Weak Galerkin finite element method for solving one-dimensional coupled Burgers’ equations

Fast algorithms for coupled Burgers' equations by sharing characteristic curves within BSLM

“…Over the last 10 years, various numerical methods have been developed to solve the viscous coupled Burgers' equations. For example, Kapoor and Joshi developed a differential quadrature method with uniform algebraic trigonometric tension B-spline [11], Başhan applied a mixed method with the finite difference and differential quadrature method based on third order modified cubic B-spline functions [4], Hussein and Kashkool used a weak Galerkin finite element method [9], Zhang et al introduced an improved backward substitution method [19], Abdullah et al developed a numerical procedure based on the cubic B-spline and the Hermite formula [1], Mohanty and Sharma presented a high accuracy two-level implicit method based on cubic spline approximation [16], Bhatt and Khaliq modified exponential time-differencing Runge-Kutta method based on Padé approximation [6], Kumar and Pandit proposed a composite scheme based on finite difference and Haar wavelets [12], Jiwari and Alshomrani developed a new collocation method based on modified cubic trigonometric B-spline function [10], Mohanty et al proposed a two-level implicit compact operator method [13]. Among the abovementioned methods, there is a backward semi-Lagrangian (BSLM) to solve the model problem.…”

Semi-discrete and fully discrete HDG methods for Burgers' equation

“…It is widely used in many physical fileds such as fluid mechanics, nonlinear acoustics, and gas dynamics. In recent decades, there have developed many finite element (FE) methods for Burgers' equation, such as conforming methods [1,4,9,10,16,25,29], B-spline methods [3,26,37], least-squares methods [23,35,39], mixed methods [12,19,20,30,33], discontinuous Galerkin (DG) methods [5,32,40], and weak Galerkin methods [14,21].…”

Semi-discrete and fully discrete HDG methods for Burgers' equation

“…In addition, the global spreading of COVID-19 is usually modeled by a system of nonlinear partial differential equations, while the Burgers' equation captures a key challenge of the nonlinear partial differential equations for modeling virus spreading. In recent decades, there have developed many finite element (FE) methods for Burgers' equation, such as conforming methods [1,4,9,10,16,26,30], B-spline methods [3,27,38], least-squares methods [24,36,40], mixed methods [12,20,21,31,34], discontinuous Galerkin (DG) methods [5,33,41], and weak Galerkin methods [14,22].…”