DiffusionEquation.ipynb

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    "# Numerical Solvers\n",
    "\n",
    "### Matthew Kafker, Jeremy Welsh-Kavan\n",
    "\n",
    "<hr>\n",
    "\n",
    "The goal of this project is to build a program for numerically solving various ODEs and PDEs. By validating various numerical methods on DEs with known analytical solutions, we hope to be able to implement these methods for a much larger set DEs.\n",
    "\n",
    "We start with a naive approach and propagate our solution in time using information from the spatial derivative. This is analogous to the the initial value problem (IVP) in which we hope to solve for $y(t)$ given\n",
    "\n",
    "$\n",
    "y'(t) = f(t, y(t)) ~~~~~ y(t_0) = y_0\n",
    "$\n",
    "\n",
    "Instead, we are solving for $u(x,t)$ given\n",
    "\n",
    "$\n",
    "\\partial_t u(x,t) = \\partial_{xx} u(x,t) \\\\\n",
    "u(x,0) = f(x) \\\\\n",
    "u(0,t) = u(L,t) = 0 \\\\\n",
    "$\n",
    "\n",
    "for $x \\in [0,L]$ and $t>0$.\n",
    "\n",
    "\n",
    "\n",
    "[//]: <> (For instructions on how to use LaTeX, see the LaTeX folder, which also contains a few other TeXShop files.)\n",
    "\n",
    "<hr>\n",
    "\n",
    "Most of what we implement here will be informed by the following resources:\n",
    "\n",
    "Asmar, Nakhlé H., and Nakhlé H. Asmar. Partial Differential Equations with Fourier Series and Boundary Value Problems. 2nd ed. Upper Saddle River, N. j: Pearson Prentice Hall, 2005. Print.\n",
    "\n",
    "Press, William H. Numerical Recipes : the Art of Scientific Computing. Cambridge ;: Cambridge University Press, 1989. Print.\n",
    "\n",
    "Kutz, Jose Nathan. Data-Driven Modeling & Scientific Computation : Methods for Complex Systems & Big Data. First edition. Oxford: Oxford University Press, 2013. Print."
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    "Libraries and plotting preferences."
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    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import norm\n",
    "\n",
    "%config InlineBackend.figure_formats = ['svg']\n",
    "plt.rcParams['figure.figsize'] = [8, 6]"
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   "source": [
    "## Simulation Set-Up\n",
    "\n",
    "Define grid in position and momentum$\\leftrightarrow$wave vector."
   ]
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    "L = 10\n",
    "xmin, xmax = 0, L\n",
    "res = 1024\n",
    "x_arr = np.linspace(xmin, xmax, res)\n",
    "dx = (x_arr[1]-x_arr[0])\n",
    "N = len(x_arr)\n",
    "\n",
    "kmax = np.pi/dx\n",
    "kmin = 2*np.pi/L\n",
    "k_pos = np.arange(0, kmax, kmin)\n",
    "k_neg = np.arange(-kmax, 0, kmin)\n",
    "k_arr = np.concatenate([k_pos, k_neg])"
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    "dx"
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   "source": [
    "Define mass distribution."
   ]
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   "source": [
    "sig = L/40\n",
    "f_arr = (2*np.pi*sig**2)**(-0.5)*np.exp(-((x_arr-L/2)**2.0)/(2*sig**2))"
   ]
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    "f_deriv_anly = (2*np.pi*sig**2)**(-0.5)*(((x_arr-L/2)**2/sig**4)*np.exp(-((x_arr-L/2)**2.0)/(2*sig**2)) -(1/sig**2)*np.exp(-((x_arr-L/2)**2.0)/(2*sig**2)))"
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   "id": "0f02065b-dcb9-4b6f-b920-43f02697708f",
   "metadata": {},
   "source": [
    "Check normalization of mass distribution."
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       "1.0000000000000007"
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   "source": [
    "(f_arr*dx).sum()"
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   "id": "6ed2fe7c-f119-4abc-970a-77ec3ae75c1e",
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   "source": [
    "Plot mass distribution."
