Parameter free hybrid numerical method for solving modified Burgers’ equations on a nonuniform mesh

Abstract

In this paper, the modified Burgers’ equation is considered. These kinds of problems come from the field of sonic boom and explosions theories. At big Reynolds’ number there is a boundary layer in the right side of the domain. From numerical point of view, the major difficulty in dealing with this type of problem is that the smooth initial data can give rise to solution varying regions i.e. boundary layer regions. To tackle this situation, we propose a numerical method on nonuniform mesh of Shishkin type, which works well at high as well as low Reynolds number. The proposed method comprises of Euler implicit scheme and hybrid scheme in time and space direction, respectively. First, we discretize the continuous problem in temporal direction by Euler implicit method, which yields a set of ode’s at each time level. The resulting set of differential equations are approximated by a hybrid scheme on Shishkin mesh i.e. upwind in regular region (nonboundary layer region) and central difference in boundary layer regions. The convergence of proposed method has been shown parameter uniform. Some numerical experiments have been carried out to corroborate the theoretical results.