \n",
" 1645.07,1023.48 1648.18,1022.08 1651.29,1020.81 1654.4,1019.67 1657.51,1018.67 1660.62,1017.81 1663.73,1017.09 1666.84,1016.52 1669.95,1016.1 1673.06,1015.82 \n",
" 1676.17,1015.7 1678.79,1015.71 1681.41,1015.82 1684.03,1016.05 1686.64,1016.37 1689.26,1016.8 1691.88,1017.34 1694.5,1017.98 1697.12,1018.72 1699.74,1019.55 \n",
" 1702.35,1020.49 1704.97,1021.52 1707.59,1022.65 1710.21,1023.86 1712.83,1025.17 1715.44,1026.55 1718.06,1028.03 1720.68,1029.58 1723.3,1031.2 1725.92,1032.9 \n",
" 1728.54,1034.67 1733.77,1038.39 1739.01,1042.34 1744.24,1046.48 1749.48,1050.77 1754.72,1055.19 1759.95,1059.68 1766.35,1065.21 1772.74,1070.73 1779.13,1076.16 \n",
" 1785.53,1081.43 1788.72,1083.99 1791.92,1086.48 1795.11,1088.9 1798.31,1091.24 1801.51,1093.49 1804.7,1095.65 1807.9,1097.7 1811.1,1099.65 1814.29,1101.47 \n",
" 1817.49,1103.18 1820.69,1104.76 1823.88,1106.21 1827.08,1107.51 1830.28,1108.68 1833.47,1109.7 1836.67,1110.58 1839.87,1111.3 1843.06,1111.87 1846.26,1112.28 \n",
" 1849.45,1112.54 1852.65,1112.64 1855.85,1112.58 1859.04,1112.36 1862.24,1111.99 1865.17,1111.51 1868.1,1110.9 1871.03,1110.16 1873.95,1109.29 1876.88,1108.31 \n",
" 1879.81,1107.2 1882.74,1105.97 1885.67,1104.63 1888.59,1103.18 1891.52,1101.62 1894.45,1099.96 1897.38,1098.2 1900.31,1096.35 1903.24,1094.41 1906.16,1092.39 \n",
" 1909.09,1090.29 1914.95,1085.88 1920.8,1081.24 1926.66,1076.4 1932.52,1071.44 1944.23,1061.32 1955.94,1051.33 1961.35,1046.88 1966.76,1042.59 1972.17,1038.5 \n",
" 1977.58,1034.65 1980.28,1032.83 1982.99,1031.08 1985.69,1029.4 1988.4,1027.81 1991.1,1026.31 1993.8,1024.89 1996.51,1023.56 1999.21,1022.33 2001.92,1021.19 \n",
" 2004.62,1020.16 2007.33,1019.23 2010.03,1018.4 2012.74,1017.68 2015.44,1017.07 2018.14,1016.57 2020.85,1016.18 2023.55,1015.91 2026.26,1015.74 2028.96,1015.69 \n",
" 2031.67,1015.75 2034.37,1015.93 2037.07,1016.21 2039.78,1016.61 2042.48,1017.12 2045.22,1017.75 2047.96,1018.49 2050.7,1019.34 2053.44,1020.3 2056.18,1021.36 \n",
" 2058.92,1022.52 2061.66,1023.79 2064.4,1025.15 2067.14,1026.6 2069.88,1028.15 2072.62,1029.78 2075.36,1031.49 2078.1,1033.28 2080.84,1035.15 2083.58,1037.08 \n",
" 2086.32,1039.08 2091.79,1043.26 2097.27,1047.63 2102.75,1052.16 2108.23,1056.81 2119.19,1066.28 2130.15,1075.66 2136.29,1080.75 2142.43,1085.63 2145.5,1087.99 \n",
" 2148.57,1090.27 2151.64,1092.47 2154.71,1094.58 2157.78,1096.61 2160.85,1098.53 2163.92,1100.35 2166.99,1102.07 2170.06,1103.67 2173.13,1105.15 2176.2,1106.51 \n",
" 2179.27,1107.74 2182.34,1108.84 2185.41,1109.8 2188.48,1110.63 2191.55,1111.32 2194.62,1111.86 2197.69,1112.26 2200.76,1112.52 2203.83,1112.63 2206.9,1112.6 \n",
" 2209.97,1112.42 2213.04,1112.1 2216.11,1111.63 2219.18,1111.01 2222.25,1110.26 2225.32,1109.37 2228.39,1108.34 2231.17,1107.3 2233.96,1106.14 2236.74,1104.89 \n",
" 2239.52,1103.53 2242.3,1102.07 2245.09,1100.53 2247.87,1098.89 2250.65,1097.16 2253.43,1095.36 2256.21,1093.48 2259,1091.52 2261.78,1089.5 2264.56,1087.41 \n",
" 2267.34,1085.27 2270.12,1083.08 2272.91,1080.84 2284.03,1071.52 2295.16,1061.9 2300.73,1057.11 2306.29,1052.38 2311.86,1047.77 2317.42,1043.33 2320.6,1040.87 \n",
" 2323.79,1038.49 2326.98,1036.19 2330.16,1033.98 2333.35,1031.88 2336.53,1029.87 2339.72,1027.98 2342.9,1026.2 2346.09,1024.55 2349.27,1023.02 2352.46,1021.63 \n",
" 2355.64,1020.38 2358.83,1019.26 2362.01,1018.3 2365.2,1017.48 2368.38,1016.81 2371.57,1016.3 2374.75,1015.94 2377.94,1015.74 2381.13,1015.69 2384.31,1015.8 \n",
" 2387.5,1016.07 2390.68,1016.5 2393.87,1017.07 2397.05,1017.81 2400.24,1018.69 2403.42,1019.71 2406.61,1020.89 2409.79,1022.2 2412.98,1023.65 2416.16,1025.23 \n",
" 2419.35,1026.93 2422.3,1028.62 2425.25,1030.41 2428.2,1032.29 2431.15,1034.25 2434.1,1036.3 2437.04,1038.43 2439.99,1040.63 2442.94,1042.9 2448.84,1047.6 \n",
" 2454.74,1052.48 2460.64,1057.5 2466.54,1062.59 2472.44,1067.69 2478.33,1072.76 2484.23,1077.73 2490.13,1082.55 2493.08,1084.89 2496.03,1087.17 2498.98,1089.39 \n",
" 2501.93,1091.53 2504.88,1093.6 2507.83,1095.59 2510.78,1097.49 2513.73,1099.3 2516.63,1100.99 2519.54,1102.57 2522.45,1104.06 2525.36,1105.43 2528.26,1106.7 \n",
" 2531.17,1107.85 2534.08,1108.88 2536.99,1109.79 2539.9,1110.58 2542.8,1111.24 2545.71,1111.78 2548.62,1112.19 2551.53,1112.47 2554.43,1112.61 2557.34,1112.63 \n",
" 2560.25,1112.52 2563.16,1112.27 2566.06,1111.9 2568.97,1111.39 2571.88,1110.76 2574.79,1110.01 2577.69,1109.13 2580.6,1108.12 2583.51,1107 2586.42,1105.77 \n",
" 2589.33,1104.42 2592.23,1102.96 2595.14,1101.4 2598.05,1099.74 2600.96,1097.98 2603.86,1096.13 2606.77,1094.2 2612.42,1090.21 2618.06,1085.96 2623.71,1081.49 \n",
