Appropriate stabilized Galerkin approaches for solving two-dimensional coupled Burgers’ equations at high Reynolds numbers.

*(English)*Zbl 1443.65198Summary: This paper aims to seek proper stabilized Galerkin methods for solving the two-dimensional coupled Burgers’ equations at high Reynolds numbers. The stabilization techniques employed here include the streamline upwind/Petrov-Galerkin (SUPG) method, the spurious oscillations at layers diminishing (SOLD) method and the characteristic Galerkin (CG) method. The first two methods are combined with the Crank-Nicolson scheme for time discretization and the last one is applied in its semi-implicit version. Different from most of the studies on the equations which are usually devoted to improving the accuracy of computed solution in the case of low Reynolds numbers, this paper mainly focuses on keeping the stability of the solution at high Reynolds numbers, which is significant in practical applications and also challenging in numerical computation. We study two problems, equipped with mixed boundary conditions and only Dirichlet boundary conditions, respectively. Numerical experiments reveal that the SUPG method is optimal for the former problem, and the SOLD method is more appropriate for the latter one. In addition, the performances of these methods demonstrate the difference between the two problems, which is seldom mentioned previously and might be helpful to other conventional methods intending to solve the problems at high Reynolds numbers. And last, since SOLD methods have rarely been utilized to solve nonlinear unsteady problems before, this study also indicates the potential of this class of methods to solve nonlinear unsteady convection-dominated problems.

##### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

76D99 | Incompressible viscous fluids |

##### Keywords:

stabilized Galerkin methods; Burgers’ equations; high Reynolds numbers; boundary conditions##### Software:

SUPG
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\textit{Y. Chai} and \textit{J. Ouyang}, Comput. Math. Appl. 79, No. 5, 1287--1301 (2020; Zbl 1443.65198)

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