{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Section Averaged 1D SIMPLE method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 基礎式"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "    \\begin{align}\n",
    "        \\frac{\\partial h}{\\partial t}+\\frac{\\partial uh}{\\partial x}  &= 0 \\\\\n",
    "        \\frac{\\partial u}{\\partial t}+\\frac{\\partial u u}{\\partial x} &= - g\\frac{\\partial h}{\\partial x} + g ( i_0  -  i_e )\n",
    "    \\end{align}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 基礎式の離散式"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "    \\begin{align}\n",
    "        \\frac{h^{n+1}_i - h^{n}_i}{\\Delta t}+\\frac{ u^{n+1}_{i+1/2} h^{n+1}_{i} - u^{n+1}_{i-1/2} h^{n+1}_{i-1} }{ \\Delta x}  &= 0 \\\\\n",
    "        \\frac{u^{n+1}_{i+1/2} - u^{n}_{i+1/2}}{\\Delta t}+\\frac{ u^{n}_{i+1} u^{n+1}_{i+1/2} - u^{n}_{i} u^{n+1}_{i-1/2} }{ \\Delta x}  &= -g\\frac{h^{n+1}_{i+1} - h^{n+1}_i}{\\Delta x}  + g ( i_0 - i_e)\\\\\n",
    "    \\end{align}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "運動方程式を以下の代数方程式の形式に書き換える"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "    \\begin{align}\n",
    "        a_{i+1/2} u^{n+1}_{i+1/2} + a_{i+3/2} u^{n+1}_{i+3/2} + a_{i-1/2} u^{n+1}_{i-1/2} &= b_{i+1/2}\n",
    "        -g\\frac{h^{n+1}_{i+1} - h^{n+1}_i}{\\Delta x}  +  g ( i_0 - i_e) \\\\\n",
    "        a_{i+1/2} &= \\dfrac{1}{\\Delta t} + \\dfrac{u^n_{i+1}}{\\Delta x} \\\\\n",
    "        a_{i+3/2} &= 0 \\\\\n",
    "        a_{i-1/2} &= - \\dfrac{u^n_{i}}{\\Delta x} \\\\\n",
    "        b_{i+1/2} &=  \\dfrac{u^n_{i+1/2}}{\\Delta t}\n",
    "    \\end{align}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ここに，$u^n_{i} = 0.5( u^n_{i+1/2} + u^n_{i-1/2})$, $u^n_{i+1} = 0.5( u^n_{i+3/2} + u^n_{i+1/2})$とする．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 基礎式の離散式の近似式"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "上式の近似式は"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "    \\begin{align}\n",
    "        \\frac{h^{*}_i - h^{n}_i}{\\Delta t}+\\frac{ u^{n+1}_{i+1/2} h^{*}_{i} - u^{n+1}_{i-1/2} h^{*}_{i-1} }{ \\Delta x}  &= 0 \n",
    "    \\end{align}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "    \\begin{align}\n",
    "        a_{i+1/2} u^{*}_{i+1/2} + a_{i+3/2} u^{*}_{i+3/2} + a_{i-1/2} u^{*}_{i-1/2} &= b_{i+1/2}\n",
    "        -g\\frac{h^{*}_{i+1} - h^{*}_i}{\\Delta x}  + g ( i_0  - i_e)\n",
    "    \\end{align}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 流速の補正式"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "    \\begin{align}\n",
    "        h^{n+1} &= h^* + \\delta h \\\\\n",
    "        u^{n+1} &= u^* + \\delta u \n",
    "    \\end{align}\n",
    "$$\n",
    "\n",
    "として，運動方程式の元式と近似式の差分をとると，"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "    \\begin{align}\n",
    "        a_{i+1/2} \\delta u_{i+1/2} + a_{i+1/2} \\delta u_{i+3/2} + a_{i+1/2} \\delta u_{i-1/2} &= \n",
    "        -g\\frac{ \\delta h_{i+1} - \\delta h_i}{\\Delta x} \n",