" 2629.36,1076.84 2640.65,1067.21 2651.94,1057.45 2657.59,1052.65 2663.23,1047.96 2668.88,1043.44 2674.53,1039.13 2677.35,1037.07 2680.17,1035.08 2683,1033.16 \n",
" 2685.82,1031.32 2688.64,1029.56 2691.46,1027.9 2694.29,1026.32 2697.11,1024.84 2700,1023.43 2702.9,1022.13 2705.79,1020.93 2708.69,1019.86 2711.58,1018.9 \n",
" 2714.48,1018.06 2717.37,1017.35 2720.26,1016.76 2723.16,1016.3 2726.05,1015.97 2728.95,1015.76 2731.84,1015.69 2734.73,1015.74 2737.63,1015.93 2740.52,1016.24 \n",
" 2743.42,1016.68 2746.31,1017.25 2749.2,1017.95 2752.1,1018.76 2754.99,1019.7 2757.89,1020.76 2760.78,1021.93 2763.67,1023.22 2766.57,1024.62 2769.46,1026.12 \n",
" 2772.36,1027.72 2775.25,1029.42 2778.14,1031.22 2781.04,1033.1 2783.93,1035.06 2786.83,1037.11 2789.72,1039.22 2796.06,1044.08 2802.4,1049.2 2808.74,1054.51 \n",
" 2815.09,1059.94 2821.43,1065.43 2827.77,1070.9 2834.11,1076.29 2840.45,1081.52 2843.62,1084.05 2846.79,1086.52 2849.96,1088.92 2853.14,1091.24 2856.31,1093.48 \n",
" 2859.48,1095.62 2862.65,1097.65 2865.82,1099.59 2868.99,1101.4 2872.16,1103.1 2875.33,1104.67 2878.5,1106.12 2881.67,1107.42 2884.84,1108.59 2888.01,1109.62 \n",
" 2891.18,1110.5 2893.49,1111.05 2895.79,1111.51 2898.09,1111.9 2900.4,1112.21 2902.7,1112.43 2905,1112.58 2907.31,1112.64 2909.61,1112.62 2911.91,1112.51 \n",
" 2914.21,1112.33 2916.52,1112.06 2918.82,1111.71 2921.12,1111.29 2923.43,1110.78 2925.73,1110.19 2928.03,1109.53 2930.34,1108.78 2932.64,1107.97 2934.94,1107.08 \n",
" 2937.24,1106.11 2939.55,1105.08 2941.85,1103.97 2944.15,1102.8 2946.46,1101.56 2951.06,1098.9 2955.67,1096 2960.27,1092.89 2964.88,1089.58 2969.49,1086.1 \n",
" 2974.09,1082.48 2978.7,1078.72 2983.31,1074.87 2992.52,1066.98 3001.73,1059.01 3006.34,1055.07 3010.94,1051.18 3015.55,1047.39 3020.15,1043.7 3024.76,1040.16 \n",
" 3029.37,1036.78 3033.97,1033.58 3038.58,1030.59 3042.36,1028.31 3046.14,1026.19 3049.92,1024.24 3053.7,1022.48 3057.48,1020.9 3061.26,1019.53 3065.04,1018.35 \n",
" 3068.82,1017.39 \n",
" \"/>\n",
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" 270.807,1064.16 275.361,1060.22 279.915,1056.31 284.469,1052.45 289.023,1048.66 294.287,1044.42 299.551,1040.35 302.183,1038.39 304.815,1036.49 307.446,1034.65 \n",
" 310.078,1032.87 312.71,1031.16 315.342,1029.53 317.974,1027.98 320.606,1026.5 323.238,1025.11 325.869,1023.8 328.501,1022.58 331.133,1021.46 333.765,1020.43 \n",
" 336.397,1019.49 339.029,1018.66 341.66,1017.92 344.292,1017.29 346.924,1016.76 349.556,1016.34 352.188,1016.02 354.82,1015.81 357.452,1015.7 360.083,1015.7 \n",
" 362.715,1015.81 365.347,1016.03 367.979,1016.35 370.611,1016.78 373.243,1017.31 375.875,1017.95 378.506,1018.69 381.138,1019.53 383.77,1020.47 386.402,1021.5 \n",
" 389.034,1022.63 391.666,1023.86 394.297,1025.17 396.929,1026.56 399.561,1028.04 402.193,1029.6 404.825,1031.24 407.457,1032.95 410.089,1034.73 412.72,1036.57 \n",
" 415.352,1038.47 420.616,1042.45 425.88,1046.61 431.143,1050.94 436.407,1055.37 446.935,1064.44 457.462,1073.5 463.788,1078.81 470.115,1083.93 473.278,1086.4 \n",
" 476.441,1088.8 479.605,1091.12 482.768,1093.35 485.931,1095.49 489.095,1097.53 492.258,1099.46 495.421,1101.28 498.584,1102.99 501.748,1104.56 504.911,1106.01 \n",
" 508.074,1107.33 511.237,1108.51 514.401,1109.54 517.564,1110.43 520.727,1111.18 523.89,1111.77 527.054,1112.21 530.217,1112.5 533.38,1112.63 536.544,1112.61 \n",
" 539.707,1112.43 542.87,1112.1 546.033,1111.62 549.197,1110.98 552.36,1110.19 555.523,1109.26 558.686,1108.19 561.361,1107.17 564.035,1106.05 566.71,1104.83 \n",
" 569.384,1103.53 572.059,1102.13 574.733,1100.65 577.408,1099.08 580.082,1097.44 585.431,1093.92 590.78,1090.13 593.455,1088.15 596.129,1086.11 598.804,1084.02 \n",
" 601.478,1081.88 606.827,1077.49 612.176,1072.98 617.525,1068.39 622.874,1063.77 628.223,1059.14 633.572,1054.56 638.921,1050.07 644.27,1045.71 647.176,1043.41 \n",
" 650.081,1041.16 652.986,1038.98 655.891,1036.86 658.797,1034.82 661.702,1032.86 664.607,1030.98 667.512,1029.19 670.418,1027.5 673.323,1025.9 676.228,1024.41 \n",
" 679.133,1023.02 682.039,1021.75 684.944,1020.59 687.849,1019.54 690.755,1018.62 693.66,1017.82 696.565,1017.15 699.47,1016.6 702.376,1016.18 705.281,1015.89 \n",
" 708.186,1015.72 711.091,1015.69 713.997,1015.79 716.902,1016.02 719.807,1016.38 722.712,1016.87 725.618,1017.49 728.523,1018.23 731.428,1019.1 734.333,1020.09 \n",
" 737.239,1021.19 739.917,1022.31 742.595,1023.53 745.274,1024.84 747.952,1026.24 750.631,1027.73 753.309,1029.3 755.987,1030.96 758.666,1032.68 761.344,1034.48 \n",
" 764.023,1036.35 766.701,1038.28 769.379,1040.27 774.736,1044.41 780.093,1048.74 790.806,1057.76 801.52,1067.02 806.877,1071.63 812.234,1076.18 817.59,1080.61 \n",
