Diffusion Problem

In this chapter, we discuss the fundamentals of how to cast a PDE into finite difference form. More specifically, the reader will learn how to discretize the diffusion equation, and learn why some discretizations work and some do not. This will be the opportunity to introduce concepts such as numerical stability, convergence, and consistency. It will be explained how to check whether a discretization is stable and consistent, and the reader will learn about explicit and implicit schemes, the rudiments of elliptic solvers, and the concept of numerical dissipation or artificial damping inherent in our discretizations.