Numerical approximation of modified burgers' equation via hybrid finite difference scheme on layer-adaptive mesh

In this paper, a numerical method is constructed for solving one-dimensional time dependent modified Burgers' equation for various values of Reynolds number. At high Reynolds number, an inviscid boundary layer is produced in the neighborhood of right part of the lateral surface of the domain and the problem can be considered as a non-linear singularly perturbed problem involving a small parameter ε. Using singular perturbation analysis, asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular components. We construct a numerical scheme that comprises of Implicit-Euler method to discretize in temporal direction on uniform mesh and a monotone hybrid finite difference operator to discretize the spatial variable with piecewise uniform Shishkin mesh. Quasi-linearization process is used to tackle the non-linearity and shown that quasi-linearization process converges quadratically. The method has been shown to be first order uniformly accurate in the temporal variable and first order parameter uniform convergent on the non-boundary layer domain and almost second order parameter uniform convergent on the boundary layer domain in the spatial variable. Uniform convergence of the method is demonstrated by numerical examples and an estimate of the error is given.