Notes on implementing the finite-volume method for physical simulations

Contents:

• Overview
  • Finite volumes vs. finite differences
• The advection-diffusion-reaction equation
  • Adaptive upwinding & exponential fitting
  • Explicit and implicit forms
  • The \(\theta\)-method
  • Discretised equation in matrix form
• Boundary conditions for the advection-diffusion-reaction equation
  • Robin boundary conditions (known flux)
  • Dirichlet boundary conditions (known value)
  • General matrix form with a transient term
  • General matrix form without a transient term
• Solving linear equations
  • Time stepping the advection-diffusion equation
• Solving nonlinear equations
  • Fisher’s equation
  • IMEX time-stepping
  • Newton iteration
  • Method of lines
• The Poisson equation
  • Naturally resolved Neumann boundary conditions
  • Dirichlet boundary conditions
  • Poisson equation in matrix form
• Solving the advection-diffusion-reaction equation in Python
  • Overview
  • The Mesh class
  • The CellVariable class
  • The Model class
• Example calculations
  • Uniform grid
  • Random grid
  • Nonuniform grid

Indices and tables

• Index
• Module Index
• Search Page