lectures/ellipticEquations.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "ad30012a",
   "metadata": {},
   "source": [
    "# Elliptic equations\n",
    "\n",
    "\n",
    "## Introduction\n",
    "\n",
    "Elliptic equations is as subclass of second-order partial differential equations. All second-order partial differential equations can be written on the form\n",
    "\n",
    "$$A \\frac{\\partial^2 f}{\\partial x^2} + B \\frac{\\partial^2 f}{\\partial x \\partial y} + C \\frac{\\partial^2 f}{\\partial y^2} + D \\frac{\\partial f}{\\partial x} +  E \\frac{\\partial f}{\\partial y} + F f + G = 0$$\n",
    "\n",
    "An <i>elliptic</i> second-order partial differential equation is an equation on the form above where $B^2-AC < 0$.\n",
    "\n",
    "Elliptic equations commonly arise in physics problems. Two typical elliptic equations are the Laplace equation's $\\nabla^2 f = 0$ and the Poisson equation's $\\nabla^2 f = f$. We see that for the Laplace equation we have $A= 1 = C$, while all the other pre-factors are zero, thus $B^2 - AC = 0 - 1 = -1 < 0$, as required to be an elliptic equation. For the Poisson equation we also have $F = -1$, however, this is not affecting the equation $B^2 - AC = -1$.\n",
    "\n",
    "In this section we will mostly investigate the Laplace equation's."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f0331fbd",
   "metadata": {},
   "source": [
    "## Laplace equation\n",
    "\n",
    "A <i>Laplace equation</i> is a PDE of the form\n",
    "\n",
    "$$\\nabla^2 f = \\frac{\\partial^2 f}{\\partial x^2} + \\frac{\\partial^2 f}{\\partial y^2} = 0$$\n",
    "\n",
    "Here $\\Delta = \\nabla^2 = \\nabla \\cdot \\nabla$ is the Laplace operator, with $\\nabla \\cdot$ being the divergence operator, and $\\nabla$ being the gradient. Thus the Laplace operator gives the divergence of the gradient of the field $f$. Remember that\n",
    "\n",
    "$$\\nabla = \\left( \\frac{\\partial}{\\partial x}, \\frac{\\partial}{\\partial y} \\right)$$\n",
    "\n",
    "with the obvious extension to higher dimensions.\n",
    "\n",
    "This type of equations where first investigate by the French mathematician Pierre-Simon Laplace, thereby its name. Steady state heat flux, steady state fluid flow (at the continuum scale in a porous medium) and steady state diffusion are some of the problems that are described by such an equation.\n",
    "\n",
    "A function that satisfy the Laplace equation, i.e., a function $f$ such that $\\nabla^2 f = 0$, is called a <i>potential function</i>.\n",
    "\n",
    "Any flux that is proportional to a potential and at steady-state is described by the Laplace equation. As an example of such a system, we will below investigate the fluid flow in a porous medium considered at the continuum scale.\n",
    "\n",
    "As a side-note: Fluid flow within the pores in a porous medium is described by the Navier-Stokes equation, where you do not have a proportional relation between potential (pressure) and flux (fluid velocity), thus, its solution is not given by a solution to the Laplace equation. In fact, the solution is so complicated that is is one of the millennium problems in mathematics to show that a smooth solution always exist."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9e003c6e",
   "metadata": {},
   "source": [
    "### Fluid flow\n",
    "\n",
    "The fluid flow in a porous medium is described by Darcy's equation:\n",
    "\n",
    "$$\\vec{q} = -\\frac{k}{\\mu} \\nabla p$$\n",
    "\n",
    "Here $q$ is the volumetric flow rate, $k$ is the permeability (a measure for how well the porous medium allows for transport of fluids), $\\mu$ is the viscosity of the fluid, and $p$ is the fluid pressure.\n",
    "\n",
    "Assume a volume element $V$ with surface area $A$. The flow rate over the surface $A$ is given by\n",
    "\n",
    "$$\\int_A \\vec{q} \\cdot \\vec{n} dA$$\n",
    "\n",
    "From Gauss divergence theorem, we have that\n",
    "\n",
    "$$\\int_A \\vec{q} \\cdot \\vec{n} dA = \\int_V \\nabla \\cdot \\vec{q} dV$$\n",
    "\n",
    "If we are at steady state, due to mass conservation, we cannot have build-up (or draw-down) of mass. Thus, when assuming constant density of the fluid, the total flow rate over the surface of any volume element $V$ must be zero. Thus\n",
    "\n",
    "$$\\int_V \\nabla \\cdot \\vec{q} dV = 0$$\n",
    "\n",
    "Since this holds for any volume element, we then have $\\nabla \\cdot q = 0$. Thus we see that a steady fluid flow is described by\n",
    "\n",
    "$$\\begin{align}\n",
    "\\nabla \\cdot q &= 0 \\\\\n",
    "\\nabla \\cdot \\left(\\frac{k}{\\mu} \\nabla p \\right) &= 0 \\\\\n",
    "\\nabla \\cdot  \\nabla p  &= 0 \\\\\n",
    "\\nabla^2 p  &= 0 \n",
    "\\end{align}$$\n",
    "\n",
    "Thus, a steady fluid flow in a porous medium is given by the Laplace equation.\n",
    "\n",
    "Here we have assumed a constant permeability $k$ and viscosity $\\mu$. Note that, in general, the viscosity is dependent on pressure. Further, for any natural porous medium the permeability is varying."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9b7ea3bf",
   "metadata": {},
   "source": [
    "## Finite difference approximation\n",
    "\n",
    "We will now investigate numerical solutions to the Laplace equation. For this we will use finite differences.\n",
    "\n",
    "The centered difference for the second order derivative is given as\n",
    "\n",
    "$$\\frac{\\partial^2 f}{\\partial x^2} \\simeq \\frac{f(x+\\Delta x) - 2 f(x) +f(x-\\Delta x)}{(\\Delta x)^2}$$\n",
    "\n",
    "Assume a 1D problem, of length $L$. We will discretize this using $n+1$ points, starting counting at zero, with the first point (index $i=0$) being the left boundary and the last point (index $i = n$) begin the right boundary. We will fix the boundary conditions, with $f(0) = f_0$ and $f(L) = f_n$ being constant values. The length between the grid vertices are $\\Delta x = L/n$. From our finite difference we then get the following equation for each internal vertex $i$:\n",
    "\n",
    "$$f_{i+1} - 2f_i + f_{i-1} \\simeq (\\Delta x)^2 f''(x_i) = 0$$\n",
    "\n",
    "where the last equality to zero is due to $f$ begin a solution to the Laplace equation.\n",
    "\n",
    "At the left boundary we have\n",
    "\n",
    "$$\\begin{align}\n",
    "f_2 - 2f_1 + f_0 &= 0 \\\\\n",
    "f_2 - 2f_1  &= -f_0 \n",
    "\\end{align}$$\n",
    "\n",
    "At the right boundary we have \n",
    "$$\\begin{align}\n",
    "f_n - 2f_{n-1} + f_{n-2} &= 0 \\\\\n",
    "-2f_{n-1} + f_{n-2}  &= -f_n \n",
    "\\end{align}$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d6bcdee8",
   "metadata": {},
   "source": [
    "### Solution by matrix inversion\n",
    "\n",
    "The set of equations above can be written on matrix form as $\\mathbf{A} \\vec{x} = \\vec{b}$, where $\\mathbf{A}$ is the $n-1 \\times n-1$ matrix\n",
    "\n",
    "$$\\mathbf{A} = \n",
    "\\begin{bmatrix}\n",
    "-2&1&0&0&\\cdots&0&0&0 \\\\\n",
    "1&-2&1&0&\\cdots&0&0&0 \\\\\n",
    "0&1&-2&1&\\cdots&0&0&0 \\\\\n",
    "\\vdots&\\vdots&\\vdots&\\vdots& & \\vdots&\\vdots&\\vdots \\\\\n",
    "0&0&0&0&\\cdots&1&-2&1 \\\\\n",
    "0&0&0&0&\\cdots&0&1&-2 \n",
    "\\end{bmatrix}$$\n",
    "\n",
    "The unknowns are given by the vector\n",
    "$$\\vec{x} = \n",
