|
| 1 | +{ |
| 2 | + "cells": [ |
| 3 | + { |
| 4 | + "cell_type": "code", |
| 5 | + "execution_count": null, |
| 6 | + "id": "cfc06113", |
| 7 | + "metadata": {}, |
| 8 | + "outputs": [], |
| 9 | + "source": [ |
| 10 | + "import numpy as np\n", |
| 11 | + "%matplotlib inline\n", |
| 12 | + "import matplotlib\n", |
| 13 | + "import matplotlib.pyplot as plt\n", |
| 14 | + "import matplotlib.animation\n", |
| 15 | + "from IPython.display import HTML\n", |
| 16 | + "font = {'size' : 15}\n", |
| 17 | + "matplotlib.rc('font', **font)" |
| 18 | + ] |
| 19 | + }, |
| 20 | + { |
| 21 | + "cell_type": "code", |
| 22 | + "execution_count": null, |
| 23 | + "id": "0209d0a0", |
| 24 | + "metadata": {}, |
| 25 | + "outputs": [], |
| 26 | + "source": [ |
| 27 | + "def rk3(u,xi,rhs):\n", |
| 28 | + " y2 = u + dt*rhs(u,xi)\n", |
| 29 | + " y3 = 0.75*u + 0.25*(y2 + dt*rhs(y2,xi))\n", |
| 30 | + " u_new = 1./3 * u + 2./3 * (y3 + dt*rhs(y3,xi))\n", |
| 31 | + " return u_new\n", |
| 32 | + "\n", |
| 33 | + "\n", |
| 34 | + "def rhs(u, xi, epsilon=1.0):\n", |
| 35 | + " uhat = np.fft.fft(u)\n", |
| 36 | + " return -u*np.real(np.fft.ifft(1j*xi*uhat)) - epsilon*np.real(np.fft.ifft(-1j*xi**3*uhat))\n", |
| 37 | + " \n", |
| 38 | + "def solve_KdV(u0,tmax=1.,m=256,epsilon=1.0, ylims=(-100,300)):\n", |
| 39 | + " \"\"\"Solve the KdV equation using Fourier spectral collocation in space\n", |
| 40 | + " and SSPRK3 in time, on the domain (-pi, pi). The input u0 should be a function.\n", |
| 41 | + " \"\"\"\n", |
| 42 | + " # Grid\n", |
| 43 | + " L = 2*np.pi\n", |
| 44 | + " x = np.arange(-m/2,m/2)*(L/m)\n", |
| 45 | + " xi = np.fft.fftfreq(m)*m*2*np.pi/L\n", |
| 46 | + "\n", |
| 47 | + " dt = 1.73/((m/2)**3)\n", |
| 48 | + " u = u0(x)\n", |
| 49 | + " uhat2 = np.abs(np.fft.fft(u))\n", |
| 50 | + "\n", |
| 51 | + " num_plots = 400\n", |
| 52 | + " nplt = np.floor((tmax/num_plots)/dt)\n", |
| 53 | + " nmax = int(round(tmax/dt))\n", |
| 54 | + "\n", |
| 55 | + " fig = plt.figure(figsize=(12,8))\n", |
| 56 | + " axes = fig.add_subplot(111)\n", |
| 57 | + " line, = axes.plot(x,u,lw=3)\n", |
| 58 | + " xi_max = np.max(np.abs(xi))\n", |
| 59 | + " axes.set_xlabel(r'$x$',fontsize=30)\n", |
| 60 | + " plt.close()\n", |
| 61 | + "\n", |
| 62 | + " frames = [u.copy()]\n", |
| 63 | + " tt = [0]\n", |
| 64 | + " uuhat = [uhat2]\n", |
| 65 | + "\n", |
| 66 | + " for n in range(1,nmax+1):\n", |
| 67 | + " u_new = rk3(u,xi,rhs)\n", |
| 68 | + "\n", |
| 69 | + " u = u_new.copy()\n", |
| 70 | + " t = n*dt\n", |
| 71 | + " # Plotting\n", |
| 72 | + " if np.mod(n,nplt) == 0:\n", |
| 73 | + " frames.append(u.copy())\n", |
| 74 | + " tt.append(t)\n", |
| 75 | + " \n", |
| 76 | + " def plot_frame(i):\n", |
| 77 | + " line.set_data(x,frames[i])\n", |
| 78 | + " axes.set_title('t= %.2e' % tt[i])\n", |
| 79 | + " axes.set_xlim((-np.pi,np.pi))\n", |
| 80 | + " axes.set_ylim(ylims)\n", |
| 81 | + "\n", |
| 82 | + " anim = matplotlib.animation.FuncAnimation(fig, plot_frame,\n", |
| 83 | + " frames=len(frames), interval=100)\n", |
| 84 | + "\n", |
