High‐order methods for differential equations with large first‐derivative terms

A method is developed to solve elliptic singular perturbation problems. Examples are presented in one and two dimensions for both linear and non‐linear problems. In particular, examples are presented for fluid flow problems with boundary layers. In the one‐dimensional case an approximating equation is developed using just three points. The method first presented is a fourth‐order approximation but is extended to become a higher‐order method. Results are included for the fourth‐, sixth‐, eighth‐ and tenth‐order methods. The results are first compared with results found by Segal in an article about elliptic singular perturbation problems. The elliptic singular perturbation problems are compared with a method by Il'in and also with central and backward difference schemes from Segal's article. There was only one case where the results in Segal's paper were as accurate as the results presented in this paper. However, in this case the method used by Segal did not give accurate values for a second problem presented. The results are also compared with results given by Spalding and by Christie. The method of this paper was also tested on the solution of some non‐linear diffusion equations with concentration‐dependent diffusion coefficients. The results were superior to results presented by Lee and by Schultz. Finally, the method is extended to several two‐dimensional problems. The method developed in this paper is accurate, easy to use and can be generalized to other problems.