" 822.947,1084.89 828.69,1089.27 834.433,1093.39 837.305,1095.34 840.176,1097.2 843.048,1098.97 845.919,1100.66 848.791,1102.25 851.662,1103.74 854.534,1105.12 \n",
" 857.405,1106.4 860.277,1107.56 863.148,1108.61 866.02,1109.55 868.891,1110.36 871.763,1111.05 874.634,1111.62 877.506,1112.07 880.377,1112.38 883.249,1112.58 \n",
" 886.12,1112.64 888.992,1112.58 891.864,1112.38 894.735,1112.07 897.607,1111.62 900.478,1111.05 903.35,1110.36 906.221,1109.55 909.093,1108.61 911.964,1107.56 \n",
" 914.836,1106.4 918.163,1104.91 921.49,1103.28 924.816,1101.5 928.143,1099.6 931.47,1097.57 934.797,1095.43 938.124,1093.17 941.451,1090.81 944.778,1088.36 \n",
" 948.105,1085.82 951.432,1083.21 954.759,1080.52 961.413,1075 968.067,1069.31 974.72,1063.56 981.374,1057.81 988.028,1052.16 994.682,1046.67 998.009,1044.02 \n",
" 1001.34,1041.43 1004.66,1038.93 1007.99,1036.51 1011.32,1034.2 1014.64,1031.99 1017.97,1029.89 1021.3,1027.91 1024.25,1026.26 1027.21,1024.72 1030.17,1023.29 \n",
" 1033.12,1021.97 1036.08,1020.77 1039.04,1019.69 1041.99,1018.73 1044.95,1017.9 1047.91,1017.2 1050.86,1016.63 1053.82,1016.2 1056.78,1015.89 1059.73,1015.73 \n",
" 1062.69,1015.69 1065.65,1015.8 1068.6,1016.03 1071.56,1016.41 1074.52,1016.91 1077.47,1017.55 1080.43,1018.32 1083.39,1019.21 1086.34,1020.23 1089.3,1021.37 \n",
" 1092.26,1022.64 1095.21,1024.02 1098.17,1025.51 1101.13,1027.1 1104.08,1028.81 1107.04,1030.61 1110,1032.5 1112.95,1034.48 1115.91,1036.55 1121.09,1040.35 \n",
" 1126.28,1044.36 1131.46,1048.54 1136.65,1052.85 1147.02,1061.73 1157.39,1070.7 1162.58,1075.11 1167.76,1079.43 1172.95,1083.63 1178.13,1087.65 1183.32,1091.47 \n",
" 1188.5,1095.06 1191.1,1096.76 1193.69,1098.39 1196.28,1099.94 1198.87,1101.42 1201.7,1102.93 1204.53,1104.35 1207.35,1105.67 1210.18,1106.88 1213.01,1107.98 \n",
" 1215.83,1108.98 1218.66,1109.85 1221.49,1110.61 1224.31,1111.25 1227.14,1111.77 1229.97,1112.17 1232.79,1112.45 1235.62,1112.6 1238.45,1112.63 1241.27,1112.54 \n",
" 1244.1,1112.33 1246.93,1111.99 1249.75,1111.52 1252.58,1110.94 1255.41,1110.24 1258.23,1109.42 1261.06,1108.49 1263.89,1107.44 1266.71,1106.28 1269.54,1105.02 \n",
" 1272.37,1103.65 1275.19,1102.17 1278.02,1100.61 1280.85,1098.95 1283.67,1097.2 1286.5,1095.36 1289.33,1093.45 1295.01,1089.38 1300.7,1085.05 1306.39,1080.51 \n",
" 1312.08,1075.79 1323.45,1066.05 1334.82,1056.24 1340.51,1051.43 1346.2,1046.75 1351.88,1042.25 1357.57,1037.97 1360.42,1035.94 1363.26,1033.97 1366.1,1032.08 \n",
" 1368.95,1030.28 1371.79,1028.56 1374.63,1026.94 1377.48,1025.41 1380.32,1023.98 1383.46,1022.52 1386.61,1021.19 1389.75,1020 1392.89,1018.95 1396.04,1018.04 \n",
" 1399.18,1017.27 1402.33,1016.66 1405.47,1016.19 1408.61,1015.88 1411.76,1015.71 1414.9,1015.7 1418.05,1015.85 1421.19,1016.14 1424.33,1016.59 1427.48,1017.19 \n",
" 1430.62,1017.93 1433.76,1018.82 1436.91,1019.86 1440.05,1021.03 1443.2,1022.34 1446.34,1023.78 1449.48,1025.35 1452.63,1027.05 1455.77,1028.85 1458.92,1030.78 \n",
" 1462.06,1032.8 1465.2,1034.93 1468.35,1037.14 1471.49,1039.45 1474.63,1041.83 1477.78,1044.28 1480.92,1046.79 1486.91,1051.72 1492.89,1056.79 1498.87,1061.94 \n",
" 1504.85,1067.12 1510.84,1072.27 1516.82,1077.32 1522.8,1082.23 1528.78,1086.92 1531.77,1089.18 1534.77,1091.36 1537.76,1093.46 1540.75,1095.49 1543.74,1097.42 \n",
" 1546.73,1099.25 1549.72,1100.99 1552.71,1102.62 1555.7,1104.14 1558.7,1105.55 1561.69,1106.84 1564.68,1108.01 1567.67,1109.05 1570.66,1109.96 1573.65,1110.75 \n",
" 1576.64,1111.4 1579.75,1111.93 1582.86,1112.31 1585.97,1112.55 1589.08,1112.64 1592.2,1112.58 1595.31,1112.36 1598.42,1112 1601.53,1111.49 1604.64,1110.84 \n",
" 1607.75,1110.04 1610.86,1109.1 1613.97,1108.02 1617.08,1106.8 1620.19,1105.46 1623.3,1103.98 1626.41,1102.39 1629.52,1100.67 1632.63,1098.84 1635.74,1096.91 \n",
" 1638.85,1094.87 1641.96,1092.74 1645.07,1090.52 1648.18,1088.22 1651.29,1085.85 1657.51,1080.9 1663.73,1075.75 1669.95,1070.46 1676.17,1065.09 1686.64,1056.06 \n",
" 1697.12,1047.31 1702.35,1043.13 1707.59,1039.14 1710.21,1037.23 1712.83,1035.37 1715.44,1033.58 1718.06,1031.86 1720.68,1030.2 1723.3,1028.62 1725.92,1027.12 \n",
" 1728.54,1025.7 1731.15,1024.36 1733.77,1023.11 1736.39,1021.95 1739.01,1020.88 1741.63,1019.91 1744.24,1019.03 1746.86,1018.25 1749.48,1017.58 1752.1,1017 \n",
" 1754.72,1016.53 1757.34,1016.16 1759.95,1015.9 1763.15,1015.72 1766.35,1015.7 1769.54,1015.84 1772.74,1016.14 1775.94,1016.59 1779.13,1017.2 1782.33,1017.96 \n",
" 1785.53,1018.87 1788.72,1019.93 1791.92,1021.13 1795.11,1022.48 1798.31,1023.96 1801.51,1025.57 1804.7,1027.31 1807.9,1029.16 1811.1,1031.14 1814.29,1033.22 \n",
