Parameter-uniform improved hybrid numerical scheme for singularly perturbed problems with interior layers
In this paper, we consider a class of singularly perturbed convection-diﬀusion boundary-value problems with discontinuous convection coeﬃcient which often occur as mathematical models for analyzing shock wave phenomena in gas dynamics. In general, interior layers appear in the solutions of this class of problems and this gives rise to diﬃculty while solving such problems using the classical numerical methods (standard central diﬀerence or standard upwind scheme) on uniform meshes when the perturbation parameter ε is small. To achieve better numerical approximation in solving this class of problems, we propose a new hybrid scheme utilizing a layer-resolving piecewise-uniform Shishkin mesh and the method is shown to be ε-uniformly stable. In addition to this, it is proved that the proposed numerical scheme is almost second-order uniformly convergent in the discrete supremum norm with respect to the parameter ε. Finally, extensive numerical experiments are conducted to support the theoretical results. Further, the numerical results obtained by the newly proposed scheme are also compared with the hybrid scheme developed in the paper [Z.Cen, Appl. Math. Comput., 169(1): 689-699, 2005]. It shows that the current hybrid scheme exhibits a signiﬁcant improvement over the hybrid scheme developed by Cen, in terms of the parameter-uniform order of convergence.
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