An implicit reconstructed discontinuous Galerkin method based on automatic differentiation for the compressible flows on tetrahedral grids

An implicit reconstructed discontinuous Galerkin method, namely IRDG(P₁P₂), based on the automatic differentiation (AD) technique, is presented for solving the compressible flows on tetrahedral grids. The key idea is to compute the flux Jacobian matrix using the AD generated code for the LU-SGS (lower-upper symmetric Gauss-Seidel) preconditioning in a GMRES (generalized minimum residual) algorithm to solve an approximate system of linear equations arising from the Newton linearization. The AD technique ensures an effcient and stable performance of the implicit time integration, and can save the great effort for algebraical derivation of the Jacobian matrix and further maintenance, which can be quite complicated depending on the complexity of the numerical flux scheme in the DG context. In this IRDG(P₁P₂) method, a Hermite WENO (HWENO) reconstruction method is used to obtain a quadratic polynomial (P₂) solution of the underlying linear (P₁) DG solution within each cell. This reconstruction scheme is able to augment the accuracy of the DG method by increasing the order of the underlying polynomial solution, and maintain the linear stability. The performance of the developed IRDG(P₁P₂) solver is assessed by computing a wide range of compressible inviscid and viscous flow problems on tetrahedral grids. The numerical results demonstrate that this automatic differentiation based implicit reconstructed discontinuous Galerkin method can improve the speed of convergence by over two orders of magnitude for all the test cases in comparison with its explicit counterpart.