" 1817.49,1035.4 1820.69,1037.67 1823.88,1040.03 1827.08,1042.47 1830.28,1044.98 1833.47,1047.55 1836.67,1050.18 1849.45,1061.06 1862.24,1072.1 1868.1,1077.05 \n",
" 1873.95,1081.86 1879.81,1086.48 1885.67,1090.85 1888.59,1092.93 1891.52,1094.93 1894.45,1096.85 1897.38,1098.68 1900.31,1100.41 1903.24,1102.04 1906.16,1103.57 \n",
" 1909.09,1105 1912.02,1106.31 1914.95,1107.5 1917.88,1108.58 1920.8,1109.53 1923.73,1110.36 1926.66,1111.07 1929.59,1111.64 1932.52,1112.09 1935.45,1112.4 \n",
" 1938.37,1112.59 1941.3,1112.64 1944.23,1112.56 1947.16,1112.34 1950.09,1111.99 1953.01,1111.52 1955.94,1110.91 1958.65,1110.23 1961.35,1109.45 1964.06,1108.56 \n",
" 1966.76,1107.57 1969.46,1106.48 1972.17,1105.29 1974.87,1104 1977.58,1102.62 1980.28,1101.15 1982.99,1099.59 1985.69,1097.95 1988.4,1096.23 1991.1,1094.44 \n",
" 1993.8,1092.57 1996.51,1090.64 1999.21,1088.65 2004.62,1084.5 2010.03,1080.16 2015.44,1075.67 2020.85,1071.07 2031.67,1061.72 2042.48,1052.47 2047.96,1047.93 \n",
" 2053.44,1043.54 2058.92,1039.35 2064.4,1035.4 2067.14,1033.52 2069.88,1031.72 2072.62,1030 2075.36,1028.36 2078.1,1026.8 2080.84,1025.33 2083.58,1023.96 \n",
" 2086.32,1022.68 2089.05,1021.51 2091.79,1020.43 2094.53,1019.46 2097.27,1018.6 2100.01,1017.84 2102.75,1017.2 2105.49,1016.67 2108.23,1016.25 2110.97,1015.95 \n",
" 2113.71,1015.76 2116.45,1015.69 2119.19,1015.74 2121.93,1015.9 2124.67,1016.17 2127.41,1016.57 2130.15,1017.07 2133.22,1017.77 2136.29,1018.61 2139.36,1019.59 \n",
" 2142.43,1020.7 2145.5,1021.95 2148.57,1023.32 2151.64,1024.81 2154.71,1026.42 2157.78,1028.14 2160.85,1029.98 2163.92,1031.91 2166.99,1033.95 2170.06,1036.07 \n",
" 2173.13,1038.28 2176.2,1040.57 2179.27,1042.92 2185.41,1047.82 2191.55,1052.92 2197.69,1058.15 2203.83,1063.45 2209.97,1068.77 2216.11,1074.02 2222.25,1079.16 \n",
" 2228.39,1084.12 2231.17,1086.29 2233.96,1088.4 2236.74,1090.46 2239.52,1092.45 2242.3,1094.37 2245.09,1096.22 2247.87,1097.99 2250.65,1099.67 2253.43,1101.27 \n",
" 2256.21,1102.77 2259,1104.18 2261.78,1105.49 2264.56,1106.7 2267.34,1107.8 2270.12,1108.8 2272.91,1109.68 2275.69,1110.45 2278.47,1111.11 2281.25,1111.65 \n",
" 2284.03,1112.08 2286.82,1112.38 2289.6,1112.57 2292.38,1112.64 2295.16,1112.59 2297.94,1112.41 2300.73,1112.12 2303.51,1111.71 2306.29,1111.19 2309.07,1110.54 \n",
" 2311.86,1109.78 2314.64,1108.91 2317.42,1107.93 2320.6,1106.68 2323.79,1105.28 2326.98,1103.75 2330.16,1102.1 2333.35,1100.32 2336.53,1098.42 2339.72,1096.42 \n",
" 2342.9,1094.31 2346.09,1092.1 2349.27,1089.8 2352.46,1087.42 2355.64,1084.96 2362.01,1079.85 2368.38,1074.54 2381.13,1063.58 2393.87,1052.66 2400.24,1047.38 \n",
" 2406.61,1042.33 2409.79,1039.9 2412.98,1037.55 2416.16,1035.29 2419.35,1033.12 2422.3,1031.2 2425.25,1029.38 2428.2,1027.65 2431.15,1026.02 2434.1,1024.49 \n",
" 2437.04,1023.08 2439.99,1021.78 2442.94,1020.6 2445.89,1019.54 2448.84,1018.61 2451.79,1017.8 2454.74,1017.12 2457.69,1016.57 2460.64,1016.15 2463.59,1015.87 \n",
" 2466.54,1015.71 2469.49,1015.7 2472.44,1015.82 2475.39,1016.07 2478.33,1016.46 2481.28,1016.98 2484.23,1017.63 2487.18,1018.41 2490.13,1019.31 2493.08,1020.34 \n",
" 2496.03,1021.5 2498.98,1022.77 2501.93,1024.15 2504.88,1025.65 2507.83,1027.26 2510.78,1028.96 2513.73,1030.77 2516.63,1032.64 2519.54,1034.59 2522.45,1036.63 \n",
" 2525.36,1038.73 2528.26,1040.91 2531.17,1043.15 2534.08,1045.45 2536.99,1047.8 2548.62,1057.57 2560.25,1067.62 2566.06,1072.61 2571.88,1077.52 2577.69,1082.28 \n",
" 2583.51,1086.85 2586.42,1089.04 2589.33,1091.17 2592.23,1093.22 2595.14,1095.2 2598.05,1097.09 2600.96,1098.89 2603.86,1100.6 2606.77,1102.22 2609.59,1103.68 \n",
" 2612.42,1105.05 2615.24,1106.31 2618.06,1107.46 2620.89,1108.51 2623.71,1109.44 2626.53,1110.25 2629.36,1110.95 2632.18,1111.53 2635,1111.99 2637.83,1112.33 \n",
" 2640.65,1112.54 2643.47,1112.64 2646.29,1112.6 2649.12,1112.45 2651.94,1112.17 2654.76,1111.77 2657.59,1111.25 2660.41,1110.61 2663.23,1109.85 2666.06,1108.98 \n",
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},
"execution_count": 127,
"metadata": {},
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}
],
"source": [
"thist, yhist = ode_rk_explicit(f2, [0., 1], h=2.9, tfinal=50)\n",
"scatter(thist, yhist')\n",
"plot!([cos, sin], size=(800, 500))"
]
},
{
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"source": [
"* Apparently it is possible to integrate this system using large time steps.\n",
"* This method evaluates $f(y)$ four times per stepso the cost is about equal when the step size $h$ is 4x larger than forward Euler."
