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"source": [
"# p01: Convergence of fourth order finite differences"
]
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"Let us compute the derivative of\n",
"$$\n",
"u(x) = \\exp(\\sin(x)), \\qquad x \\in [-\\pi,\\pi]\n",
"$$\n",
"using fourth order finite difference scheme\n",
"$$\n",
"u'(x_j) \\approx w_j = \\frac{1}{h} \\left( \\frac{1}{12} u_{j-2} - \\frac{2}{3} u_{j-1} + \\frac{2}{3} u_{j+1} - \\frac{1}{12} u_{j+2} \\right)\n",
"$$\n",
"using periodic boundary conditions."
]
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"%matplotlib inline\n",
"%config InlineBackend.figure_format='svg'\n",
"from scipy.sparse import coo_matrix\n",
"from numpy import arange,pi,exp,sin,cos,ones,inf\n",
"from numpy.linalg import norm\n",
"from matplotlib.pyplot import figure,loglog,semilogy,text,grid,xlabel,ylabel,title"
]
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"\n",
"\n",
"\n"
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""
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"source": [
"Nvec = 2**arange(3,13)\n",
"for N in Nvec:\n",
" h = 2*pi/N\n",
" x = -pi + arange(1,N+1)*h\n",
" u = exp(sin(x))\n",
" uprime = cos(x)*u\n",
" e = ones(N)\n",
" e1 = arange(0,N)\n",
" e2 = arange(1,N+1); e2[N-1]=0\n",
" e3 = arange(2,N+2); e3[N-2]=0; e3[N-1]=1;\n",
" D = coo_matrix((2*e/3,(e1,e2)),shape=(N,N)) \\\n",
" - coo_matrix((e/12,(e1,e3)),shape=(N,N))\n",
" D = (D - D.T)/h\n",
" error = norm(D.dot(u)-uprime,inf)\n",
" loglog(N,error,'or')\n",
" \n",
"semilogy(Nvec,Nvec**(-4.0),'--')\n",
"text(105,5e-8,'$N^{-4}$')\n",
"grid(True); xlabel('N'); ylabel('error')\n",
"title('Convergence of fourth-order finite difference');"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise\n",
"\n",
"Find sixth and eigth order finite differences, implement them and make error plot as above.\n",
"\n",
"## Exercise\n",
"\n",
"Construct the differentiation matrix using `toeplitz` function. You need to pass the first column and first row."
]
}
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