    "\\begin{bmatrix}\n",
    "f_1\\\\\n",
    "f_2\\\\\n",
    "\\vdots \\\\\n",
    "f_{n-1}\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "Finally, the boundary conditions are collected in the vector\n",
    "$$\\vec{x} = \n",
    "\\begin{bmatrix}\n",
    "-f_0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\vdots \\\\\n",
    "0\\\\\n",
    "-f_{n}\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "Such a system can be solved in Python as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "b84f43b0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-2.  1.  0.  0.  0.  0.  0.  0.  0.]\n",
      " [ 1. -2.  1.  0.  0.  0.  0.  0.  0.]\n",
      " [ 0.  1. -2.  1.  0.  0.  0.  0.  0.]\n",
      " [ 0.  0.  1. -2.  1.  0.  0.  0.  0.]\n",
      " [ 0.  0.  0.  1. -2.  1.  0.  0.  0.]\n",
      " [ 0.  0.  0.  0.  1. -2.  1.  0.  0.]\n",
      " [ 0.  0.  0.  0.  0.  1. -2.  1.  0.]\n",
      " [ 0.  0.  0.  0.  0.  0.  1. -2.  1.]\n",
      " [ 0.  0.  0.  0.  0.  0.  0.  1. -2.]]\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "#Creating a class for our problem\n",
    "class Celliptic:\n",
    "    def __init__(self):\n",
    "        #Length\n",
    "        self.fL=10.0\n",
    "        #Number of grid cells\n",
    "        self.iN=11\n",
    "        #Left boundary condition\n",
    "        self.fBl=1.0\n",
    "        #Right boundary condition\n",
    "        self.fBr=0.0\n",
    "\n",
    "        #Length vector\n",
    "        self.afLv= np.linspace(0.0,self.fL,self.iN)\n",
    "        \n",
    "    def matrixSolution(self):\n",
    "        #Bondary conditions vector\n",
    "        afBc=np.zeros(self.iN-2)\n",
    "        afBc[0]=-self.fBl\n",
    "        afBc[-1]=-self.fBr\n",
    "\n",
    "        #Matrix\n",
    "        aafA=np.zeros((self.iN-2,self.iN-2))\n",
    "        for ii in range(1,self.iN-3):\n",
    "            aafA[ii,ii-1]=1\n",
    "            aafA[ii,ii]=-2\n",
    "            aafA[ii,ii+1]=1\n",
    "        aafA[0,0]=-2\n",
    "        aafA[0,1]=1\n",
    "        aafA[-1,-1]=-2\n",
    "        aafA[-1,-2]=1\n",
    "\n",
    "        print(aafA)\n",
    "\n",
    "        #Solve system using numpy\n",
    "        afx=np.dot(np.linalg.inv(aafA),afBc)\n",
    "        \n",
    "        return afx\n",
    "    \n",
    "    def plotMatrixSolution(self):\n",
    "        afx=self.matrixSolution()\n",
    "\n",
    "        #Plot solution values\n",
    "        fig = plt.figure()\n",
    "        plt.plot(self.afLv[1:-1],afx)\n",
    "        plt.show()\n",
    "\n",
    "tOneDim=Celliptic()\n",
    "tOneDim.plotMatrixSolution()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "92b5a2ae",
   "metadata": {},
   "source": [
    "The solution for such a one-dimensional system is quite simple, however, whenever you want a solution in a complex geometry you will also get non-trivial and interesting solutions. You will also observe that creating the matrix $A$ can become quite complicated.\n",
    "\n",
    "Matrix inversion is an operation that is needed in a lot of numerical problems, and can be computational heavy. It is often the most computational demanding part of a numerical code. Often we only approximate the inverse by an iterative method, such as the conjugate gradient method for matrix inversion. We will return to the problem of matrix inversion later in these notes.\n",
    "\n",
    "\n",
    "### Relaxation methods\n",
    "\n",
    "From our finite difference approximation, we have the following equation\n",
    "\n",
    "$$f_{i+1} - 2f_i + f_{i-1} = 0$$\n",
    "\n",
    "which can be rearranged as follows:\n",
    "\n",
    "\n",
