A robust implicit high-order discontinuous Galerkin method for solving compressible Navier-Stokes equations on arbitrary grids

The primary impediments impeding the implementation of high-order methods in simulating viscous flow over complex configurations are robustness and convergence. These challenges impose significant constraints on computational efficiency, particularly in the domain of engineering applications. To address these concerns, this paper proposes a robust implicit high-order discontinuous Galerkin (DG) method for solving compressible Navier-Stokes (NS) equations on arbitrary grids. The method achieves a favorable equilibrium between computational stability and efficiency. To solve the linear system, an exact Jacobian matrix solving strategy is employed for preconditioning and matrix-vector generation in the generalized minimal residual (GMRES) method. This approach mitigates numerical errors in Jacobian solution during implicit calculations and facilitates the implementation of an adaptive Courant-Friedrichs-Lewy (CFL) number increasing strategy, with the aim of improving convergence and robustness. To further enhance the applicability of the proposed method for intricate grid distortions, all simulations are performed in the reference domain. This practice significantly improves the reversibility of the mass matrix in implicit calculations. A comprehensive analysis of various parameters influencing computational stability and efficiency is conducted, including CFL number, Krylov subspace size, and GMRES convergence criteria. The computed results from a series of numerical test cases demonstrate the promising results achieved by combining the DG method, GMRES solver, exact Jacobian matrix, adaptive CFL number, and reference domain calculations in terms of robustness, convergence, and accuracy. These analysis results can serve as a reference for implicit computation in high-order calculations. (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(DG)(sic)(sic). (sic)(sic)(sic)(sic) (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic). (sic)(sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic) (sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(GMRES)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic). (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic), (sic)(sic) (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic). (sic)(sic), (sic)(sic)(sic)(sic)(sic)CFL(sic)(sic)(sic)(sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic). (sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic) (sic)(sic)(sic)(sic)(sic)(sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic). (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic). (sic)(sic) (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic), (sic)(sic)CFL(sic),Krylov(sic)(sic)(sic)(sic)(sic)(sic)GMRES(sic)(sic)(sic)(sic). (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic), (sic)(sic)(sic)(sic)DG(sic)(sic),GMRES(sic)(sic),(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic),(sic)(sic)(sic)CFL(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic),(sic)(sic)(sic)(sic) (sic)(sic)(sic)(sic). (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic).