Lecture 2

Introduction to Finite Difference Methods (FDM)

This lecture covers slides 11 .

Reading Material

• Ferziger, J.H., Perić, M., Street, R.L. (2020). Finite Difference Methods. In: Computational Methods for Fluid Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-99693-6_3, Chapter 3.

Finite difference method basics

• Sampling on a regular grid
• Finite difference approximations in 1D
• Higher-order finite difference approximations
• Covers: slides 11 (pages 7-13)

Going to higher dimensions

• Extending finite differences to higher-order spatial dimensions
• 2D toy example
• Covers: slides 11 (pages 14-16)

Solutions for a 2D toy example

• What is a stencil?
• Definition of the computational domain
• Update formulas
• Covers: slides 11 (pages 17-19)

Boundary conditions

• Dirichlet and von Neumann boundary conditions
• Covers: slides 11 (pages 21-22)

Boundary approaches

• Applying a von Neumann boundary condition on a computational domain
• Eliminating dependent variables
• Using ghost nodes
• Covers: slides 11 (pages 23-28)

Matrix form

• What is the matrix form?
• Conversion between update-equations and the matrix form
• Covers: slides 11 (pages 29-34)

Overview of the approaches

• Using update-equations vs. matrix form
• Ghost nodes vs. elimination of variables
• Covers: slides 11 (page 38)