Author's Review of Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations

Chapter 8 of Book 2 covers the topic of the stability of initial-boundary value problems as developed by Gustafsson-Kreiss-Sundstrom-Osher---we are most interested in the stability of the boundary conditions. These techniques are most important for hyperbolic problems. Often when you solve hyperbolic problems by finite difference methods, the numerical problem needs more boundary conditions than the analytic problem provides---you need to include numerical boundary conditions. The "extra" boundary conditions are common sources of instabilities. This material is difficult so we tried to present these methods in their simplest forms. We do not include proofs.

In Chapter 9 we have a fairly complete treatment of the numerical treatment of conservation law equations. We include a theoretical treatment of conservation laws and basic numerical schemes for conservation laws including TVD and high resolution schemes. Of course we emphasize the use of conservative difference schemes and work to find schemes that will select the vanishing viscosity solution. We also include systems of conservation laws and some work with two dimensional conservation laws.

In Chapter 10 we tried to give a fairly complete description of methods for elliptic problems. We give a very complete treatment of relaxation methods---including some great results that we found for problems with Neumann boundary conditions. We include a very basic description of multigrid methods that seems to be very easily understood and shows exactly why and how the method works. We also include a treatment of precondition conjugate gradient methods but it is only an introduction to the subject. In addition we show how to treat elliptic problems by using finite Fourier transforms.

By this time we had approximately 1000 pages of solutions of problems on the interval [0,1], the square [0,1]x[0,1] or even the cube [0,1]x[0,1]x[0,1]---we did give a technique for solving some elliptic problems in irregular domains. We conclude Book 2 with a short discussion of solution techniques for irregular domains and problems that require some sort of irregular meshes.