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   "source": [
    "plt.plot(x_arr, f_arr,label=\"u(x)\")\n",
    "plt.plot(x_arr, f_deriv_anly,label=r\"$u_{xx}(x)$\")\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"$u(x)$\")\n",
    "plt.title(\"Mass Density Profile\")\n",
    "plt.legend()\n",
    "plt.show()"
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  },
  {
   "cell_type": "markdown",
   "id": "7c358bb8-d206-44c6-a8d1-15a0974e6271",
   "metadata": {},
   "source": [
    "Perform spatial derivatives of mass distribution using FFT."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "af660f65-9fd0-45bc-8265-216d7fa54acf",
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   "source": [
    "f_arr_fft = np.fft.fft(f_arr) \n",
    "deriv_f_arr = np.fft.ifft(1j*k_arr*f_arr_fft)\n",
    "\n",
    "plt.plot(x_arr, np.real(deriv_f_arr),label=\"Real Part\")\n",
    "plt.plot(x_arr, np.imag(deriv_f_arr),label=\"Imaginary Part\")\n",
    "plt.title(\"First Derivative\")\n",
    "plt.legend()\n",
    "plt.show()\n",
    "\n",
    "deriv_f_arr_2 = np.fft.ifft(-(k_arr**2)*f_arr_fft)\n",
    "plt.plot(x_arr, np.real(deriv_f_arr_2),label=\"Real Part\")\n",
    "plt.plot(x_arr, np.imag(deriv_f_arr_2),label=\"Imaginary Part\")\n",
    "plt.title(\"Second Derivative\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d0cf9166-31da-4de1-8df3-5c5f161c94ba",
   "metadata": {},
   "source": [
    "## Time-Stepping\n",
    "\n",
    "We wish to solve the diffusion equation, $u_t=D u_{xx}$ in our simulation box for some interval of time.\n",
    "\n",
    "Function to evaluate RHS."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "8d6bf916-f938-4c19-8ec6-cc1f48fc5994",
   "metadata": {},
   "outputs": [],
   "source": [
    "def deriv(u,D):\n",
    "    return D*np.fft.ifft(-(k_arr**2)*np.fft.fft(u))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "02d0603f-fd37-46e2-83c8-b9b9c27cf863",
   "metadata": {},
   "source": [
    "### Solver 1: Forward Euler\n",
    "\n",
    "The Forward Euler technique implements the time-stepping scheme $$u(x,t+dt) = u(x,t) + D u_{xx}(x,t)dt$$\n",
    "\n",
    "Time-stepper parameters and initial conditions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "eef1fd35-3503-45ff-8864-6dcf05f92e4f",
   "metadata": {},
   "outputs": [],
   "source": [
    "dt_FE = 0.001\n",
    "T_FE = 14.4072\n",
    "s_FE = int(np.ceil(T_FE/dt_FE))\n",
    "u_FE = np.zeros([s_FE,len(x_arr)],dtype = 'complex_') # Fixes \"ComplexWarning: Casting complex values to real discards the imaginary part\"\n",
    "u_FE[0] = f_arr\n",
    "D_FE = 0.01"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c3cf4cc-3dbe-4d04-9077-385961b5c712",
   "metadata": {},
   "source": [
    "Perform numerical integration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "c4c1d1bc-cb3a-45f2-b7cc-583a8d318af0",
   "metadata": {},
   "outputs": [],
   "source": [
    "for t in range(0,s_FE-1):\n",
    "    u_FE[t+1] = u_FE[t]+dt_FE*deriv(u_FE[t],D_FE)\n",
    "    \n",
    "# NOTE: somewhere we appear to be throwing out complex part. Shouldn't do that!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "af1eb2ef-04ff-4dd5-9c32-b17ee8ed1018",
   "metadata": {},
   "outputs": [
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   "source": [
    "plt.imshow(np.real(u_FE),aspect=\"0.5\",extent=[0,L,T_FE,0])\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Time\")\n",
    "plt.title(\"u(x) Over Time\")\n",
    "plt.colorbar()\n",
    "plt.show()"
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  },
  {
   "cell_type": "markdown",
   "id": "9c6f4d09-bab2-4513-b5bd-fcbe21ae49e2",
   "metadata": {},
   "source": [
    "Test mass conservation"
   ]
  },
  {
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   "execution_count": 37,
   "id": "274222a1-d632-486b-98b3-2ecd5a6330f0",
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      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
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   ],
   "source": [
    "norm = np.array([(u_FE[t]*dx).sum() for t in range(s_FE)], dtype=\"complex_\")\n",
    "plt.plot(np.real(norm) - 1)\n",
    "plt.xlabel(\"Time Step\")\n",
    "plt.ylabel(\"$\\int u(x)dx -1$\")\n",
    "plt.title(\"Normalization vs. Time\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "f62882b3-ea4e-4a4c-983e-d31536511a36",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.42861286636753e-17"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.max(np.imag(norm))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7916db4a-b69b-4c30-b93b-dbbf278b7fc5",
   "metadata": {},
   "source": [
    "### Solver 2: Runge-Kutta 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "47b0763b-14fc-4b08-9094-7b466114f8d4",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "dt_RK4 = 0.001\n",
    "T_RK4 = 14.4072\n",
    "s_RK4 = int(np.ceil(T_RK4/dt_RK4))\n",
    "u_RK4 = np.zeros([s_RK4,len(x_arr)],dtype=\"complex_\")\n",
    "D_RK4 = 0.01\n",
    "# Need 4 initial values now\n",
    "u_RK4[0]=f_arr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "59057d1d-b30f-4974-b5e2-cce4a1ccc6b7",
   "metadata": {},
   "outputs": [],
   "source": [
    "for t in range(0,s_RK4-1):\n",
    "    f1 = deriv(u_RK4[t],D_RK4)\n",
    "    f2 = deriv(u_RK4[t] + (dt_RK4/2)*f1,D_RK4)\n",
    "    f3 = deriv(u_RK4[t] + (dt_RK4/2)*f2,D_RK4)\n",
    "    f4 = deriv(u_RK4[t] + dt_RK4*f3,D_RK4)\n",
    "    u_RK4[t+1] = u_RK4[t] + (dt_RK4/6)*(f1 + 2*f2 + 2*f3 + f4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "506c94ae-b770-490a-ad4d-5d7f0f1cc8d9",
   "metadata": {
    "tags": []
   },
   "outputs": [
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   "source": [
    "plt.imshow(np.real(u_RK4),aspect=\"0.5\",extent=[0,L,T_RK4,0])\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Time\")\n",
    "plt.title(\"u(x) Over Time\")\n",
    "plt.colorbar()\n",
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   "source": [
    "$\\textbf{Analysis}$\n",
    "\n",
    "We still cannot make the diffusion constant arbitrarily large. $D \\approx 0.02$ appears to be close to the upper bound where the solver does not crash. Perhaps this has something to do with the the density becoming non-negligible at the boundary?\n",
    "\n",
    "Also, even with 4th order Runge-Kutta, with truncation error of order $\\Delta t ^5$, we still cannot significantly increase $dt > 0.001$ without the solver failing. Why?"
