A hybrid numerical method approach for the two-dimensional coupled Burgers' equations with nonlinearity preservation

Abstract

In this study, we develop a novel hybrid numerical method based on the finite element method and finite difference method for solving the two-dimensional coupled Burgers' equations. The time discretization is achieved by applying a backward finite difference approximation. The proposed hybrid method is evaluated using three challenging shock wave problems of Burgers' equation with different values of parameters. The most outstanding feature of the combined method is that the proposed hybrid method is applied without linearization of the two-dimensional coupled Burgers' equations and preserving the nonlinearity of the model equations. To the authors' knowledge, this study represents the first successful application of the proposed numerical method capable of solving the governing equations for large time levels and extremely low viscosity values (epsilon=0.00001) with minimal computational cost. Numerical experiments validate the accuracy and convergence order of the method, demonstrating its advantage over exact and previously reported solutions.