]
},
{
"cell_type": "markdown",
"id": "6201b60c",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# Linear Stability Analysis\n",
"\n",
"Why did Euler diverge (even if slowly) while RK4 solved this problem accurately?\n",
"And why do both methods diverge if the step size is too large?\n",
"We can understand the convergence of methods by analyzing the test problem\n",
"$$ y' = \\lambda y $$\n",
"for different values of $\\lambda$ in the complex plane.\n",
"One step of the Euler method with step size $h$ maps\n",
"$$ y \\mapsto y + h \\lambda y = \\underbrace{(1 + h \\lambda)}_{R(h \\lambda)} y .$$\n",
"When does this map cause solutions to \"blow up\" and when is it stable?"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "295efcde",
"metadata": {
"cell_style": "split"
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"outputs": [
{
"data": {
"text/plain": [
"plot_stability (generic function with 1 method)"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"function plot_stability(Rz, method; xlim=(-3, 2), ylim=(-1.5, 1.5))\n",
" x = xlim[1]:.02:xlim[2]\n",
" y = ylim[1]:.02:ylim[2]\n",
" plot(title=\"Stability: $method\", aspect_ratio=:equal, xlim=xlim, ylim=ylim)\n",
" heatmap!(x, y, (x, y) -> abs(Rz(x + 1im*y)), c=:bwr, clims=(0, 2))\n",
" contour!(x, y, (x, y) -> abs(Rz(x + 1im*y)), color=:black, linewidth=2, levels=[1.])\n",
" plot!(x->0, color=:black, linewidth=1, label=:none)\n",
" plot!([0, 0], [ylim...], color=:black, linewidth=1, label=:none)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "f233ed2c",
"metadata": {
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"outputs": [
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},
"execution_count": 8,
"metadata": {},
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}
],
"source": [
"plot_stability(z -> 1 + z, \"Forward Eulor\")"
]
},
{
"cell_type": "markdown",
"id": "708a2cb7",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# Stability for RK4"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "82563216",
"metadata": {
"cell_style": "split"
},
"outputs": [
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},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot_stability(z -> rk_stability(4z, rk4), \"RK4\")\n"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "653d9295",
"metadata": {
"cell_style": "split"
},
"outputs": [
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],
"source": [
"plot_stability(z -> rk_stability(2z, heun), \"Heun's method\")"
]
},
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"source": [
"# Implicit methods"
]
},
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"cell_type": "markdown",
"id": "290a4108",
"metadata": {
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"source": [
"Recall that forward Euler is the step\n",
"$$ \\tilde y(h) = y(0) + h f(0, y(0)) . $$\n",
"This can be evaluated **explicitly**; all the terms on the right hand side are known so the approximation $\\tilde y(h)$ is computed merely by evaluating the right hand side.\n",
"Let's consider an alternative, **backward Euler** (or \"implicit Euler\"),\n",
"$$ \\tilde y(h) = y(0) + h f(h, \\tilde y(h)) . $$\n",
"This is a (generally) nonlinear equation for $\\tilde y(h)$.\n",
"For the test equation $y' = \\lambda y$, the backward Euler method is\n",
"$$ \\tilde y(h) = y(0) + h \\lambda \\tilde y(h) $$\n",
"or\n",
"$$ \\tilde y(h) = \\underbrace{\\frac{1}{1 - h \\lambda}}_{R(h\\lambda)} y(0) . $$"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "c9e9d867",
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]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot_stability(z -> 1/(1-z), \"Backward Euler\")"
]
},
{
"cell_type": "markdown",
"id": "f3a71738",
"metadata": {
"cell_style": "split",
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# Computing with implicit methods\n",
"\n",
"$$ \\tilde y(h) = y(0) + h f\\big(\\tilde y(h) \\big) $$"
]
},
{
"cell_type": "markdown",
"id": "0ece15d3",
"metadata": {
"cell_style": "split"
},
"source": [
"* Linear solve for linear problem\n",
"* Nonlinear (often Newton) solve for nonlinear problem\n",
"* Need Jacobian or finite differencing"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "c479bcf4",
"metadata": {
"cell_style": "split"
},
"outputs": [
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"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot_stability(z -> Rz_theta(z, .5), \"Midpoint method\")"
]
},
{
"cell_type": "markdown",
"id": "6631b25b",
"metadata": {
"slideshow": {
"slide_type": "slide"
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"source": [
"# The $\\theta$ method\n"
]
},
{
"cell_type": "markdown",
"id": "4d01c9a3",
"metadata": {
"cell_style": "split",
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"source": [
"Forward and backward Euler are bookends of the family known as $\\theta$ methods.\n",
"\n",
"$$ \\tilde u(h) = u(0) + h f\\Big(\\theta h, \\theta\\tilde u(h) + (1-\\theta)u(0) \\Big) $$\n",
"\n",
"which, for linear problems, is solved as\n",
"\n",
"$$ (I - h \\theta A) u(h) = \\Big(I + h (1-\\theta) A \\Big) u(0) . $$\n",
"\n",
"$\\theta=0$ is explicit Euler, $\\theta=1$ is implicit Euler, and $\\theta=1/2$ are the midpoint or trapezoid rules (equivalent for linear problems).\n",
"The stability function is\n",
"$$ R(z) = \\frac{1 + (1-\\theta)z}{1 - \\theta z}. $$"
]
},
{
"cell_type": "code",
"execution_count": 132,
"id": "a035183c",
"metadata": {
"cell_style": "split"
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"outputs": [
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]
},
"execution_count": 132,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Rz_theta(z, theta) = (1 + (1-theta)*z) / (1 - theta*z)\n",
"theta = .5\n",
"plot_stability(z -> Rz_theta(z, theta), \"\\$\\\\theta=$theta\\$\", \n",
" xlim=(-20, 1))"
]
},
{
"cell_type": "markdown",
"id": "85a81669",
"metadata": {
"cell_style": "center",
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# $\\theta$ method for the oscillator"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "f9ac0b5e",
"metadata": {
"cell_style": "split"
},
"outputs": [
{
"data": {
"text/plain": [
"ode_theta_linear (generic function with 1 method)"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function ode_theta_linear(A, u0; forcing=zero, tfinal=1, h=0.1, theta=.5)\n",
" u = copy(u0)\n",
" t = 0.\n",
" thist = [t]\n",
" uhist = [u0]\n",
" while t < tfinal\n",
" tnext = min(t+h, tfinal)\n",
" h = tnext - t\n",
" rhs = (I + h*(1-theta)*A) * u .+ h*forcing(t+h*theta)\n",
" u = (I - h*theta*A) \\ rhs\n",
" t = tnext\n",
" push!(thist, t)\n",
" push!(uhist, u)\n",
" end\n",
" thist, hcat(uhist...)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 140,
"id": "8ecc9503",
"metadata": {
"cell_style": "split"
},
"outputs": [
{
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]
},
"execution_count": 140,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Test on oscillator\n",
"A = [0 1; -1 0]\n",
"theta = .5\n",
"thist, uhist = ode_theta_linear(A, [0., 1], h=.6, theta=theta, tfinal=20)\n",
"scatter(thist, uhist')\n",
"plot!([cos, sin])"
]
},
{
"cell_type": "markdown",
"id": "3a83dbf3",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# $\\theta$ method for the cosine decay"
]
},
{
"cell_type": "code",
"execution_count": 169,
"id": "fdf09c98",
"metadata": {},
"outputs": [
{
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},
"execution_count": 169,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"k = 500000\n",
"theta = .7\n",
"thist, uhist = ode_theta_linear(-k, [.2], forcing=t -> k*cos(t), tfinal=5, h=.1, theta=theta)\n",
"scatter(thist, uhist[1,:], title=\"\\$\\\\theta = $theta\\$\")\n",
"plot!(cos, size=(800, 500))"
]
},
{
"cell_type": "markdown",
"id": "c536eb9d",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# Stability classes and the $\\theta$ method\n"
]
},
{
"cell_type": "markdown",
"id": "b984e88a",
"metadata": {
"cell_style": "split"
},
"source": [
"## Definition: $A$-stability\n",
"A method is $A$-stable if the stability region\n",
"$$ \\{ z : |R(z)| \\le 1 \\} $$\n",
"contains the entire left half plane $$ \\Re[z] \\le 0 .$$\n",
"This means that the method can take arbitrarily large time steps without becoming unstable (diverging) for any problem that is indeed physically stable."
]
},
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"id": "232b9cdf",
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"source": [
"## Definition: $L$-stability\n",
"A time integrator with stability function $R(z)$ is $L$-stable if\n",
"$$ \\lim_{z\\to\\infty} R(z) = 0 .$$\n",
"For the $\\theta$ method, we have\n",
"$$ \\lim_{z\\to \\infty} \\frac{1 + (1-\\theta)z}{1 - \\theta z} = \\frac{1-\\theta}{\\theta} . $$\n",
"Evidently only $\\theta=1$ is $L$-stable."
]
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"id": "f8c8e7fa",
"metadata": {
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"slide_type": "slide"
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"source": [
"# Heat equation as linear ODE\n",
"\n",
"* How do different $\\theta \\in [0, 1]$ compare in terms of stability?\n",
"* Are there artifacts even when the solution is stable?"
]
},
{
"cell_type": "code",
"execution_count": 107,
"id": "2ea7e708",
"metadata": {
"cell_style": "split"
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"outputs": [
{
"data": {
"text/plain": [
"5×5 SparseMatrixCSC{Float64, Int64} with 15 stored entries:\n",
" -12.5 6.25 â‹… â‹… 6.25\n",
" 6.25 -12.5 6.25 â‹… â‹… \n",
" â‹… 6.25 -12.5 6.25 â‹… \n",
" â‹… â‹… 6.25 -12.5 6.25\n",
" 6.25 â‹… â‹… 6.25 -12.5"
]
},
"execution_count": 107,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using SparseArrays\n",
"\n",
"function heat_matrix(n)\n",
" dx = 2 / n\n",
" rows = [1]\n",
" cols = [1]\n",
" vals = [0.]\n",
" wrap(j) = (j + n - 1) % n + 1\n",
" for i in 1:n\n",
" append!(rows, [i, i, i])\n",
" append!(cols, wrap.([i-1, i, i+1]))\n",
" append!(vals, [1, -2, 1] ./ dx^2)\n",
" end\n",
" sparse(rows, cols, vals)\n",
"end\n",
"heat_matrix(5)"
]
},
{
"cell_type": "code",
"execution_count": 111,
"id": "8b832a79",
"metadata": {
"cell_style": "split"
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 0.001373 seconds (846 allocations: 1.483 MiB)\n"
]
},
