11. PDE Solvers: Finite Difference Methods 2#
Chapter 11 covers PDE solvers using finite difference methods. The lecture will review what we’ve seen several times already on finite difference approximations to derivatives, how to derive them and how to estimate errors. The lecture will also review the numerical solution of advection-diffusion problems (steady and unsteady) using upwind and central difference methods and the very distinct accuracy/stability properties of each. By the end of this chapter you should be able to:
Understand and recognise some of the issues that result from rapid (relative to the mesh resolution) variations in the solution - e.g. the presence of sharp boundary layers, and how this related to stability. To understand how a move away from uniform mesh resolution is one way to alleviate these issues.
To understand three key stability issues - where these come from and how they can be “managed”. One of these is the cell Peclet number related issue from the previous lecture, while the other two relate to the time derivative and limiting the speed of information propagation on the discrete mesh.
To consider how eigenvalue analysis of the discretisation matrix can be combined with theory we saw for ODE solvers to consider stability issues for PDEs.