The direct discontinuous Galerkin method with explicit-implicit-null time discretizations for the compressible Navier-Stokes equations

In this paper, we discuss the direct discontinuous Galerkin (DDG) method combined with two specific high-order explicit-implicit-null (EIN) time discretizations for solving the compressible Navier-Stokes (CNS) equations. This paper presents the EIN method whose basic idea is to add and subtract an identical Laplacian operator on the right-hand side of the considered equations, and then apply the implicit-explicit (IMEX) time-marching method to the equivalent equations. More specifically, the added term is treated implicitly while the rest of the terms are treated explicitly. The EIN method is designed to eliminate the severe time step restriction associated with explicit methods, without requiring any nonlinear iterative solver. Based on the Fourier method, we analyze the stability of the EIN-DDG schemes for the one-dimensional CNS equations, and further validate numerically that the stability criteria can be extended to the two-dimensional case. The numerical results demonstrate that our schemes achieve both stability and optimal orders of accuracy under a relaxed time-step restriction, provided that an appropriate coefficient is used for the Laplacian operator. Furthermore, the computational efficiency of different time discretizations, such as the strong stability-preserving Runge-Kutta (SSP-RK) and EIN methods, is evaluated and compared, demonstrating the advantages of the proposed schemes.