An implicit Hermite WENO reconstruction-based discontinuous Galerkin method on tetrahedral grids

Abstract

An Implicit Reconstructed Discontinuous Galerkin method, IRDG (P1P2), is presented for solving the compressible Euler equations on tetrahedral grids. In this method, a quadratic polynomial (P2) solution is first reconstructed using a least-squares method from the underlying linear polynomial (P1) DG solution. By taking advantage of the derivatives in the DG formulation, the stencils used in the reconstruction involve only von Neumann neighborhood (adjacent face-neighboring cells) and thus are compact and consistent with the underlying DG method. The final P2 solution is then obtained using a WENO reconstruction, which is necessary to ensure stability of the RDG (P1P2) method. A matrix-free GMRES (generalized minimum residual) algorithm is presented to solve the approximate system of linear equations arising from Newton linearization. The LU-SGS (lower–upper symmetric Gauss–Seidel) preconditioner is applied with both the simplified and approximate Jacobian matrices. The numerical experiments on a variety of flow problems demonstrate that the developed IRDG (P1P2) method is able to obtain a speedup of at least two orders of magnitude than its explicit counterpart, maintain the linear stability, and achieve the designed third order of accuracy: one order of accuracy higher than the underlying second-order DG (P1) method without significant increase in computing costs and storage requirements. It is also found that a well approximated Jacobian matrix is essential for the IRDG method to achieve fast converging speed and maintain robustness on large-scale problems.