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   "id": "1cf1c8f5-00b9-4938-ab99-b71ab9a26c16",
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   "source": [
    "### Fourier Series solution\n"
   ]
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   "execution_count": 42,
   "id": "e5cddeb3-830d-4060-aada-120b1df2be6e",
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   "source": [
    "M = 120\n",
    "n_arr = np.arange(1,M+1)\n",
    "lambda_n = np.sqrt(D_RK4)*n_arr*np.pi/L\n",
    "xn_arr = n_arr[:,np.newaxis]@x_arr[np.newaxis]\n",
    "t_arr  = np.arange(0, T_RK4, dt_RK4)\n",
    "sin_arr = np.sin(np.pi*xn_arr/L)\n",
    "b_n = (2/L)*np.array([np.trapz(f_arr*line,dx=dx) for line in sin_arr])\n",
    "exp_arr = np.exp(-(lambda_n**2)[:,np.newaxis]@t_arr[np.newaxis])"
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   "cell_type": "code",
   "execution_count": 43,
   "id": "62aba197-6942-4012-803f-4b271894f0ee",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(14408, 1024)"
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     "execution_count": 43,
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   "source": [
    "u_ANLY = ((b_n*exp_arr.T)@sin_arr)\n",
    "u_ANLY.shape"
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   "id": "caf8525a-9dd9-4879-8aa7-b387f7c23e01",
   "metadata": {},
   "source": [
    "Comparing Fourier Series IC with exact."
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   "execution_count": 44,
   "id": "c35a5585-91d3-4ba7-8e5d-2bbc0020073e",
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   "outputs": [
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       "2.9110710469221243e-15"
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     "execution_count": 44,
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   "source": [
    "np.max(np.abs(u_ANLY[0] - f_arr))/np.max(f_arr)"
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   "id": "d8d02515-22e5-437b-804d-9f85f3732f7b",
   "metadata": {},
   "source": [
    "Although the Fourier series is accurate at short times, it eventually deviates from the infinite-space solution, which we can see by monitoring its normalization vs. time."
   ]
  },
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   "execution_count": 45,
   "id": "d4598c32-ef75-47d7-adf9-40d5c1a365f7",
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     "output_type": "display_data"
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   ],
   "source": [
    "plt.plot(t_arr,[-1+(u_ANLY[j].sum())*dx for j in range(len(u_ANLY))])\n",
    "plt.xlabel(\"Time (t)\")\n",
    "plt.ylabel(\"$\\int u(x,t)dx -1$\")\n",
    "plt.title(\"Normalization of Analytic Solution vs. Time\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "580f5ad0-c835-4957-bb4a-bc66e34f443e",
   "metadata": {},
   "outputs": [
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    "plt.imshow(u_ANLY,aspect=\"0.1\",extent=[0,L,T_RK4,0])\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Time\")\n",
    "plt.title(f\"Analytic u(x) Over Time, {M} Terms\")\n",
    "plt.colorbar()\n",
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   "metadata": {},
   "source": [
    "Comparing analytical with numerical results, keeping in mind that we know the analytical results are not trustworthy beyond ~ t = 8."
