Partial differential equations are at the foundation of much of computational science. Most physical phenomena depend in complex ways on space and time. Examples include fluid flow, heat transfer, nuclear and chemical and reactions and population dynamics. Computational scientists often seek to gain understanding of such phenomena by casting fundamental principles, such as conservation of mass, momentum and energy in the form of mathematical models of the underlying physical phenomena. Usually a mathematical model requires more than one independent variable to characterize the state of the physical system. For example, to describe a general fluid flow usually requires that the physical variables of interest, say pressure, density and velocity, be dependent on time and three space variables. If a mathematical model involves more than one independent and if at least one of the physical variables of interest is nonconstant with respect to space or time, then the mathematical model will involve a partial differential equation (PDE).

This chapter is not intended to be a complete discussion of partial differential equations. Instead, its aim is to serve as an introduction to a minimal amount of terminology from the field of PDEs, followed by some examples of issues that are likely to confront a computational scientist. Thus, the emphasis will be placed on

- Notation and Terminology
- Introduction to the Finite Difference Method
- Selected Numerical Algorithms for Solving Finite Difference Equations
- Performance Programming and Algorithm to Architecture Mapping