{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## BVPs, BCs, nonlinear, nonuniform grids\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Boundary conditions\n", "* Dirichlet conditions are straightforward\n", " * Unknowns are interior points. Points near the boundaries are written in terms of the boundaries.\n", "* Neumann and Robin conditions:\n", " * $y^{\\prime} = \\alpha$ is given at the boundary (or $y^{\\prime} + \\beta y = \\alpha$ for Robin).\n", " * Interior cells need $y$ on the boundary, not $y^{\\prime}$.\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Two Approaches\n", "**Approach 1. Ghost Cell Method** Include the boundary point in the list of unknowns (like interior points).\n", "* Discretize the boundary point exactly as if it was an interior point.\n", "* For central differences, this will reference one point past the boundary.\n", " ``` \n", " \\\\|\n", " \\\\|\n", " \\\\|\n", " * * * *\n", " i-1 \\\\|i i+1 \n", " -1 \\\\|0 1 2\n", " \\\\|\n", " \\\\|\n", " ```\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "* Use the BC $y^{\\prime}=\\alpha$ as an equation for the new unknown $y_{-1}$.\n", " $$y_0^{\\prime} = \\alpha \\rightarrow \\frac{y_1-y_{-1}}{2\\Delta x} = \\alpha \\rightarrow$$\n", " \n", " $$y_{-1}=y_{1} - 2\\Delta x\\alpha.$$\n", " \n", "* Substitute this equation in for $y_{i-1} = y_{-1}$ appearing in the finite difference equation (FDE) at point $i=0$. The first two equations below are the FDE at $i=0$. The last equation has the substitution for $y_{-1}$:\n", " $$l_iy_{i-1} + a_iy_i + u_{i}y_{i+1} = F_i$$\n", " $$\\mbox{or,}$$\n", " $$l_0y_{-1} + a_0y_0 + u_0y_{1} = F_0, $$\n", " $$\\rightarrow a_0y_0 + (l_0+u_0)y_1 = F_0 + 2\\Delta x\\alpha l_0. $$\n", "* That is, the unknown $y_{-1}$ is in terms of $y_1$, so when we substitute it into the FDE at $i=0$, the coefficient of $y_1$ and the RHS are modified in the $i=0$ equation. \n", "* This is called the **Ghost Cell Method** since a false or *ghost* point arises, which is then handled with the boundary condition equation.\n", " \n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "\n", "* **Advantages**\n", " * Solves directly for the unknown boundary value.\n", " * Uses a uniform stencil (central difference everywhere, even at the boundaries.\n", "\n", "* **Disadvantage**\n", " * Higher order $y^{\\prime\\prime\\prime}$ or higher order FDA can lead to multiple outside points, which can be awkward.\n", " * Remedy this using one-sided differences." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "**Method 2** Don't include the boundary point in the list of unkowns. \n", "\n", " ``` \n", " \\\\|\n", " \\\\|\n", " \\\\|\n", " * * * * \n", " \\\\|i-1 i i+1\n", " \\\\|-1 0 1 2 \n", " \\\\|\n", " ```\n", "* The first unknown point is $i=0$.\n", "* The FDE at this point is in terms of point $i-1=-1$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "* Use the Boundary condition at $i-1$ with a one-sided difference to get another equation to write $y_{-1}$ in terms of the interior points we are solving for ($y_0$, $y_{1}$, etc.)\n", "* At the left side for the $i=0$ FDE (which has a $y_{-1}$ term):\n", " * Write a one sided difference for $y^{\\prime}_{-1}$.\n", " $$y^{\\prime}_{-1} = \\alpha = \\frac{-\\frac{3}{2}y_{-1} + 2y_0 - \\frac{1}{2}y_1}{\\Delta x}.$$\n", " * Solve for $y_{-1}$:\n", " \n", " $$ y_{-1} = \\frac{\\alpha\\Delta x - 2y_0 + \\frac{1}{2}y_{1}}{-3/2}.$$\n", " \n", " * The FDE at point $i=0$ is \n", " $$l_0y_{-1} + a_0y_0 + u_0y_1 = F_0.$$\n", " * Insert the above BC equation for $y_{-1}$ (blue) into this FDE to get the final FDE at the $i=0$ point.\n", " $$ (a_0 + \\frac{4}{3}l_0)y_0 + (u_0 - \\frac{1}{3}l_0)y_1 = F_0 + \\frac{2}{3}\\alpha l_0\\Delta x.$$\n", "* A similar procedure is done at the right side of the domain.\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "* **Advantage**\n", " * Extends easily to higher order\n", " * On the homework, if we exclude the BC point, then we don't have to divide by $r=0$ at the cylindrical centerline.\n", "* **Disadvantage**\n", " * Two different stencils are needed. One for the fully interior points $i=1,\\,2,\\ldots$, and one for the point $i=0$ next to the boundary. (Similarly for the right side of the domain.)\n", " * The boundary point remains unknown.\n", " * Once the solution to the interior points $i=0,\\,1,\\ldots$ is found, we can find $y_{-1}$, (the boundary point), using the above blue equation." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Nonlinear relaxation methods\n", "* The usual approach is to **iterate**\n", " * Use a linearized form of the equation.\n", " * Guess a solution.\n", " * Iterate to improve it.\n", "* Apply the FDA to $y^{\\prime\\prime}$, $y^{\\prime}$ as before.\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "\n", "* But consier a term like $y^\\prime y$\n", " - This results in nonlinear terms:\n", " $$y^{\\prime}y \\rightarrow \\left(\\frac{y_{i+1}-y_{i-1}}{2\\Delta x}\\right)\\cdot y_i.