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   "id": "a41086e8-3e18-4456-add8-568f07b31da2",
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    "np.max(np.abs(u_ANLY - u_RK4))/np.max(u_ANLY)"
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       "(2.097204790197692e-05+0j)"
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    "np.max(np.abs(u_RK4 - u_FE))/np.max(u_RK4)"
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   "execution_count": 49,
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    "np.max(np.abs(u_ANLY[0:8000] - u_RK4[0:8000]))/np.max(u_ANLY[0:8000])"
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    "np.max(np.abs(u_ANLY[0:8000] - u_FE[0:8000]))/np.max(u_ANLY[0:8000])"
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   "execution_count": 51,
   "id": "1d93fc65-93a0-40db-8d1b-32d75ace6415",
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      "text/plain": [
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     },
     "metadata": {
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     "output_type": "display_data"
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   ],
   "source": [
    "plt.imshow(np.abs(u_ANLY - u_RK4)/np.max(u_ANLY),aspect=\"0.1\",extent=[0,L,T_RK4,0])\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Time\")\n",
    "plt.title(\"RK4 Error Over Time\")\n",
    "plt.colorbar()\n",
    "plt.hlines(8,0,L,\"w\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "e7c7c3c5-f804-4811-a0fe-afd07d37292a",
   "metadata": {},
   "outputs": [
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   "source": [
    "plt.imshow(np.abs(u_ANLY - u_FE)/np.max(u_ANLY),aspect=\"0.1\",extent=[0,L,T_RK4,0])\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Time\")\n",
    "plt.title(\"FE Error Over Time\")\n",
    "plt.colorbar()\n",
    "plt.hlines(8,0,L,\"w\")\n",
    "plt.show()"
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  {
   "cell_type": "markdown",
   "id": "d7c24786-2b4f-46cc-8646-19ddedc27eb9",
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   "source": [
    "### Attempt 4: Using replacing derivative with matrix from PHYSICAL REVIEW C 87, 051301(R) (2013)\n",
    "\n",
    "This essentially comes from unwrapping the FFT second derivative.\n",
    "\n",
    "$$\\Bigl(\\frac{d^2}{dx^2}\\Bigr)_{i,j}= \\begin{cases}-\\frac{2 \\pi^2}{N^2 dx^2} \\frac{(-1)^{i-j}}{\\text{sin}^2(\\frac{\\pi (i-j)}{N})} \\\\ -\\frac{\\pi^2}{3 dx^2}(1+\\frac{2}{N^2}) \\end{cases}$$"
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   "id": "f6ef71d6-2239-4358-bf9b-faca5c13d807",
   "metadata": {},
   "source": [
    "Defining the derivative matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "e03ab650-7b6e-466a-88a2-f28bfa602f41",
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   "source": [
    "derivmat = np.zeros([N,N])\n",
    "for i in range(N):\n",
    "    for j in range(N):\n",
    "        if i!=j:\n",
    "            derivmat[i,j] = -( 2*(np.pi**2)/(N**2 * dx**2) ) * (-1)**(i-j)/np.sin(np.pi*(i-j)/N)**2\n",
    "        else:\n",
    "            derivmat[i,j] = -( (np.pi**2)/(3 * dx**2) ) * (1 + 2/N**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2e9379b0-25b2-4e81-87cb-8fbd2e5d0f24",
   "metadata": {},
   "source": [
    "Comparing second derivative with that due to FFT."
   ]
  },
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   "execution_count": 54,
   "id": "91be3f70-ec81-44c4-99ab-535ef5a7cebb",
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     "text": [
      "Mean Squared Error\n",
      "fft derivative error: 0.07681687102742213\n",
      "matrix derivative error: 2.7690104357280847e-23\n",
      "Max Absolute Error\n",
      "fft derivative error: 0.05776177553843428\n",
      "matrix derivative error: 2.5443483862777175e-12\n"
     ]
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   ],
   "source": [
    "plt.plot(f_deriv_anly, linewidth=2.0, c=\"xkcd:red\" ,label=\"analytic derivative\")\n",
    "plt.plot(derivmat@f_arr, '--',c=\"xkcd:green\" , label=\"matrix derivative\")\n",
    "plt.plot(np.real(deriv_f_arr_2),'--',c=\"xkcd:purple\" ,label=\"fft derivative\")\n",
    "plt.plot(f_arr, c=\"xkcd:black\")\n",
    "plt.xlim((400,600))\n",
    "plt.legend()\n",
    "plt.title(\"Numerical Derivative Comparisons\",size=14)\n",
    "plt.show()\n",
    "\n",
    "fft_mse_err = ((np.real(deriv_f_arr_2) - f_deriv_anly)**2).mean()\n",
    "mat_mse_err = ((derivmat@f_arr - f_deriv_anly)**2).mean()\n",
    "print(f\"Mean Squared Error\")\n",
    "print(f\"fft derivative error: {fft_mse_err}\")\n",
    "print(f\"matrix derivative error: {mat_mse_err}\")\n",
    "\n",
    "\n",
    "fft_mae_err = np.max(np.abs(np.real(deriv_f_arr_2) - f_deriv_anly))/np.max(np.abs(f_deriv_anly))\n",
    "mat_mae_err = np.max(np.abs(derivmat@f_arr - f_deriv_anly))/np.max(np.abs(f_deriv_anly))\n",
    "print(f\"Max Absolute Error\")\n",
    "print(f\"fft derivative error: {fft_mae_err}\")\n",
    "print(f\"matrix derivative error: {mat_mae_err}\")"
   ]
  },
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   "cell_type": "markdown",
   "id": "6683decd-820c-4a91-bbb3-ca71c72ae343",
   "metadata": {},
   "source": [
    "This definitely suggests that the matrix derivative will be a better choice. Nice."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "14dd29fb-b61d-4428-a16c-6318353cd511",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compile with jit for faster matmul. WARNING: Jit is extremely temperamental\n",
    "from numba import jit\n",
    "@jit(nopython=True)\n",
    "def deriv2(u,D):\n",