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" 1278.14,440.014 1288.38,455.261 1298.62,499.069 1308.85,566.018 1319.09,648.245 1329.33,736.877 1339.56,823.494 1349.8,901.305 1360.04,965.824 1370.28,1015.01 \n",
" 1380.51,1048.93 1390.75,1069.21 1400.99,1078.33 1411.23,1079.03 1421.46,1073.88 1431.7,1065.05 1441.94,1054.18 1452.18,1042.42 1462.41,1030.56 1472.65,1019.04 \n",
" 1482.89,1008.11 1493.13,997.891 1503.36,988.422 1513.6,979.697 1523.84,971.69 1534.07,964.362 1544.31,957.674 1554.55,951.585 1564.79,946.056 1575.02,941.05 \n",
" 1585.26,936.531 1595.5,932.466 1605.74,928.823 1615.97,925.573 1626.21,922.688 1636.45,920.142 1646.69,917.91 1656.92,915.969 1667.16,914.298 1677.4,912.876 \n",
" 1687.63,911.683 1697.87,910.703 1708.11,909.918 1718.35,909.312 1728.58,908.871 1738.82,908.58 1749.06,908.428 1759.3,908.4 1769.53,908.487 1779.77,908.678 \n",
" 1790.01,908.962 1800.25,909.33 1810.48,909.774 1820.72,910.285 1830.96,910.856 1841.2,911.48 1851.43,912.15 1861.67,912.86 1871.91,913.604 1882.14,914.377 \n",
" 1892.38,915.174 1902.62,915.99 1912.86,916.821 1923.09,917.663 1933.33,918.512 1943.57,919.365 1953.81,920.219 1964.04,921.071 1974.28,921.917 1984.52,922.756 \n",
" 1994.76,923.585 2004.99,924.402 2015.23,925.205 2025.47,925.992 2035.7,926.761 2045.94,927.512 2056.18,928.241 2066.42,928.949 2076.65,929.633 2086.89,930.294 \n",
" 2097.13,930.929 2107.37,931.538 2117.6,932.12 2127.84,932.674 2138.08,933.2 2148.32,933.696 2158.55,934.163 2168.79,934.6 2179.03,935.006 2189.27,935.381 \n",
" 2199.5,935.724 2209.74,936.036 2219.98,936.316 2230.21,936.563 2240.45,936.778 2250.69,936.96 2260.93,937.109 2271.16,937.225 2281.4,937.308 2291.64,937.358 \n",
" \n",
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" 459.152,923.805 469.389,923.947 479.627,924.102 489.864,924.269 500.101,924.451 510.339,924.647 520.576,924.857 530.814,925.083 541.051,925.325 551.288,925.583 \n",
" 561.526,925.857 571.763,926.148 582,926.456 592.238,926.78 602.475,927.121 612.712,927.478 622.95,927.851 633.187,928.238 643.424,928.639 653.662,929.053 \n",
" 663.899,929.478 674.137,929.913 684.374,930.354 694.611,930.8 704.849,931.248 715.086,931.694 725.323,932.134 735.561,932.565 745.798,932.979 756.035,933.373 \n",
" 766.273,933.74 776.51,934.072 786.747,934.361 796.985,934.598 807.222,934.773 817.459,934.876 827.697,934.893 837.934,934.812 848.172,934.617 858.409,934.293 \n",
" 868.646,933.82 878.884,933.179 889.121,932.349 899.358,931.307 909.596,930.025 919.833,928.477 930.07,926.63 940.308,924.453 950.545,921.909 960.782,918.959 \n",
" 971.02,915.558 981.257,911.662 991.495,907.221 1001.73,902.178 1011.97,896.477 1022.21,890.055 1032.44,882.844 1042.68,874.775 1052.92,865.779 1063.16,855.794 \n",
" 1073.39,844.776 1083.63,832.72 1093.87,819.699 1104.11,805.912 1114.34,791.768 1124.58,777.974 1134.82,765.646 1145.05,756.389 1155.29,752.333 1165.53,756.045 \n",
" 1175.77,770.294 1186,797.628 1196.24,839.784 1206.48,897.026 1216.72,967.551 1226.95,1047.17 1237.19,1129.43 1247.43,1206.24 1257.67,1269.05 1267.9,1310.26 \n",
" 1278.14,1324.63 1288.38,1310.26 1298.62,1269.05 1308.85,1206.24 1319.09,1129.43 1329.33,1047.17 1339.56,967.551 1349.8,897.026 1360.04,839.784 1370.28,797.628 \n",
" 1380.51,770.294 1390.75,756.045 1400.99,752.333 1411.23,756.389 1421.46,765.646 1431.7,777.974 1441.94,791.768 1452.18,805.912 1462.41,819.699 1472.65,832.72 \n",
" 1482.89,844.776 1493.13,855.794 1503.36,865.779 1513.6,874.775 1523.84,882.844 1534.07,890.055 1544.31,896.477 1554.55,902.178 1564.79,907.221 1575.02,911.662 \n",
" 1585.26,915.558 1595.5,918.959 1605.74,921.909 1615.97,924.453 1626.21,926.63 1636.45,928.477 1646.69,930.025 1656.92,931.307 1667.16,932.349 1677.4,933.179 \n",
" 1687.63,933.82 1697.87,934.293 1708.11,934.617 1718.35,934.812 1728.58,934.893 1738.82,934.876 1749.06,934.773 1759.3,934.598 1769.53,934.361 1779.77,934.072 \n",
" 1790.01,933.74 1800.25,933.373 1810.48,932.979 1820.72,932.565 1830.96,932.134 1841.2,931.694 1851.43,931.248 1861.67,930.8 1871.91,930.354 1882.14,929.913 \n",
" 1892.38,929.478 1902.62,929.053 1912.86,928.639 1923.09,928.238 1933.33,927.851 1943.57,927.478 1953.81,927.121 1964.04,926.78 1974.28,926.456 1984.52,926.148 \n",
" 1994.76,925.857 2004.99,925.583 2015.23,925.325 2025.47,925.083 2035.7,924.857 2045.94,924.647 2056.18,924.451 2066.42,924.269 2076.65,924.102 2086.89,923.947 \n",
" 2097.13,923.805 2107.37,923.675 2117.6,923.556 2127.84,923.448 2138.08,923.35 2148.32,923.261 2158.55,923.181 2168.79,923.11 2179.03,923.046 2189.27,922.989 \n",
" 2199.5,922.939 2209.74,922.895 2219.98,922.857 2230.21,922.824 2240.45,922.796 2250.69,922.773 2260.93,922.755 2271.16,922.741 2281.4,922.731 2291.64,922.725 \n",
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" 663.899,919.296 674.137,918.774 684.374,918.223 694.611,917.643 704.849,917.035 715.086,916.4 725.323,915.74 735.561,915.058 745.798,914.356 756.035,913.637 \n",
" 766.273,912.907 776.51,912.17 786.747,911.431 796.985,910.699 807.222,909.98 817.459,909.285 827.697,908.624 837.934,908.009 848.172,907.454 858.409,906.974 \n",
" 868.646,906.588 878.884,906.315 889.121,906.178 899.358,906.201 909.596,906.413 919.833,906.845 930.07,907.531 940.308,908.509 950.545,909.823 960.782,911.519 \n",
