Uniformly convergent novel finite difference methods for singularly perturbed reaction-diffusion equations

In this paper, we construct a kind of novel finite difference (NFD) method for solving singularly perturbed reaction-diffusion problems. Different from directly truncating the high-order derivative terms of the Taylor's series in the traditional finite difference method, we rearrange the Taylor's expansion in a more elaborate way based on the original equation to develop the NFD scheme for 1D problems. It is proved that this approach not only can highly improve the calculation accuracy but also is uniformly convergent. Then, applying alternating direction implicit technique, the newly deduced schemes are extended to 2D equations, and the uniform error estimation based on Shishkin mesh is derived, too. Finally, numerical experiments are presented to verify the high computational accuracy and theoretical prediction.