Author's Review of Numerical Partial Differential Equations: Finite Difference Methods

The Finite Differences book came out of our one semester course on finite difference solutions to partial differential equations. The clientele consisted of terminal Mathematics Masters students, graduate engineering students and a small number of PHD students. Originally the course consisted of methods for parabolic, hyperbolic and elliptic equations. Because there was plenty of work to do with parabolic and hyperbolic equations, I put the methods for elliptic equations in another semester and in the second book .

Numerically I tried to include all types of methods available and treatments of all different types of boundary conditions. I wanted to make my students comfortable facing a wide variety of problems.

I didn't want to make my course too theoretical because I wanted to include the Masters students and the engineers. However, I think that it is not smart to try to solve partial differential equations numerically without understanding what is happening. I tried to include a reasonable level of theory of finite difference methods. I placed the theory in the sequence spaces because this seemed to be more accessible to the students that I had in my class.

Other than a reasonably complete treatment of numerical schemes and a solid theoretical basis, there are three aspects of Book 1 that I thought were enjoyable: the treatment of nonlinear problems related to the first four home work problems, HW0.1-0.4, the treatment of systems of hyperbolic equations and the last chapter on dissipation and dispersion of solutions. I included the work on the nonlinear problems because I wanted to stress that most solutions to nonlinear problems in the literature used some linearization---essentially a linear scheme, carefully and essentially experimentally. I wanted to convince the students that these methods might or might not work on nonlinear problems---you try them, you use methods that are more apt to work and you use them very carefully.

The work on numerical schemes for systems of hyperbolic equations came out of the fact that I had never seen them done carefully. Systems of hyperbolic equations are so very important.

The last chapter on dissipation and dispersion came out of several homework problems that I gave to a class in a fit of anger. I was intrigued by how badly the methods behaved when you run them for long time periods. Since I wrote that chapter, I have often wondered how many "interesting" numerical results are due to dispersion occurring in one or more modes of a hyperbolic system, or what bad results do not appear because they have just been damped away.