Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method

In this paper a numerical method is proposed to approximate the solution of the nonlinear Burgers' equation. The method is based on collocation of modified cubic B-splines over finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply modified cubic B-splines for spatial variable and derivatives which produce a system of first order ordinary differential equations. We solve this system by using SSP-RK43 or SSP-RK54 scheme. This method needs less storage space that causes less accumulation of numerical errors. The numerical approximate solutions to the Burgers' equation have been computed without transforming the equation and without using the linearization. Illustrative eleven examples are included to demonstrate the validity and applicability of the technique. Easy and economical implementation is the strength of this method.