The Multidomain Hybrid Discontinuous Galerkin Method and Finite Difference Method for Solving Compressible Navier-Stokes Equations on Hybrid Meshes

Abstract

In this work, we develop a multidomain hybrid discontinuous Galerkin (DG) method and finite difference(FD) method for solving two-dimensional compressible Navier-Stokes equations on the hybrid meshes. The direct discontinuous Galerkin (DDG) method and central difference(CD) scheme are utilized to discretize the viscous fluxes respectively. This approach combines the flexibility for the complex geometries of the DG method on the unstructured meshes, and the computational efficiency of the FD method on Cartesian grids. At the artificial interfaces between the DG subdomain and FD subdomain, the square ghost cells are generated and the weighted essentially non-oscillatory (WENO) interpolation is employed to reconstruct the degrees of freedom of these ghost cells. To ensure the accuracy in smooth regions and the correct position of the shock wave, the troubled cell indicator is adopted to determine the nonconservative or conservative coupling modes. The construction process of the numerical fluxes at the artificial interfaces is described specifically and the corresponding WENO interpolation coefficients are given in detail. Numerous numerical results demonstrate that the multidomain hybrid method achieves high-order accuracy in smooth regions, robustness in shock simulations, flexibility in handling complex geometries, and significant computational cost savings compared to the traditional DG method on the hybrid meshes.