Numerical solutions of linear and nonlinear singular perturbation problems

A new method is developed by detecting the boundary layer of the solution of a singular perturbation problem. On the non-boundary layer domain, the singular perturbation problem is dominated by the reduced equation which is solved with standard techniques for initial value problems. While on the boundary layer domain, it is controlled by the singular perturbation. Its numerical solution is provided with finite difference methods, of which up to sixth order methods are developed. The numerical error is maintained at the same level with a constant number of mesh points for a family of singular perturbation problems. Numerical experiments support the analytical results.