Parameter Free Hybrid Numerical Method for Solving Modified Burgers' Equation on a Non-uniform Mesh
by
Mohan Kadalbajoo
Department . of Mathematics & Statistics, Indian Institute of Technology Kanpur,India 208016

In this presentation,we consider the one-dimensional modified Burgers' equation in the finite domain.These problems arise in the field of sonic boom and explosion theory.At the high Reynolds' number there is a boundary layer in the right side of the domain.From the numerical point of view,one of the difficulties in dealing with this problem is that even smooth initial data can give rise to solution varying regions,i.e.,boundary layer regions.To tackle this problem,we propose a numerical method on non-uniform mesh of Shishkin type,which works well at high as well as low Reynolds' numbers.The proposed numerical method comprises of Euler implicit and upwind difference scheme.First we discretize in the temporal direction by means of Euler implicit method which yields the set of ordinary differential equations at each time level.The resulting set of differential equations are approximated by hybrid scheme on Shishkin mesh.i.e.,upwind scheme in smooth regions and central difference in boundary layer regions.The proposed method has been shown to be parameter uniform and of first order accurate in space and time.An extensive amount of analysis has been carried out in order to prove parameter uniform convergence of the method.Some test examples have been solved to corroborate the theoretical results.

Date received: February 5, 2007