$$\n", "* We can linearize these terms by splitting the product and *lagging* part of it. For example\n", " $$y_i^2\\rightarrow y_i\\cdot y_i \\rightarrow y_i^{new}\\cdot y_i^{old},$$\n", " where $y_i^{old}$ is the value from the previous iteration, which is known.\n", "* For $$y^{\\prime\\prime} + P(x,y)y^{\\prime} + Q(x,y)y = F(x),$$\n", "use $$y^{\\prime\\prime} + P(x,y^{old})y^{\\prime} + Q(x,y^{old})y = F(x),$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "\n", "* Example\n", " $$\\nabla\\cdot\\vec{q} = F \\rightarrow -\\nabla\\cdot(k\\nabla T) = F.$$\n", " $$-\\frac{d}{dx}\\left(k\\frac{dT}{dx}\\right) = F.$$\n", " $$\\frac{dT}{dx}\\frac{dk}{dx} + k\\frac{d^2T}{dx^2} = -F.$$\n", " * $k = k(T)$ so the above equation is nonlinear. So lag $T$ when evaluating $k$. Use $k(T^{old})$.\n", " * This is now linear, but we have to iterate.\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "**Question** What are other approaches that could (or should?) be used?\n", "- Newton's method?\n", "- What about a Taylor Series linearization instead of the linearization shown?\n", " - Consider $yy$\n", " - above linearization: $yy\\approx y_0y$\n", " - Taylor series: $yy\\approx y_0^2 + 2y_0(y-y_0)$\n", " - Plot these for some arbitrary $y_0$" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "import Plots as plt" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y = LinRange(0,10,100)\n", "\n", "y0 = 3.\n", "y2 = y.^2\n", "y0y = y0 * y\n", "yta = y0^2 .+ 2*y0*(y.-y0)\n", "\n", "plt.resetfontsizes(); plt.scalefontsizes(1.5)\n", "plt.plot( y,y2, color=\"grey\", lw=5, label=\"exact\")\n", "plt.plot!(y,yta, color=\"black\", lw=2, label=\"Taylor\")\n", "plt.plot!(y,y0y, color=\"blue\", lw=2, linestyle=:dash)\n", "plt.plot!(xlabel=\"y\", ylabel=\"yy\", foreground_color_legend=nothing)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Nonuniform grids\n", "#### Method 1\n", "* Set an arbitrary grid with arbitrary $\\Delta x$ spacing between points.\n", "\n", "```\n", " * * * \n", "i-1 i i+1\n", "```\n", "\n", "* Let $\\Delta x_{i-1} = x_i - x_{i-1}$, and $\\Delta x_{i} = x_{i+1}-x_i$.\n", "* Then a central difference approximation is \n", "$$f^{\\prime}(x_i) = f_i^{\\prime} \\approx \\frac{f_{i+1}-f_{i-1}}{\\Delta x_{i-1} + \\Delta x_i}.$$\n", " * A Taylor Series gives second order when $\\Delta x_{i-1}=\\Delta x_i$, but only first order when $\\Delta x_{i-1}\\ne \\Delta x_i$.\n", " * These are second and first order **asymptotically** as $\\Delta x\\rightarrow 0$. \n", " * In practice, accuracy is not severely compromized if $\\Delta x_{i-1}$ is not too different from $\\Delta x_i$.\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "\n", "* For a second derivative, we have\n", "$$f^{\\prime\\prime}_i\\approx \\frac{\\frac{f_{i+1}-f_i}{\\Delta x_i}-\\frac{f_i-f_{i-1}}{\\Delta x_{i-1}}}{\\frac{\\Delta x_{i-1}+\\Delta x_i}{2}}.$$\n", " " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Method 2\n", "* Stretch the grid **analytically**\n", "* Let $x$ be the nonuniform grid and\n", "* Let $\\eta$ be a corresponding uniform grid.\n", "* For example\n", "$$\\eta = \\ln(x+1) \\rightarrow x = e^{\\eta} - 1.$$\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "\n", "* Then \n", " $$ \\frac{dy}{dx} = \\frac{dy}{d\\eta}\\frac{d\\eta}{dx} = \\frac{dy}{d\\eta}\\left(\\frac{1}{x+1}\\right).$$\n", " * This allows us to transform all the $dy/dx$ derivatives on the nonuniform grid to $dy/d\\eta$ derivatives on a uniform grid times some known factor $1/(x+1)$ at any given point. \n", " * So, if we are evaluating $dy/dx$ at point $i$, we would have \n", " $$y^{\\prime}_i = \\frac{y_{i+1}-y_{i-1}}{\\Delta \\eta}\\left(\\frac{1}{x_i+1}\\right).$$\n", "* This works great, and is easy to use. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Questions\n", "- What would you do to have an arbitrary grid spacing that is not defined by some known function?\n", "- Could you develop a second order approximation to a central derivative on a nonuniform grid?" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "celltoolbar": "Slideshow", "kernelspec": { "display_name": "Julia 1.8.3", "language": "julia", "name": "julia-1.8" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.8.3" }, "varInspector": { "cols": { "lenName": 16, "lenType": 16, "lenVar": 40 }, "kernels_config": { "python": { "delete_cmd_postfix": "", "delete_cmd_prefix": "del ", "library": "var_list.py", "varRefreshCmd": "print(var_dic_list())" }, "r": { "delete_cmd_postfix": ") ", "delete_cmd_prefix": "rm(", "library": "var_list.r", "varRefreshCmd": "cat(var_dic_list()) " } }, "types_to_exclude": [ "module", "function", "builtin_function_or_method", "instance", "_Feature" ], "window_display": false } }, "nbformat": 4, "nbformat_minor": 4 }