{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Inspired by https://perso.ensta-paris.fr/~becache/COURS-ONDES/Poly-num-0209.pdf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{u_{ij}^{n+1} - 2u_{ij}^{n} + u_{ij}^{n-1}}{\\Delta t^2} - \\frac{u_{i+1j}^{n} - 2u_{ij}^{n} + u_{i-1j}^{n}}{\\Delta x^2} - \\frac{u_{ij+1}^{n} - 2u_{ij}^{n} + u_{ij-1}^{n}}{\\Delta x^2} = g(n\\Delta t) f(x_i, y_j)\n",
    "$$\n",
    "\n",
    "Rearranging for explicit terms, we get:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import holoviews as hv\n",
    "hv.extension('bokeh', logo=False)\n",
    "\n",
    "import numba\n",
    "import numpy as np\n",
    "\n",
    "@numba.jit\n",
    "def step(U, V, W, dt, dx, dy, g, t, f):\n",
    "    \"\"\"U = u^n+1, V = u^n, W = u^n-1.\"\"\"\n",
    "    N, M = U.shape\n",
    "    for i in range(1, N-1):\n",
    "        for j in range(1, M-1):\n",
    "            U[i, j] = dt**2 / dx ** 2 * (V[i+1, j] - 2 * V[i, j] + V[i-1, j]) + \\\n",
    "                      dt**2 / dy ** 2 * (V[i, j+1] - 2 * V[i, j] + V[i, j-1]) + \\\n",
    "                      dt**2 * g(t) * f(i * dx, j * dy) + \\\n",
    "                      2 * V[i, j] - W[i, j]            \n",
    "    return U\n",
    "\n",
    "\n",
    "\n",
    "@numba.jit\n",
    "def g(t):\n",
    "    return np.sin(2 * np.pi * 15e-2 * t)\n",
    "\n",
    "@numba.jit\n",
    "def f(x, y):\n",
    "    x0, y0 = (62.5, 32.5)\n",
    "    return np.exp(- np.abs((x - x0)**2 + (y - y0)**2))\n",
    "\n",
    "def n_steps(U, V, W, dt, dx, dy, g, t0, f, nsteps=100):\n",
    "    for i in range(nsteps):\n",
    "        t = t0 + i * dt\n",
    "        U = step(U, V, W, dt, dx, dy, g, t, f)\n",
    "        W = V.copy()\n",
    "        V = U.copy()\n",
    "    return U, V, W, t"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "N, M = 250, 260\n",
    "U = np.zeros((N, M))\n",
    "V = np.zeros((N, M))\n",
    "W = np.zeros((N, M))\n",
    "dt = 0.1\n",
    "dx = 0.5\n",
    "dy = dx\n",
    "t = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "N * dx / 2, M * dy / 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "snapshots = []\n",
    "for _ in range(30):\n",
    "    U, V, W, t = n_steps(U, V, W, dt, dx, dy, g, t, f, nsteps=50)\n",
    "    snapshots.append(U.copy())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%output holomap='scrubber'\n",
    "hv.HoloMap({ind: hv.Image(U.T).opts(cmap='seismic', tools=['hover'], colorbar=True, width=700, height=600).redim.range(z=(-1e-1, 1e-1)) for ind, U in enumerate(snapshots)})"
   ]
  }
 ],
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