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Finite difference methods for diffusion processes¶

Contents:

  • Finite difference methods for diffusion processes
  • 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

Index¶

  • Index

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©2015, Hans Petter Langtangen.