Numerical solution of the viscous Burgers’ equation using Localized Differential Quadrature method

Abstract

In this article, we propose a numerical method to solve the viscous Burgers’ equation in one and two dimensions using the Localized Differential Quadrature (LDQ) scheme. To approximate spatial derivatives, we use the differential quadrature technique locally in a neighborhood of each node using Lagrangian weights and then march the numerical solution in time using the Taylor’s series approximation. Convergence analysis is done by observing that the error between the numerical and exact solutions goes to zero as the number of subdivisions of the spatial domain increases. The localization technique of the differential quadrature method maintains the stability and accuracy of our proposed numerical scheme. We test our proposed numerical scheme using LDQ for several examples. In comparison with several other known methods in the literature, our numerical results confirm the effectiveness of our proposed method and demonstrate their advantages over other known methods in terms of accuracy and computational complexity.