    "$$f_i = \\frac{f_{i+1} + f_{i-1}}{2}$$\n",
    "\n",
    "We now want to obtain a solution by iteratively trying to converge towards the right solution by using the equation above. \n",
    "\n",
    "#### Jacobi relaxation\n",
    "\n",
    "We start by a relaxation method called <i>Jacobi relaxation</i>. Let $f_i^m$ be the solution at iteration $m$, then we calculate the value in the next iteration as\n",
    "\n",
    "$$f_i^{m+1} = \\frac{f_{i+1}^m + f_{i-1}^m}{2}$$\n",
    "\n",
    "This method is named after the German mathematician Carl Gustav Jacob Jacobi. The method can be implemented in Python as follows:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "d64856b8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def jacobiRelaxation(self,iIterations):\n",
    "    #Solution vector\n",
    "    afJR=np.zeros(self.iN)\n",
    "    #Set boundary conditions\n",
    "    afJR[0]=self.fBl\n",
    "    afJR[-1]=self.fBr\n",
    "    afJRnew=np.copy(afJR)\n",
    "\n",
    "    #Number of iterations\n",
    "    itn=iIterations\n",
    "\n",
    "    for it in range(0,itn):\n",
    "        for ii in range(1,self.iN-1):\n",
    "            afJRnew[ii]=(afJR[ii-1]+afJR[ii+1])/2\n",
    "            #Alternative (faster) way by slicing:\n",
    "            #    fvnew[1:N-1]=0.5*(fv[:N-2]+fv[2:])\n",
    "        afJR=np.copy(afJRnew)\n",
    "    return afJR\n",
    "\n",
    "def plotJacobiRelaxation(self,iIterations):\n",
    "    afSol=self.jacobiRelaxation(iIterations)\n",
    "    #Plot solution values\n",
    "    fig = plt.figure()\n",
    "    plt.plot(self.afLv,afSol)\n",
    "    plt.show()\n",
    "\n",
    "#Adding Jacobi relaxation function to class\n",
    "Celliptic.jacobiRelaxation=jacobiRelaxation\n",
    "Celliptic.plotJacobiRelaxation=plotJacobiRelaxation\n",
    "\n",
    "\n",
    "tOneDim.plotJacobiRelaxation(5)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2c3a882b",
   "metadata": {},
   "source": [
    "The Jacobi relaxation needs two copies of each grid cell in its memory, as we cannot overwrite $f^m$ before we are finished calculating $f^{m+1}$. The next method is almost the same as the Jacobi relaxation, however, it continuously overwrites values.\n",
    "\n",
    "#### Gauss-Seidel relaxation\n",
    "\n",
    "The next method, called <i>Gauss-Seidel relaxation</i>, continuously use the current values in a single copy of the grid.\n",
    "\n",
    "It has been shown that one obtain a faster convergence by using the current best value when updating the cell values. Gauss-Seidel converges twice as fast as Jacobi [1]. Thus, we calculate\n",
    "\n",
    "$$f_i = \\frac{f_{i+1} + f_{i-1}}{2}$$\n",
    "\n",
    "with whatever value we currently have for $f_{i+1}$ and $f_{i-1}$. This can be done in Python as follows:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "c509768f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def gaussSeidel(self,iIterations):\n",
    "    #Solution vector\n",
    "    afGS=np.zeros(self.iN)\n",
    "    #Set boundary conditions\n",
    "    afGS[0]=self.fBl\n",
    "    afGS[-1]=self.fBr\n",
    "\n",
    "    #Number of iterations\n",
    "    itn=iIterations\n",
    "\n",
    "    for it in range(0,itn):\n",
    "        for ii in range(1,self.iN-1):\n",
    "            afGS[ii]=(afGS[ii-1]+afGS[ii+1])/2\n",
    "    \n",
    "    return afGS\n",
    "\n",
    "def plotGaussSeidel(self,iIterations):\n",
    "    afSol=self.gaussSeidel(iIterations)\n",
    "    #Plot solution values\n",
    "    fig = plt.figure()\n",
    "    plt.plot(self.afLv,afSol)\n",
    "    plt.show()\n",
    "\n",
    "#Adding the Gauss-Seidel function to our class\n",