    "    global derivmat\n",
    "    return D*derivmat@u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "8da82c62-a4cb-4086-8bbf-05125f14342b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.061323033167564295"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "np.max(np.abs(deriv2(f_arr,D_RK4)-deriv(f_arr,D_RK4)))/np.max(np.abs(deriv(f_arr,D_RK4)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "4fdc3227-72b7-487a-84ae-1fad053aa127",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 13.7 ms, sys: 518 µs, total: 14.3 ms\n",
      "Wall time: 1.88 ms\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([-6.93889390e-18, -2.77555756e-17, -1.38777878e-17, ...,\n",
       "       -2.77555756e-17, -4.16333634e-17, -4.16333634e-17])"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "%time deriv2(f_arr,D_RK4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "1571e910-6538-44ee-b924-dac03882a2eb",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 36 ms, sys: 4.82 ms, total: 40.8 ms\n",
      "Wall time: 4.95 ms\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([-6.93889390e-18, -2.77555756e-17, -1.38777878e-17, ...,\n",
       "       -2.77555756e-17, -4.16333634e-17, -4.16333634e-17])"
      ]
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     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time D_RK4*derivmat@f_arr"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e44d936-f4db-4957-9dd6-e539aae3a4ac",
   "metadata": {},
   "source": [
    "#### RK4 again, but with matrix 2nd derivative\n",
    "\n",
    "This is computationally very expensive, requiring 4 matrix multiplications per time step, where the matrix is $N \\times N$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "7492e99b-e382-4cbd-abaa-cecdc4e56749",
   "metadata": {},
   "outputs": [],
   "source": [
    "dt_FE_2 = 0.001\n",
    "T_FE_2 = 14.4072\n",
    "s_FE_2 = int(np.ceil(T_FE_2/dt_FE_2))\n",
    "u_FE_2 = np.zeros([s_FE_2,N])\n",
    "D_FE_2 = 0.01\n",
    "# Need 4 initial values now\n",
    "u_FE_2[0]=f_arr"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0fe29858-5d44-4786-a2bc-60180be29df4",
   "metadata": {},
   "source": [
    "Time evolution is now $\\textbf{extremely expensive}$, and the accuracy decreases. (Takes like 10-20 minutes to run.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "b5168be0-8cb7-460c-a893-4e8d2fd6e35e",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|██████████| 14407/14407 [00:13<00:00, 1063.36it/s]\n"
     ]
    }
   ],
   "source": [
    "from tqdm import tqdm\n",
    "for t in tqdm(range(0,s_FE_2-1)):\n",
    "     u_FE_2[t+1] = u_FE_2[t] + dt_FE_2*deriv2(u_FE_2[t],D_FE_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "id": "fce6d2a8-ffce-4edb-9fa7-0a749bd93a7a",
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   "outputs": [
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    "plt.imshow(u_FE_2,aspect=\"0.5\",extent=[0,L,T_FE_2,0])\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Time\")\n",
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  {
   "cell_type": "code",
   "execution_count": 62,
   "id": "833e160c-0fef-41b3-a96a-68d93e54099e",
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    {
     "ename": "NameError",
     "evalue": "name 'u_FS' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-62-593241ac4df9>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmax\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu_FS\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0ms_FE_2\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mu_FE_2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmax\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu_FS\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0ms_FE_2\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m: name 'u_FS' is not defined"
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   "source": [
    "np.max(np.abs(u_FS[0:s_FE_2+1] - u_FE_2))/np.max(np.abs(u_FS[0:s_FE_2+1]))"
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  },
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   "execution_count": 63,
   "id": "bc765036-d2c2-40a3-8714-f9f3b8cdb6f2",
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   "source": [
    "dt_RK4_2 = 0.001\n",
    "T_RK4_2 = 14.4072\n",
    "s_RK4_2 = int(np.ceil(T_RK4_2/dt_RK4_2))\n",
    "u_RK4_2 = np.zeros([s_RK4_2,N])\n",
    "D_RK4_2 = 0.01\n",
    "u_RK4_2[0]=f_arr"
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  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "07091f1b-2379-407f-a7ff-dda980d8a662",
   "metadata": {
    "tags": []
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   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|██████████| 14407/14407 [01:01<00:00, 234.67it/s]\n"
     ]
    }
   ],
   "source": [
    "for t in tqdm(range(0,s_RK4_2-1)):\n",
    "     f1 = deriv2(u_RK4_2[t],D_RK4_2)\n",
    "     f2 = deriv2(u_RK4_2[t] + (dt_RK4_2/2)*f1,D_RK4_2)\n",
    "     f3 = deriv2(u_RK4_2[t] + (dt_RK4_2/2)*f2,D_RK4_2)\n",
    "     f4 = deriv2(u_RK4_2[t] + dt_RK4_2*f3,D_RK4_2)\n",
    "     u_RK4_2[t+1] = u_RK4_2[t] + (dt_RK4_2/6)*(f1 + 2*f2 + 2*f3 + f4)"
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  {
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   "id": "7f876850-6d2a-403f-88dc-0a03775dd135",
   "metadata": {},
   "source": [
    "Added Numba compiler to deriv2() function to maybe speed up the matrix multiplication. This seems to make the for-loop above run a bit faster (~5 minutes). Commented out print statements because Python print statements slow things down a ton. Added tqdm progress bar."
