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
    • Similarity solution
    • Solution for a Gaussian pulse
    • Solution for a sine component
  • Example: Diffusion of a discontinues profile
  • Analysis of discrete equations
  • Analysis of the finite difference schemes
    • Stability (1)
    • Accuracy (1)
  • Analysis of the Forward Euler scheme
    • Stability (2)
    • Accuracy (2)
  • Analysis of the Backward Euler scheme
    • Stability (3)
  • Analysis of the Crank-Nicolson scheme
    • Stability (4)
  • Summary of accuracy of amplification factors
  • Exercise 1: Use an analytical solution to formulate a 1D test
  • Exercise 2: Use an analytical solution to formulate a 2D test
  • Exercise 3: Examine stability of a diffusion model with a source term
• Diffusion in heterogeneous media
  • Stationary solution
  • Piecewise constant medium
  • Implementation
• Exercises
  • Exercise 4: Stabilizing the Crank-Nicolson method by Rannacher time stepping
  • Project 5: Energy estimates for diffusion problems
• Bibliography

Indices and tables

• Index
• Module Index
• Search Page