julia/ODE2.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "今週の講義は常微分方程式（ODEs: ordinary differential equations）の数値解法のベクトル編です．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**課題**：以下のsymplectic Euler法の関数を完成させてください．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "時刻$T=3$まで数値計算することを考えます．いくつか変数を準備します．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "T =6.0;\n",
    "N = 60; # TやNは適当に変更して色々試してみてください\n",
    "h = T / N;\n",
    "t = 0:h:T;\n",
    "y0 = [1.0;0.0];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f (generic function with 1 method)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#右辺の関数を定義します．\n",
    "function f(y)\n",
    "    return [y[2] ; -y[1]]\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 数値解を格納する配列\n",
    "yEE = zeros(2, N+1); # 陽的Euler用\n",
    "yEE[:, 1] = y0; # 初期値\n",
    "yRK = zeros(2, N+1); # Runge--Kutta用\n",
    "yRK[:, 1] = y0; # 初期値\n",
    "ysE = zeros(2, N+1); # symplectic Euler用\n",
    "ysE[:, 1] = y0; # 初期値"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ExEuler (generic function with 1 method)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 陽的Euler法の一ステップ\n",
    "function ExEuler(y,h)\n",
    "    return y + h * f(y)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "RK (generic function with 1 method)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Runge-Kutta法の一ステップ\n",
    "function RK(y,h)\n",
    "    Y1 = y\n",
    "    Y2 = y + h/2 * f(Y1)\n",
    "    Y3 = y + h/2 * f(Y2)\n",
    "    Y4 = y + h   * f(Y3)\n",
    "    return y + h/6 * (f(Y1) + 2*f(Y2) + 2*f(Y3) + f(Y4) )\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "SymEuler (generic function with 1 method)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# symplectic Euler法の一ステップ (fは使わずに書く)\n",
    "function SymEuler(y,h)\n",
    "#     \n",
    "#     return \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "for n = 1 : N\n",
    "    yEE[:, n+1] = ExEuler(yEE[:, n],h)\n",
    "    yRK[:, n+1] = RK(yRK[:, n],h) \n",
    "#     ysE[:, n+1] = SymEuler(ysE[:, n],h) # 上のsymplectic Euler法の関数が完成したら一番左の # を消す．\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(cos.(t),-sin.(t),label=\"exact\")\n",
    "scatter!(yEE[1,:],yEE[2,:],label=\"explicit Euler\")\n",
    "scatter!(yRK[1,:],yRK[2,:],label=\"Runge-Kutta\") \n",
    "# scatter!(ysE[1,:],ysE[2,:],label=\"symplectic Euler\") # 上のsymplectic Euler法の関数が完成したら一番左の # を消す．\n",
    "xlabel!(\"q\")\n",
    "ylabel!(\"p\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "symplectic Euler法の結果はどうなったでしょうか．4次精度のRK法が厳密解に近いのは直感的にも納得できると思いますが，symplectic Euler法は陽的Euler法と同じ1次精度解法です．ちょっと不思議な結果になったと思います．関連する問題をレポート課題で出題しています．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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