   ]
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   "cell_type": "code",
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   "id": "9f13cb35-7dbc-467a-9f59-60511e88dc73",
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   "source": [
    "plt.imshow(u_RK4_2,aspect=\"0.5\",extent=[0,L,T_RK4,0])\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Time\")\n",
    "plt.title(\"u(x) Over Time\")\n",
    "plt.colorbar()\n",
    "plt.show()"
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  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ec92f8e4-7990-456e-9c80-68fee998eb72",
   "metadata": {},
   "outputs": [],
   "source": [
    "np.max(np.abs(u_FS - u_RK4_2))/np.max(np.abs(u_FS[0:s_RK4_2+1]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "ec8fe616-9a54-404a-be79-f4ac63ccc5fd",
   "metadata": {},
   "outputs": [],
   "source": [
    "dt_RK_2 = 0.001\n",
    "T_RK_2 = 14.4072\n",
    "s_RK_2 = int(np.ceil(T_RK_2/dt_RK_2))\n",
    "u_RK_2 = np.zeros([s_RK_2,N])\n",
    "D_RK_2 = 0.01\n",
    "# Need 4 initial values now\n",
    "u_RK_2[0]=f_arr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "d678a089-63da-4d16-8075-badb124039c6",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|██████████| 14407/14407 [00:30<00:00, 479.53it/s]\n"
     ]
    }
   ],
   "source": [
    "for t in tqdm(range(0,s_RK_2-1)):\n",
    "    u_RK_2[t+1] = u_RK_2[t]+dt_RK_2*deriv2( (u_RK_2[t]+(dt_RK_2/2)*deriv2(u_RK_2[t],D_RK_2)), D_RK_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8d4339ae-96a5-48ed-941f-0d325aebea45",
   "metadata": {},
   "outputs": [],
   "source": [
    "np.max(np.abs(u_FS - u_RK_2))/np.max(np.abs(u_FS))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a549f14b-35e7-4f4f-8dfe-1cb1ae629252",
   "metadata": {},
   "source": [
    "### Attempt 5: scipy.integrate.solve_ivp\n",
    "\n",
    "Use an actual ODE library. If we don't specify method, I'm pretty sure it uses RK4. We should probably expect the best possible accuracy, given a differentiation method and spatiotemporal resolution from this integrator."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "id": "cde7c717-3834-4692-a1d0-5df3cf25b158",
   "metadata": {},
   "outputs": [],
   "source": [
    "D_ivp = D_RK4\n",
    "def deriv3(t,u):\n",
    "    return D_ivp*np.real(np.fft.ifft(-(k_arr**2)*np.fft.fft(u)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "id": "0854f4a8-8e53-4610-89a7-6da98aff79d2",
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.integrate import solve_ivp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "id": "828fba43-daa3-404c-be87-c6295d22ea3e",
   "metadata": {},
   "outputs": [],
   "source": [
    "sol = solve_ivp(deriv3, [0, T_RK4], f_arr,\n",
    "                t_eval=t_arr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "id": "eee45ca2-733b-45db-939c-f66546a79ead",
   "metadata": {},
   "outputs": [
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       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 576x432 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(sol.y.T,aspect=\"0.1\",extent=[0,L,T_RK4,0])\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Time\")\n",
    "plt.title(\"u(x) Over Time\")\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "id": "ce03c736-d0bf-4be3-8a0e-5f10f3522fb4",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "    <dc:date>2022-08-26T18:37:15.808519</dc:date>\n",
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   "source": [
    "plt.imshow(np.abs(sol.y.T-u_ANLY)/np.max(np.abs(u_ANLY)),aspect=\"0.1\",extent=[0,L,T_RK4,0])\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Time\")\n",
    "plt.title(\"u(x) Over Time\")\n",
    "plt.colorbar()\n",
    "plt.hlines(8,0,L,\"w\")\n",
    "plt.show()"
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   "cell_type": "code",
   "execution_count": null,
   "id": "86b69d06-745c-4677-ba5c-9407d2e546b4",
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   "outputs": [],
   "source": [
    "sol.y.T.shape"
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   "id": "028a812c-8bed-4e92-9b46-5af2c2f4b7a0",
   "metadata": {},
   "source": [
    "Seems that we need to make changes to the grid or the differentiator if we are going to get better than ~2.5% accuracy."
