An Implicit Method for a Reconstructed Discontinuous Galerkin Method on Tetrahedron Grids

An implicit method for a reconstructed discontinuous Galerkin (RDG) method is presented to solve compressible flow problems on tetrahedron grids. The idea is to combine the accuracy of the RDG method and the efficiency of implicit methods to obtain a better numerical algorithm in computational fluid dynamics. A least-squares reconstruction method is presented to obtain a quadratic polynomial representation of the underlying linear discontinuous Galerkin solution on each cell via an in-cell reconstruction process. The devised in-cell reconstruction is able to augment the accuracy of the DG method by increasing the order of the underlying polynomial solution. A matrix-free GMRES (generalized minimum residual) algorithm with an LU-SGS (lower-upper symmetric Gauss- Seidel) preconditioner is presented to solve an approximate system of linear equations arising from the Newton linearization. The implicit method is used to compute a variety of three-dimensional problems on tetrahedron grids to assess its accuracy and robustness. The numerical experiments demonstrate that the implicit reconstructed discontinuous Galerkin method can obtain an overall speedup of more than two orders of magnitude for all test cases compared with multi-stage Runge-Kutta reconstructed DG methods. The numerical results also indicate that this implicit RDG(P1P2) method can deliver the desired third-order accuracy, while maintaining advantage in cost over the implicit DG(P2) method.