Shock Calculations and the Numerical Solution of Singular Perturbation Problems

This chapter presents shock calculations and discusses the numerical solution of singular perturbation problems. These methods can also be used for shock calculations. D. Brown has developed a much more sophisticated way to deal with shock calculations. He solves these problems using the Lax–Wendroff method on a fixed relatively coarse grid. Then the computer isolates those intervals where the gradient of the solution is large. A locally moving coordinate system is generated, and local singular perturbation problems are solved. In particular, a local mesh is constructed, which resolves the large gradients. D. Brown has also generalized the above technique to two space dimensions. The crudest way is to use the implicit Euler method in every direction.