Boundary value problems (BVPs) involve the solution of ODEs or partial differential equations (PDEs) on a spatial domain, subject to boundary conditions that hold on the domain boundary. Many problems from solid and fluid mechanics, electromagnetics, and heat and mass transfer are expressed naturally as BVPs. The forms of these differential equations often resemble each other because they arise from similar conservation principles. Here the emphasis is upon BVPs that arise from problems in transport phenomena.

This chapter focuses upon real-space methods, in which a computational grid is overlaid upon the domain. The BVP is then converted into a set of ODEs for a time-dependent problem or a set of algebraic equations for a steady problem. This technique can be used even when no analytical solution exists, and can be extended to BVPs with multiple equations or complex domain geometries. Here, the focus is upon the methods of finite differences, finite volumes, and finite elements. These methods have many characteristics in common; therefore, particular attention is paid to the finite difference method, as it is the easiest to code. The finite volume and finite element methods also are discussed; however, as the reader is most likely to use these in the context of prewritten software, the emphasis is upon conceptual understanding as opposed to implementation.