A fast numerical method for solving coupled Burgers' equations. Issue 6 (21st April 2017)

• Record Type: Journal Article
• Authors: Shi, Feng; Zheng, Haibiao; Cao, Yong; Li, Jingzhi; Zhao, Ren
• Is Part Of: Numerical methods for partial differential equations. Volume 33:Issue 6(2017:Nov.)
• Page Start: 1823
• Page End: 1838
• Publication Date: 2017-04-21
• Subjects: coupled Burgers' equations -- operator splitting -- finite elements -- multistep scheme; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22160
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract : A new fast numerical scheme is proposed for solving time‐dependent coupled Burgers' equations. The idea of operator splitting is used to decompose the original problem into nonlinear pure convection subproblems and diffusion subproblems at each time step. Using Taylor's expansion, the nonlinearity in convection subproblems is explicitly treated by resolving a linear convection system with artificial inflow boundary conditions that can be independently solved. A multistep technique is proposed to rescue the possible instability caused by the explicit treatment of the convection system. Meanwhile, the diffusion subproblems are always self‐adjoint and coercive at each time step, and they can be efficiently solved by some existing preconditioned iterative solvers like the preconditioned conjugate galerkin method, and so forth. With the help of finite element discretization, all the major stiffness matrices remain invariant during the time marching process, which makes the present approach extremely fast for the time‐dependent nonlinear problems. Finally, several numerical examples are performed to verify the stability, convergence and performance of the new method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1823–1838, 2017