Higher order weak Galerkin methods for the Navier-Stokes equations with large Reynolds number

In this article, we provide a class of numerical schemes to solve the steady state and non-steady state Navier-Stokes equations with large Reynolds number. The high order weak Galerkin methods are numerically proposed and tested. We numerically find that the stabilizing term has a great impact on the robustness of the Jacobian matrices associated with the Newton's method in the procedure of solving the discrete nonlinear systems generated by weak Galerkin discretizations, especially for the case with large Reynolds number. The numerical investigations of several benchmarks are then carried out to demonstrate the efficiency and robustness of our proposed numerical methods. The errors for the Taylor-Green vortex flow are computed and compared with different weak Galerkin techniques. The lid driven flow and the flow around a cylinder are further numerically investigated and compared with existing references. The Navier-Stokes equations with large Reynolds number can be efficiently solved by the high order weak Galerkin methods with proper stabilizers.