    "Celliptic.gaussSeidel=gaussSeidel\n",
    "Celliptic.plotGaussSeidel=plotGaussSeidel\n",
    "\n",
    "\n",
    "tOneDim.plotGaussSeidel(5)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "578545c8",
   "metadata": {},
   "source": [
    "Let us now compare the efficiency of the Jacobi iteration versus the Gauss-Seidel."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "eea98215",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "afJacobi=tOneDim.jacobiRelaxation(5)\n",
    "afGaussSeidel=tOneDim.gaussSeidel(5)\n",
    "\n",
    "\n",
    "#Plot solution values\n",
    "fig = plt.figure()\n",
    "plt.plot(tOneDim.afLv,afJacobi,label='Jacobi')\n",
    "plt.plot(tOneDim.afLv,afGaussSeidel,label='Gauss-Seidel')\n",
    "plt.legend(loc='best')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "29ac3f0d",
   "metadata": {},
   "source": [
    "#### Over-relaxation\n",
    "\n",
    "Instead of using the average of the surrounding grid-cells, another method is to use a weighted sum of the current value and the average of the surrounding values. \n",
    "\n",
    "$$f_i^{m+1} = (1-\\alpha) f_i^m + \\alpha\\frac{f_{i+1}^m + f_{i-1}^m}{2}$$\n",
    "\n",
    "We call $\\alpha$ the relaxation factor. Thus, we see that if $\\alpha = 1$, then this gives the Jacobi relaxation.\n",
    "\n",
    "Using the current values, we also have the Gauss-Seidel equivalent \n",
    "\n",
    "$$f_i^n = (1-\\alpha) f_i^c + \\alpha\\frac{f_{i+1} + f_{i-1}}{2}$$\n",
    "\n",
    "Here $f_i^c$ is the current value in cell $i$, while $f_i^n$ is the updated <i>next</i> value. Again, we see that a relaxation factor of $\\alpha=1$ yield the Gauss-Seidel method.\n",
    "\n",
    "If $\\alpha < 1$ then we expect a slower convergence than for the traditional $\\alpha=1$ methods. However, it turns out that for $\\alpha>1$ we obtain a faster convergence. Using $\\alpha>1$ is called over-relaxation. Note that the method only converge for $\\alpha<2$.\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "d46a9b68",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def overRelaxation(self,iIterations,fAlpha):\n",
    "    #Solution vector\n",
    "    afOR=np.zeros(self.iN)\n",
    "    #Set boundary conditions\n",
    "    afOR[0]=self.fBl\n",
    "    afOR[-1]=self.fBr\n",
    "\n",
    "    #Number of iterations\n",
    "    itn=iIterations\n",
    "\n",
    "    for it in range(0,itn):\n",
    "        for ii in range(1,self.iN-1):\n",
    "            afOR[ii]=(1-fAlpha)*afOR[ii]+fAlpha*(afOR[ii-1]+afOR[ii+1])/2\n",
    "    \n",
    "    return afOR\n",
    "\n",
    "#Adding the over relaxation function to our class\n",
    "Celliptic.overRelaxation=overRelaxation\n",
    "\n",
    "\n",
    "afJacobi=tOneDim.jacobiRelaxation(5)\n",
    "afGaussSeidel=tOneDim.gaussSeidel(5)\n",
    "afOverRelax=tOneDim.overRelaxation(5,1.5)\n",
    "\n",
    "#Plot solution values\n",
    "fig = plt.figure()\n",
    "plt.plot(tOneDim.afLv,afJacobi,label='Jacobi')\n",
    "plt.plot(tOneDim.afLv,afGaussSeidel,label='Gauss-Seidel')\n",
    "plt.plot(tOneDim.afLv,afOverRelax,label='Over-relaxation',linestyle='dashed')\n",
    "plt.legend(loc='best')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bf168298",
   "metadata": {},
   "source": [
    "## References\n",
    "\n",
    "[1] PRESS, W. H., FLANNERY, B. P., TEUKOLSKY, S. A., and VETTERLING, W. T. 1989. Numerical Recipes in Pascal: The Art of Scientific Computing. Cambridge University Press, Cambridge, MA.\n",
    "\n",
    "[2] Hansen, Per Brinch. \"Numerical solution of Laplace's equation.\" (1992)."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}