A new stable finite difference scheme and its error analysis for two‐dimensional singularly perturbed convection–diffusion equations

Abstract

AbstractThis work focuses on the numerical solution of two‐dimensional singularly perturbed convection–diffusion equations via a new stable finite difference (NSFD) scheme on a tensor product of two piecewise‐uniform Shishkin meshes. First, we convert the two‐dimensional equation into two one‐dimensional equations using the alternating direction implicit technique. A NSFD scheme has been developed using Taylor's series and the one‐dimensional equations. Here the truncation of Taylor's series is different from the classical finite difference scheme. The convergence analysis is also studied on a tensor product of two piecewise‐uniform Shishkin meshes. Numerical simulations confirm the theory. Moreover, from the numerical illustrations, it is also observed that the method is parameter uniform convergent.