High order fitted operator numerical method for self-adjoint singular perturbation problems

We consider self-adjoint singularly perturbed two-point boundary value problems in conservation form. Highest possible order of uniform convergence for such problems achieved hitherto, via fitted operator methods, was one (see, e.g., [Doolan et al. Uniform numerical methods for problems with initial and boundary layers, Boole Press, Dublin, 1980], p. 121]). Reducing the original problem into the normal form and then using the theory of inverse monotone matrices, a fitted operator finite difference method is derived via the standard Numerov’s method. The scheme thus derived is fourth order accurate for moderate values of the perturbation parameter ε whereas for very small values of this parameter the method is “ε-uniformly convergent with order two”. Numerical examples are given in support of the theory.