| 85 | + " return HTML(anim.to_jshtml())" |
| 86 | + ] |
| 87 | + }, |
| 88 | + { |
| 89 | + "cell_type": "markdown", |
| 90 | + "id": "3f0cfe80", |
| 91 | + "metadata": {}, |
| 92 | + "source": [ |
| 93 | + "## Initial sinusoid\n", |
| 94 | + "\n", |
| 95 | + "Here we set up something similar to the FPUT experiment, with a single low-frequency mode as initial condition on a periodic domain. Notice how, at some later times, the solution comes close to the initial condition." |
| 96 | + ] |
| 97 | + }, |
| 98 | + { |
| 99 | + "cell_type": "code", |
| 100 | + "execution_count": null, |
| 101 | + "id": "7bd939ca", |
| 102 | + "metadata": {}, |
| 103 | + "outputs": [], |
| 104 | + "source": [ |
| 105 | + "def u0(x):\n", |
| 106 | + " return 100*np.sin(x)\n", |
| 107 | + "solve_KdV(u0)" |
| 108 | + ] |
| 109 | + }, |
| 110 | + { |
| 111 | + "cell_type": "markdown", |
| 112 | + "id": "784f0474", |
| 113 | + "metadata": {}, |
| 114 | + "source": [ |
| 115 | + "## Formation of a soliton train from an initial positive pulse." |
| 116 | + ] |
| 117 | + }, |
| 118 | + { |
| 119 | + "cell_type": "code", |
| 120 | + "execution_count": null, |
| 121 | + "id": "c38c986f", |
| 122 | + "metadata": {}, |
| 123 | + "outputs": [], |
| 124 | + "source": [ |
| 125 | + "def u0(x):\n", |
| 126 | + " return 2000*np.exp(-10*(x+2)**2)\n", |
| 127 | + "solve_KdV(u0, tmax=0.005, ylims=(-100,3000))" |
| 128 | + ] |
| 129 | + }, |
| 130 | + { |
| 131 | + "cell_type": "markdown", |
| 132 | + "id": "f191670a", |
| 133 | + "metadata": {}, |
| 134 | + "source": [ |
| 135 | + "# Interaction of two solitons" |
| 136 | + ] |
| 137 | + }, |
| 138 | + { |
| 139 | + "cell_type": "code", |
| 140 | + "execution_count": null, |
| 141 | + "id": "76b69166", |
| 142 | + "metadata": {}, |
| 143 | + "outputs": [], |
| 144 | + "source": [ |
| 145 | + "A = 25; B = 16;\n", |
| 146 | + "def u0(x):\n", |
| 147 | + " return 3*A**2/np.cosh(0.5*(A*(x+2.)))**2 + 3*B**2/np.cosh(0.5*(B*(x+1)))**2\n", |
| 148 | + "solve_KdV(u0,tmax = 0.006, ylims=(-10,3000))" |
| 149 | + ] |
| 150 | + }, |
| 151 | + { |
| 152 | + "cell_type": "markdown", |
| 153 | + "id": "fa18aa7b", |
| 154 | + "metadata": {}, |
| 155 | + "source": [ |
| 156 | + "The next simulation shows a comparison between the propagation of a single soliton versus the interaction of two solitons." |
| 157 | + ] |
| 158 | + }, |
| 159 | + { |
| 160 | + "cell_type": "code", |
| 161 | + "execution_count": null, |
| 162 | + "id": "b24c8e5d", |
| 163 | + "metadata": {}, |
| 164 | + "outputs": [], |
| 165 | + "source": [ |
| 166 | + "# Grid\n", |
| 167 | + "m = 256\n", |
| 168 | + "L = 2*np.pi\n", |
| 169 | + "x = np.arange(-m/2,m/2)*(L/m)\n", |
| 170 | + "xi = np.fft.fftfreq(m)*m*2*np.pi/L\n", |
| 171 | + "\n", |
| 172 | + "dt = 1.73/((m/2)**3)\n", |
| 173 | + "\n", |
| 174 | + "A = 25; B = 16;\n", |
| 175 | + "u = 3*A**2/np.cosh(0.5*(A*(x+2.)))**2 + 3*B**2/np.cosh(0.5*(B*(x+1)))**2\n", |
| 176 | + "v = 3*A**2/np.cosh(0.5*(A*(x+2.)))**2\n", |
| 177 | + "\n", |
| 178 | + "tmax = 0.006\n", |
| 179 | + "\n", |
