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