    "    \\end{align}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "左辺第2,3項をキャンセルして，"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "    \\begin{align}\n",
    "         \\delta u_{i+1/2}  &= \n",
    "        -\\dfrac{g}{a_{i+1/2}}\n",
    "        \\frac{ \\delta h_{i+1} - \\delta h_i}{\\Delta x} \n",
    "    \\end{align}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 圧力の補正式"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "これらを連続式の近似式に代入すると，"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "    \\begin{align}\n",
    "        \\frac{h^{*}_i - h^{n}_i}{\\Delta t}\n",
    "       +\\dfrac{1}{\\Delta x} \n",
    "        \\left[ (u^*_{i+1/2} + \\delta u_{i+1/2}) h^*_{i}  - (u^*_{i-1/2} + \\delta u_{i-1/2}) h^*_{i-1}  \\right]\n",
    "        = 0\n",
    "%         h^{n+1}_{i} - u^{n+1}_{i-1/2} h^{n+1}_{i-1} }{ \\Delta x}  &= 0 \\\\\n",
    "%         \\frac{u^{n+1}_{i+1/2} - u^{n}_{i+1/2}}{\\Delta t}+\\frac{ u^{n}_{i+1} u^{n+1}_{i+1/2} - u^{n}_{i} u^{n+1}_{i-1/2} }{ \\Delta x}  &= -g\\frac{H^{n+1}_{i+1} - H^{n+1}_i}{\\Delta x}  - g i_0 \\\\\n",
    "    \\end{align}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "代数方程式の形式に書き換えると，"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "    \\begin{align}\n",
    "        A_{i} \\delta h_{i} + A_{i+1} \\delta h_{i+1} + A_{i-1} \\delta h_{i-1} &= B_i \\\\\n",
    "        A_{i} &= \\dfrac{g}{\\Delta x^2} \\left( \\dfrac{h^*_i}{a_{i+1/2}} + \\dfrac{h^*_{i-1}}{a_{i-1/2}} \\right) \\\\ \n",
    "        A_{i+1} &= - \\dfrac{g}{\\Delta x^2} \\dfrac{h^*_{i}}{a_{i+1/2}}  \\\\\n",
    "        A_{i-1} &= - \\dfrac{g}{\\Delta x^2} \\dfrac{h^*_{i-1}}{a_{i-1/2}}  \\\\\n",
    "        B_{i} &=  -\\frac{h^{*}_i - h^{n}_i}{\\Delta t} - \\dfrac{  u^*_{i+1/2} h^*_{i} - u^*_{i-1/2} h^*_{i-1} }{ \\Delta x}\n",
    "    \\end{align}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 緩和係数の導入"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "緩和係数$\\alpha_p$,$\\alpha_u$を導入すると，水深の更新式および運動方程式の近似式は以下となる．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "    \\begin{align}\n",
    "        h^{n+1} &= h^* + \\alpha_p \\delta h \\\\\n",
    "    \\end{align}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "    \\begin{align}\n",
    "        \\dfrac{a_{i+1/2}}{\\alpha_u} u^{*}_{i+1/2} + a_{i+3/2} u^{*}_{i+3/2} + a_{i-1/2} u^{*}_{i-1/2} &= b_{i+1/2}\n",
    "        -g\\frac{h^{*}_{i+1} - h^{*}_i}{\\Delta x}  + g ( i_0  - i_e) + (1-\\alpha_u)\\dfrac{a_{i+1/2}}{\\alpha_u} u^{*old}_{i+1/2}\n",
    "    \\end{align}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ここに，$u^{*old}$は，反復計算の前のステップの値を示す．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 計算手順\n",
    "\n",
    "1. $h^*$の仮値を設定\n",
    "1. 運動方程式の近似式より$u^*$を計算\n",
    "1. 圧力の補正式より$\\delta h$を計算\n",
    "1. 流速の補正式より$\\delta u$を計算\n",
    "1. 収束判定：$\\delta h$が十分小さいか\n",
    "1. 小さくない場合．$h^{n+1}$を$h^*$として計算を進める\n"
   ]
  }
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