A Reynolds uniform scheme for singularly perturbed parabolic differential equation

Abstract

Time dependent convection diffusion problems with large Reynolds number are considered. Such a problem has been considered by using Shishkin's scheme, which was uniformly convergent with respect to large Reynolds number in order O(N -1 log 2 N+M -1 ), where N and M are number of intervals in x direction and t direction respectively. A three-transition points scheme, four piecewise-uniform mesh, is introduced. The mesh partition, the barrier function, the estimate of truncation error and the techniques of proof are different from others. The new scheme is non-equidistant. It is proved uniformly convergent with respect to large Reynolds number in order O(N -1 +M -1 ). Our work is better than Shishkin's traditional scheme, while the computational procedure is as simple as Shishkin's scheme. This novel method also has the same accurate result as Bakhvalov--Shishkin's scheme, while the computational procedure is simpler than Bakhvalov--Shishkin's scheme. Shishkin's scheme and Bakhvalov--Shishkin's scheme are compared with the new method. Finally, numerical results support the theoretical results.