$$ \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\Aex}{{A_{\small\mbox{e}}}} \newcommand{\Oof}[1]{\mathcal{O}(#1)} \newcommand{\dfc}{\alpha} % diffusion coefficient \newcommand{\dx}{\, \mathrm{d}x} \newcommand{\ds}{\, \mathrm{d}s} $$

Finite difference methods for diffusion processes

Hans Petter Langtangen [1, 2]

[1] Center for Biomedical Computing, Simula Research Laboratory
[2] Department of Informatics, University of Oslo

Oct 17, 2015


Note: PRELIMINARY VERSION

Table of contents

The 1D diffusion equation
      The initial-boundary value problem for 1D diffusion
      Forward Euler scheme
      Backward Euler scheme
      Sparse matrix implementation
      Crank-Nicolson scheme
      The \( \theta \) rule
      The Laplace and Poisson equation
      Extensions
Analysis of schemes for the diffusion equation
      Properties of the solution
      Example: Diffusion of a discontinues profile
      Analysis of discrete equations
      Analysis of the finite difference schemes
      Analysis of the Forward Euler scheme
      Analysis of the Backward Euler scheme
      Analysis of the Crank-Nicolson scheme
      Summary of accuracy of amplification factors
      Exercise 1: Explore symmetry in a 1D problem
      Exercise 2: Investigate approximation errors from a \( u_x=0 \) boundary condition
      Exercise 3: Experiment with open boundary conditions in 1D
      Exercise 4: Simulate a diffused Gaussian peak in 2D/3D
      Exercise 5: Examine stability of a diffusion model with a source term
Diffusion in heterogeneous media
      Stationary solution
      Piecewise constant medium
      Implementation
      Diffusion equation in axi-symmetric geometries
      Diffusion equation in spherically-symmetric geometries
Exercises
      Exercise 6: Stabilizing the Crank-Nicolson method by Rannacher time stepping
      Project 7: Energy estimates for diffusion problems
Bibliography