ajout calcul diff et ode

ode/TP/ode-raide-etudiant.ipynb

+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Problème raide (stiff)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Plots.PyPlotBackend()"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# import Pkg; Pkg.add(\"DifferentialEquations\")\n",
+    "# Pkg.add(\"Plots\")\n",
+    "# import Pkg; Pkg.add(\"ODEInterfaceDiffEq\")\n",
+    "using DifferentialEquations\n",
+    "using Plots\n",
+    "using ODEInterfaceDiffEq              # pour accéder à radau5\n",
+    "pyplot()\n",
+    "# gr()\n",
+    "# gr(dpi=300,size=(600,400),thickness_scaling=1)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Exemple curtiss et Hirschfelder\n",
+    "Cet exemple provient du tome 2 du livre d'Hairer, page 2. Le problème de Cauchy est\n",
+    "$$(IVP)_2\\left\\{\\begin{array}{l}\n",
+    "\\dot{y}(t)=-50(y(t)-\\cos(t))\\\\\n",
+    "y(0)=0\n",
+    "\\end{array}\\right.\n",
+    "$$\n",
+    "avec $[t_0\\;\\; t_f]=[0\\;\\; 1.5]$.\n",
+    "\n",
+    "On demande de résoudre numériquement ce problème avec les algorithmes : \n",
+    "- d'Euler explicite, alg = Euler(), et avec un pas de 1.974/50 et un pas de 1.875/50 (option dt). On prendra aussi comme option dense = false afin de n'avoir que les points aux instant $t_i$.\n",
+    "- d'Euler implicite, algo = ImplicitEuler(), avec un pas de 0.5. On prendra aussi comme options dense = false et adaptive = false afin de n'avoir que les points aux instant $t_i$.\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\u001b[36mODEProblem\u001b[0m with uType \u001b[36mArray{Float64,1}\u001b[0m and tType \u001b[36mFloat64\u001b[0m. In-place: \u001b[36mfalse\u001b[0m\n",
+       "timespan: (0.0, 1.5)\n",
+       "u0: [0.0]"
+      ]
+     },
+     "execution_count": 9,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "function curtiss(x,p,t)\n",
+    "# deuxieme membre de l'equation differentiel de l'equation de Curtiss et Hirschfelder\n",
+    "# ref: Hairer page 2 tome 2\n",
+    "# Input\n",
+    "# x : state\n",
+    "#     real(2)\n",
+    "# p : parameter vector\n",
+    "# t : time\n",
+    "#     real\n",
+    "# Output\n",
+    "# xpoint : vector of velocity\n",
+    "#          same as x\n",
+    "    xpoint    = similar(x)\n",
+    "    \n",
+    "# A compléter\n",
+    "    return xpoint\n",
+    "end\n",
+    "\n",
+    "x0 = [0.];\n",
+    "t0 = 0.;\n",
+    "tf = 1.5;\n",
+    "p = [1.]\n",
+    "tspan = (t0,tf)\n",
+    "prob = ODEProblem(curtiss,x0,tspan,p)\n",
+    "# A compléter\n",
+    "# ..."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Exemple de Roberston\n",
+    "On considère la réaction chimique \n",
+    "$$\\begin{array}{cccl}\n",
+    "A & \\stackrel{0.04}{\\longrightarrow} & B & \\textrm{(lente)},\\\\\n",
+    "B+B & \\stackrel{3.10^7}{\\longrightarrow} & C+B & \\textrm{(très rapide)},\\\\\n",
+    "B+C & \\stackrel{10^4}{\\longrightarrow} & A+C & \\textrm{(rapide)}.\\\\\n",
+    "\\end{array}\n",
+    "$$\n",
+    "Le système différentiel associé à cette réaction chimique est donnée par\n",
+    "$$(IVP)\\left\\{\\begin{array}{lrr}\n",
+    "\\dot{x}_1(t)= &-0.04x_1(t)+10^4x_2(t)x_3(t) &\\\\\n",
+    "\\dot{x}_2(t)= & 0.04x_1(t)-10^4x_2(t)x_3(t) &-3.10^7x_2^2(t)\\\\\n",
+    "\\dot{x}_3(t)= &                             & 3.10^7x_2^2(t)\\\\\n",
+    "x_1(0)=1,\n",
+    "x_2(0)=0,\n",
+    "x_3(0)=0.\n",
+    "\\end{array}\\right.\n",
+    "$$\n",
+    "avec $[t_0\\;\\; t_f]=[0\\;\\; 0.3]$.\n",
+    "\n",
+    "On demande de résoudre numériquement ce problème avec les algorithmes : \n",
+    "- algo = DP5() (Dormand-Prince's 5/4 Runge-Kutta method, algorithme utilisé dans ode45 de matlab et dopri5 du Professeur Hairer) et les options reltol = 1e-2, abstol = 1.e-6, dense = false\n",
+    "- algo = DP5() (Dormand-Prince's 5/4 Runge-Kutta method, algorithme utilisé dans ode45 de matlab et dopri5 du Professeur Hairer) et les options reltol = 1e-3, abstol = 1.e-6, dense = false, dense = false\n",
+    "- algo = radau5() : pragramme Radau5 du Professeur Hairer pour les problème raides.\n",
+    "\n",
+    "Et d'afficher sur le même graphique la deuxième composante\n",
+    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\u001b[36mODEProblem\u001b[0m with uType \u001b[36mArray{Float64,1}\u001b[0m and tType \u001b[36mFloat64\u001b[0m. In-place: \u001b[36mfalse\u001b[0m\n",
+       "timespan: (0.0, 0.3)\n",
+       "u0: [1.0, 0.0, 0.0]"
+      ]
+     },
+     "execution_count": 10,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "function chimie(x,p,t)\n",
+    "# deuxieme membre de l'equation differentiel de l'equation de Robertson (1966)\n",
+    "# ref: Hairer page 3 tome 2\n",
+    "# Input\n",
+    "# x : state\n",
+    "#     real(2)\n",
+    "# p : parameter vector\n",
+    "# t : time\n",
+    "#     real\n",
+    "# Output\n",
+    "# xpoint : vector of velocity\n",
+    "#          same as x\n",
+    "    xpoint    = similar(x)\n",
+    "# A compléter\n",
+    "    \n",
+    "    return xpoint\n",
+    "end\n",
+    "\n",
+    "x0 = [1., 0., 0.];\n",
+    "t0 = 0.;\n",
+    "tf = 0.3;\n",
+    "p = []\n",
+    "tspan = (t0,tf)\n",
+    "xpoint = similar(x0)\n",
+    "prob = ODEProblem(chimie,x0,tspan,p)\n",
+    "# A compléter\n",
+    "# ..."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Julia 1.4.1",
+   "language": "julia",
+   "name": "julia-1.4"
+  },
+  "language_info": {
+   "file_extension": ".jl",
+   "mimetype": "application/julia",
+   "name": "julia",
+   "version": "1.4.1"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}