| 180 | + "uhat2 = np.abs(np.fft.fft(u))\n", |
| 181 | + "\n", |
| 182 | + "num_plots = 400\n", |
| 183 | + "nplt = np.floor((tmax/num_plots)/dt)\n", |
| 184 | + "nmax = int(round(tmax/dt))\n", |
| 185 | + "\n", |
| 186 | + "fig = plt.figure(figsize=(12,8))\n", |
| 187 | + "axes = fig.add_subplot(111)\n", |
| 188 | + "line, = axes.plot(x,u,lw=3)\n", |
| 189 | + "line2, = axes.plot(x,v,lw=3)\n", |
| 190 | + "xi_max = np.max(np.abs(xi))\n", |
| 191 | + "axes.set_xlabel(r'$x$',fontsize=30)\n", |
| 192 | + "plt.close()\n", |
| 193 | + "\n", |
| 194 | + "frames = [u.copy()]\n", |
| 195 | + "vframes = [v.copy()]\n", |
| 196 | + "tt = [0]\n", |
| 197 | + "uuhat = [uhat2]\n", |
| 198 | + "\n", |
| 199 | + "for n in range(1,nmax+1):\n", |
| 200 | + " u_new = rk3(u,xi,rhs)\n", |
| 201 | + " v_new = rk3(v,xi,rhs)\n", |
| 202 | + "\n", |
| 203 | + " u = u_new.copy()\n", |
| 204 | + " v = v_new.copy()\n", |
| 205 | + " t = n*dt\n", |
| 206 | + " # Plotting\n", |
| 207 | + " if np.mod(n,nplt) == 0:\n", |
| 208 | + " frames.append(u.copy())\n", |
| 209 | + " vframes.append(v.copy())\n", |
| 210 | + " tt.append(t)\n", |
| 211 | + " uhat2 = np.abs(np.fft.fft(u))\n", |
| 212 | + " uuhat.append(uhat2)\n", |
| 213 | + " \n", |
| 214 | + "def plot_frame(i):\n", |
| 215 | + " line.set_data(x,frames[i])\n", |
| 216 | + " line2.set_data(x,vframes[i])\n", |
| 217 | + " power_spectrum = np.abs(uuhat[i])**2\n", |
| 218 | + " axes.set_title('t= %.2e' % tt[i])\n", |
| 219 | + " axes.set_xlim((-np.pi,np.pi))\n", |
| 220 | + " axes.set_ylim((-10,3000))\n", |
| 221 | + " \n", |
| 222 | + "anim = matplotlib.animation.FuncAnimation(fig, plot_frame,\n", |
| 223 | + " frames=len(frames), interval=100)\n", |
| 224 | + "\n", |
| 225 | + "HTML(anim.to_jshtml())" |
| 226 | + ] |
| 227 | + }, |
| 228 | + { |
| 229 | + "cell_type": "markdown", |
| 230 | + "id": "70b44acc", |
| 231 | + "metadata": {}, |
| 232 | + "source": [ |
| 233 | + "## Formation of a dispersive shockwave" |
| 234 | + ] |
| 235 | + }, |
| 236 | + { |
| 237 | + "cell_type": "code", |
| 238 | + "execution_count": null, |
| 239 | + "id": "758b2d7e", |
| 240 | + "metadata": {}, |
| 241 | + "outputs": [], |
| 242 | + "source": [ |
| 243 | + "def u0(x):\n", |
| 244 | + " return -500*np.exp(-10*(x-2)**2)\n", |
| 245 | + "solve_KdV(u0, tmax=0.005, epsilon=0.1, ylims=(-600,300))" |
| 246 | + ] |
| 247 | + } |
| 248 | + ], |
| 249 | + "metadata": { |
| 250 | + "kernelspec": { |
| 251 | + "display_name": "Python 3 (ipykernel)", |
| 252 | + "language": "python", |
| 253 | + "name": "python3" |
| 254 | + }, |
| 255 | + "language_info": { |
| 256 | + "codemirror_mode": { |
| 257 | + "name": "ipython", |
| 258 | + "version": 3 |
| 259 | + }, |
| 260 | + "file_extension": ".py", |
| 261 | + "mimetype": "text/x-python", |
| 262 | + "name": "python", |
| 263 | + "nbconvert_exporter": "python", |
| 264 | + "pygments_lexer": "ipython3", |
| 265 | + "version": "3.10.4" |
| 266 | + } |
| 267 | + }, |
| 268 | + "nbformat": 4, |
| 269 | + "nbformat_minor": 5 |
| 270 | +} |
0 commit comments