" 971.02,913.65 981.257,916.273 991.495,919.451 1001.73,923.254 1011.97,927.758 1022.21,933.046 1032.44,939.208 1042.68,946.341 1052.92,954.544 1063.16,963.913 \n",
" 1073.39,974.533 1083.63,986.449 1093.87,999.638 1104.11,1013.95 1114.34,1029.04 1124.58,1044.25 1134.82,1058.53 1145.05,1070.32 1155.29,1077.55 1165.53,1077.65 \n",
" 1175.77,1067.86 1186,1045.58 1196.24,1008.97 1206.48,957.616 1216.72,893.1 1226.95,819.34 1237.19,742.49 1247.43,670.308 1257.67,611.053 1267.9,572.082 \n",
" 1278.14,558.486 1288.38,572.082 1298.62,611.053 1308.85,670.308 1319.09,742.49 1329.33,819.34 1339.56,893.1 1349.8,957.616 1360.04,1008.97 1370.28,1045.58 \n",
" 1380.51,1067.86 1390.75,1077.65 1400.99,1077.55 1411.23,1070.32 1421.46,1058.53 1431.7,1044.25 1441.94,1029.04 1452.18,1013.95 1462.41,999.638 1472.65,986.449 \n",
" 1482.89,974.533 1493.13,963.913 1503.36,954.544 1513.6,946.341 1523.84,939.208 1534.07,933.046 1544.31,927.758 1554.55,923.254 1564.79,919.451 1575.02,916.273 \n",
" 1585.26,913.65 1595.5,911.519 1605.74,909.823 1615.97,908.509 1626.21,907.531 1636.45,906.845 1646.69,906.413 1656.92,906.201 1667.16,906.178 1677.4,906.315 \n",
" 1687.63,906.588 1697.87,906.974 1708.11,907.454 1718.35,908.009 1728.58,908.624 1738.82,909.285 1749.06,909.98 1759.3,910.699 1769.53,911.431 1779.77,912.17 \n",
" 1790.01,912.907 1800.25,913.637 1810.48,914.356 1820.72,915.058 1830.96,915.74 1841.2,916.4 1851.43,917.035 1861.67,917.643 1871.91,918.223 1882.14,918.774 \n",
" 1892.38,919.296 1902.62,919.789 1912.86,920.251 1923.09,920.684 1933.33,921.088 1943.57,921.464 1953.81,921.812 1964.04,922.133 1974.28,922.429 1984.52,922.7 \n",
" 1994.76,922.947 2004.99,923.172 2015.23,923.376 2025.47,923.559 2035.7,923.724 2045.94,923.871 2056.18,924.002 2066.42,924.118 2076.65,924.219 2086.89,924.307 \n",
" 2097.13,924.384 2107.37,924.45 2117.6,924.506 2127.84,924.554 2138.08,924.593 2148.32,924.626 2158.55,924.653 2168.79,924.674 2179.03,924.691 2189.27,924.704 \n",
" 2199.5,924.713 2209.74,924.72 2219.98,924.725 2230.21,924.728 2240.45,924.73 2250.69,924.731 2260.93,924.732 2271.16,924.732 2281.4,924.732 2291.64,924.732 \n",
" \n",
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]
},
"execution_count": 111,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"n = 200\n",
"A = heat_matrix(n)\n",
"x = LinRange(-1, 1, n+1)[1:end-1]\n",
"u0 = exp.(-200 * x .^ 2)\n",
"@time thist, uhist = ode_theta_linear(A, u0, h=.1, theta=.5, tfinal=1);\n",
"nsteps = size(uhist, 2)\n",
"plot(x, uhist[:, 1:5])"
]
},
{
"cell_type": "markdown",
"id": "0ac0a049",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# Advection as a linear ODE"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "f73784a3",
"metadata": {
"cell_style": "split",
"slideshow": {
"slide_type": "skip"
}
},
"outputs": [
{
"data": {
"text/plain": [
"5×5 SparseMatrixCSC{Float64, Int64} with 11 stored entries:\n",
" 0.0 -1.25 â‹… â‹… 1.25\n",
" 1.25 â‹… -1.25 â‹… â‹… \n",
" â‹… 1.25 â‹… -1.25 â‹… \n",
" â‹… â‹… 1.25 â‹… -1.25\n",
" -1.25 â‹… â‹… 1.25 â‹… "
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function advect_matrix(n; upwind=false)\n",
" dx = 2 / n\n",
" rows = [1]\n",
" cols = [1]\n",
" vals = [0.]\n",
" wrap(j) = (j + n - 1) % n + 1\n",
" for i in 1:n\n",
" append!(rows, [i, i])\n",
" if upwind\n",
" append!(cols, wrap.([i-1, i]))\n",
" append!(vals, [1., -1] ./ dx)\n",
" else\n",
" append!(cols, wrap.([i-1, i+1]))\n",
" append!(vals, [1., -1] ./ 2dx)\n",
" end\n",
" end\n",
" sparse(rows, cols, vals)\n",
"end\n",
"advect_matrix(5)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "0545bfb7",
"metadata": {
"cell_style": "split"
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 0.075686 seconds (154.41 k allocations: 9.739 MiB, 97.59% compilation time)\n"
]
},
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},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"n = 50\n",
"A = advect_matrix(n, upwind=false)\n",
"x = LinRange(-1, 1, n+1)[1:end-1]\n",
"u0 = exp.(-9 * x .^ 2)\n",
"@time thist, uhist = ode_theta_linear(A, u0, h=.04, theta=1, tfinal=1.);\n",
"nsteps = size(uhist, 2)\n",
"plot(x, uhist[:, 1:(nsteps÷8):end])"
]
},
{
"cell_type": "markdown",
"id": "4c61e8fb",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# Spectrum of operators"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "3f736202",
"metadata": {
"cell_style": "split"
},
"outputs": [
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},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"theta=.5\n",
"h = .1\n",
"plot_stability(z -> Rz_theta(z, theta), \"\\$\\\\theta=$theta, h=$h\\$\")\n",
"ev = eigvals(Matrix(h*advect_matrix(20, upwind=true)))\n",
"scatter!(real(ev), imag(ev))"
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "833e0984",
"metadata": {
"cell_style": "split"
},
"outputs": [
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},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"theta=.5\n",
"h = .2 / 4\n",
"plot_stability(z -> Rz_theta(z, theta), \"\\$\\\\theta=$theta, h=$h\\$\")\n",
"ev = eigvals(Matrix(h*heat_matrix(20)))\n",
"scatter!(real(ev), imag(ev))"
]
}
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