p27.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p27: Solve KdV equation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solve the KdV equation using FFT\n",
    "$$\n",
    "u_t + u u_x + u_{xxx} = 0, \\qquad x \\in [-\\pi,\\pi]\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib.collections import LineCollection\n",
    "from numpy import pi,cosh,exp,round,zeros,arange,real\n",
    "from numpy.fft import fft,ifft\n",
    "from matplotlib.pyplot import figure"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
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   "source": [
    "# Set up grid and differentiation matrix:\n",
    "N = 256; dt = 0.4/N**2; x = (2*pi/N)*arange(-N/2,N/2);\n",
    "A, B = 25.0, 16.0\n",
    "u = 3*A**2/cosh(0.5*A*(x+2))**2 + 3*B**2/cosh(0.5*B*(x+1))**2\n",
    "v = fft(u); \n",
    "k = zeros(N); k[0:N//2] = arange(0,N/2); k[N//2+1:] = arange(-N/2+1,0,1)\n",
    "ik3 = 1j*k**3\n",
    "\n",
    "# Time-stepping by Runge-Kutta\n",
    "tmax = 0.006; nplt = int(round((tmax/25)/dt))\n",
    "nmax = int(round(tmax/dt))\n",
    "udata = []; udata.append(list(zip(x, u)))\n",
    "tdata = [0.0]\n",
    "for n in range(1,nmax+1):\n",
    "    t = n*dt; g = -0.5j*dt*k\n",
    "    E = exp(dt*ik3/2); E2 = E**2\n",
    "    a = g * fft(real(ifft( v         ))**2)\n",
    "    b = g * fft(real(ifft( E*(v+a/2) ))**2)\n",
    "    c = g * fft(real(ifft( E*v+b/2   ))**2)\n",
    "    d = g * fft(real(ifft( E2*v+E*c  ))**2)\n",
    "    v = E2*v + (E2*a + 2*E*(b+c) + d)/6\n",
    "    if n%nplt == 0:\n",
    "        u = real(ifft(v))\n",
    "        udata.append(list(zip(x, u)))\n",
    "        tdata.append(t);\n",
    "\n",
    "fig = figure(figsize=(12,10))\n",
    "ax = fig.add_subplot(111,projection='3d')\n",
    "poly = LineCollection(udata)\n",
    "poly.set_alpha(0.5)\n",
    "ax.add_collection3d(poly, zs=tdata, zdir='y')\n",
    "ax.set_xlabel('x')\n",
    "ax.set_xlim3d(-pi, pi)\n",
    "ax.set_ylabel('t')\n",
    "ax.set_ylim3d(0, tmax)\n",
    "ax.set_zlabel('u')\n",
    "ax.set_zlim3d(0, 2000)\n",
    "ax.view_init(80,-115);"
   ]
  }
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