   ]
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   "execution_count": null,
   "id": "f9ac6afb-30b6-47bd-8331-28d54ccdbd43",
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    "np.max(np.abs(sol.y.T[0:8000]-u_ANLY[0:8000]))/np.max(np.abs(u_ANLY[0:8000]))"
   ]
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   "source": [
    "plt.plot(t_arr,[-1+(sol.y.T[j].sum())*dx for j in range(len(sol.y.T))])\n",
    "plt.show()"
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   "id": "17c01e7c-5f45-4238-8ceb-ad66d831c367",
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   "source": [
    "plt.plot([dt_RK4*t for t in range(8000)],np.array([np.max(np.abs(sol.y.T[t]-u_ANLY[t]))/np.max(np.abs(u_ANLY[t])) for t in range(8000)]))\n",
    "plt.show()\n",
    "\n",
    "plt.loglog([dt_RK4*t for t in range(8000)],np.array([np.max(np.abs(sol.y.T[t]-u_ANLY[t]))/np.max(np.abs(u_ANLY[t])) for t in range(8000)]))\n",
    "plt.show()\n",
    "\n",
    "plt.plot([t for t in range(10)],np.array([np.max(np.abs(sol.y.T[t]-u_ANLY[t]))/np.max(np.abs(u_ANLY[t])) for t in range(10)]))\n",
    "plt.yscale(\"log\")\n",
    "plt.show()"
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  {
   "cell_type": "markdown",
   "id": "6e21753e-a7bf-473f-a567-5fae32a18ba9",
   "metadata": {},
   "source": [
    "Let's try to implement $4^\\text{th}$ order Runge-Kutta while enforcing the BCs in the analytic solution above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "eed7a019-b02f-4087-8550-5911b009b952",
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   "source": [
    "dt_RK4_BC = 0.001\n",
    "T_RK4_BC = 100\n",
    "s_RK4_BC = int(np.ceil(T_RK4_BC/dt_RK4_BC))\n",
    "u_RK4_BC = np.zeros([s_RK4_BC,N])\n",
    "D_RK4_BC = 0.01\n",
    "# Need 4 initial values now\n",
    "u_RK4_BC[0]=f_arr"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69df4f16-26eb-4833-b106-3353b502d3c1",
   "metadata": {},
   "source": [
    "We'll just do this super naively and set u_RK4_BC = 0 on the boundary at every timestep."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6cd64183-b612-4c27-87f2-14574f1c1c5b",
   "metadata": {},
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   "source": [
    "from tqdm import tqdm\n",
    "for t in tqdm(range(0,s_RK4_BC-1)):\n",
    "    #print(\"\\r\",end=\"\")\n",
    "    #print(f\"Integrating PDE: {int(100*t/s_RK4_2)+1}%  \",end=\"\",flush=True)\n",
    "    f1 = deriv2(u_RK4_BC[t],D_RK4_BC)\n",
    "    f2 = deriv2(u_RK4_BC[t] + (dt_RK4_BC/2)*f1,D_RK4_BC)\n",
    "    f3 = deriv2(u_RK4_BC[t] + (dt_RK4_BC/2)*f2,D_RK4_BC)\n",
    "    f4 = deriv2(u_RK4_BC[t] + dt_RK4_BC*f3,D_RK4_BC)\n",
    "    u_RK4_BC[t+1] = u_RK4_BC[t] + (dt_RK4_BC/6)*(f1 + 2*f2 + 2*f3 + f4)\n",
    "    u_RK4_BC[t+1, 0] = 0.0\n",
    "    u_RK4_BC[t+1,-1] = 0.0"
   ]
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   "execution_count": null,
   "id": "b77679e5-8cb9-4af3-94a8-282d4d889676",
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.real(u_RK4_BC),aspect=\"0.1\",extent=[-L/2,L/2,T_RK,0])\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Time\")\n",
    "plt.title(\"u(x) Over Time\")\n",
    "plt.colorbar()\n",
    "plt.show()"
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   "execution_count": null,
   "id": "aafbe233-43d6-4476-a4b8-b4eaee4dac3b",
   "metadata": {},
   "outputs": [],
   "source": [
    "np.max(np.abs(u_ANLY - u_RK4_BC))/np.max(np.abs(u_ANLY))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4a92aee4-bb88-419b-b381-4730851e656b",
   "metadata": {},
   "outputs": [],
   "source": [
    "np.max(np.abs(sol.y.T - u_RK4_BC))/np.max(np.abs(sol.y.T))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1f056201-e612-4db8-b2ec-c470961ef0fb",
   "metadata": {},
   "source": [
    "Whelp, thats not very good."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4585c779-988b-4dff-abe6-a6ce22149fa8",
   "metadata": {},
   "source": [
    "## Free-space solution to the diffusion equation.\n",
    "\n",
    "Following Tong Kinetic theory page 6, I claim that the analytical solution for $$u(x,0) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}}e^{-\\frac{(x-L/2)^2}{2 \\sigma^2}}$$\n",
    "\n",
    "is given by $$u(x,t) = \\frac{1}{\\sqrt{2 \\pi (2 D t + \\sigma^2)}}e^{-\\frac{(x-L/2)^2}{2 (2 D t + \\sigma^2)}}.$$\n",
    "\n",
    "Tong gives the solution for a delta-function initial condition which immediately evolves into a gaussian. I just redefine $t=0$ to be when Tong's gaussian looks like our initial condition.\n",
    "\n",
    "I also claim that we should stop the time evolution when the free-space solution exceeds machine-epsilon at the boundaries. This can be found in mathematica: for $D = 0.01, \\sigma = 0.25, L = 10$, we have $$t_{final} = 14.4072.$$"
