The accurate numerical solution of singular perturbation problems by finite difference methods is considered. (For efficient computations of this type, refinement of the finite difference mesh is important. The technique of solution-adaptive mesh refinement, in which the mesh is refined iteratively by looking at the properties of a computed solution, can be the simplest method by which to implement a mesh refinement.) The theoretical justification of solution-adaptive mesh refinement for singularly perturbed systems of first order ordinary differential equations (ODEs) is discussed. It is shown that a posteriori error estimates can be found for weighted one-sided difference approximations to systems of ODEs without turning points and to systems of ODEs with turning points that can be transformed to a typical normal form. These error estimates essentially depend only on the local meshwidths and on lower order divided differences of the computed solution, and so can be used in the implementation of solution-adaptive mesh refinement. It is pointed out, however, that not all systems with turning points fall into these categories, and solution-adaptive mesh refinement can sometimes be inadequate for the accurate resolution of solutions of these systems.
Numerical examples are presented in which the solutions of some model equations of fluid dynamics are resolved by transforming the problems to singularly perturbed ODEs and applying weighted one-sided difference approximations with solution-adaptive mesh refinement. In particular, well-resolved steady and moving shock solutions to Burgers' equation and to the equations of one-dimensional isentropic gas dynamics are obtained numerically. The method is further extended to problems in two space dimensions by using the method of dimensional splitting together with careful interpolation. In particular, in this extension the mesh refinement is only used to resolve the one-dimensional problems which are solved within the splitting algorithm. Numerical examples are presented in which two-dimensional oblique shocks are resolved.
