Numerical solutions of nonlinear Burgers equation with modified cubic B-splines collocation method. (English) Zbl 1242.65209

Summary: 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 the SSP-RK43 or SSP-RK54. These methods need less storage space that causes less accumulation of numerical errors. The numerical approximate solutions to the Burgers’ equation are 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.


65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
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