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   "execution_count": 73,
   "id": "bca57f74-5ec6-4e8b-b282-dcae45f84f64",
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   "outputs": [],
   "source": [
    "u_FS = np.array([(2*np.pi*(2*D_RK4*t + sig**2))**(-0.5)*np.exp(-((x_arr-L/2)**2.0)/(2*(2*D_RK4*t + sig**2))) for t in t_arr])"
   ]
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   "id": "0b6c68e9-ec35-4cae-b132-4e91f72ea454",
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   "source": [
    "plt.plot(t_arr,[-1+(u_FS[j].sum())*dx for j in range(len(t_arr))])\n",
    "plt.ylim([-10**(-13),10**(-13)])\n",
    "plt.vlines(14.4,-0.01,0.001,\"k\")\n",
    "plt.show()"
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   "cell_type": "code",
   "execution_count": 75,
   "id": "35473958-9a89-4fff-b48f-0223cc22973a",
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      ],
      "text/plain": [
       "<Figure size 576x432 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(u_FS,aspect=\"0.1\",extent=[0,L,T_RK4,0])\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Time\")\n",
    "plt.title(\"u(x) Over Time\")\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "id": "6c8ff683-0059-40f3-b7c1-d4ef0143d6e2",
   "metadata": {},
   "outputs": [
    {
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    "plt.imshow(np.abs(u_FS-u_ANLY)/np.max(np.abs(u_FS)),aspect=\"0.5\",extent=[0,L,T_RK4,0])\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Time\")\n",
    "plt.title(\"Diff between PBC and Free-Space Sols\")\n",
    "plt.colorbar()\n",
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   "source": [
    "plt.loglog(t_arr,np.array([np.max(np.abs(u_FE[t]-u_FS[t]))/np.max(np.abs(u_FS[t])) for t in range(s_RK4_2)]),\"y\",label=\"Forward Euler\")\n",
    "plt.loglog(t_arr,np.array([np.max(np.abs(sol.y.T[t]-u_FS[t]))/np.max(np.abs(u_FS[t])) for t in range(s_RK4_2)]),\"k\",label=\"scipy\")\n",
    "plt.loglog(t_arr,np.array([np.max(np.abs(u_RK4[t]-u_FS[t]))/np.max(np.abs(u_FS[t])) for t in range(s_RK4_2)]),\"b\",label=\"Runge-Kutta 4\")\n",
    "plt.loglog(t_arr,np.array([np.max(np.abs(u_FE_2[t]-u_FS[t]))/np.max(np.abs(u_FS[t])) for t in range(s_RK4_2)]),\"r\",label=\"Forward Euler (matrix)\")\n",
    "plt.loglog(t_arr,np.array([np.max(np.abs(u_RK_2[t]-u_FS[t]))/np.max(np.abs(u_FS[t])) for t in range(s_RK4_2)]),\"c\",label=\"Runge-Kutta 2 (matrix)\")\n",
    "plt.loglog(t_arr,np.array([np.max(np.abs(u_RK4_2[t]-u_FS[t]))/np.max(np.abs(u_FS[t])) for t in range(s_RK4_2)]),\"g\",label=\"Runge-Kutta 4 (matrix)\")\n",
    "\n",
    "plt.ylabel(\"Error\")\n",
    "plt.xlabel(\"Time\")\n",
    "plt.title(\"Errors of Integration Schemes vs. Time\")\n",
    "plt.legend(loc=\"upper left\")\n",
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   "source": [
    "$\\textbf{Double check this!!!!!!!}$. "
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   "id": "ac9c88fb-9a5e-4fda-9f87-b57d621c19c9",
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    "np.max(np.abs(u_FS[0:14408] - u_RK4[0:14408]))/np.max(u_FS[0:14408])"
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   "cell_type": "code",
   "execution_count": 62,
   "id": "ac764fe8-7b0e-45b0-8cc1-c58be76eef59",
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   "outputs": [
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     "execution_count": 62,
     "metadata": {},
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    "np.max(np.abs(u_FS[0:14408] - u_RK4_2[0:14408]))/np.max(u_FS[0:14408])"
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   "execution_count": 183,
   "id": "b46e29d7-ac11-4b6f-ac0b-ac269c5a0b58",
   "metadata": {},
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       "0.024839727251512962"
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     "execution_count": 183,
     "metadata": {},
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   "source": [
    "np.max(np.abs(u_FS[0:14408] - u_FE[0:14408]))/np.max(u_FS[0:14408])"
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   "cell_type": "code",
   "execution_count": 184,
   "id": "29e721ea-909e-4e83-85dc-7f9e196562d0",
   "metadata": {},
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    {
     "data": {
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       "0.023647744265167647"
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     "execution_count": 184,
     "metadata": {},
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    "np.max(np.abs(u_FS[0:14408] - sol.y.T[0:14408]))/np.max(u_FS[0:14408])"
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