An efficient solver for the high-order accurate time-discontinuous Galerkin (TDG) method for second-order hyperbolic systems

An efficient predictor/multi-corrector algorithm for the high-order accurate time-discontinuous Galerkin (TDG) method is presented. The algorithm overcomes the demanding storage and computational effort of the direct solution of the TDG matrix equations. The success of the block matrix iterative solution method is a result of a specific form of time approximation. An analysis shows that the algorithm needs only a few iteration passes with a single matrix factorization of size equal to the number of spatial degrees-of-freedom to reach the 2p+1 order of accuracy of the parent TDG solution obtained from a direct solve. For linear (p=1) and quadratic (p=2) approximations in time, each iteration pass retains the unconditional and asymptotic stability of the TDG parent solution. Numerical results demonstrate the efficiency and accuracy of the time-integration algorithm over second-order accurate single-step/single-solve (SS/SS) methods.