Runge-Kutta discontinuous Galerkin methods with compact stencils for hyperbolic conservation laws
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In this talk, we present a new type of Runge-Kutta (RK) discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws. Compared with the standard RKDG methods, the new methods feature improved compactness and allow simple boundary treatment. Our main idea is to hybridize two different spatial operators in an explicit RK scheme, with local projected derivatives used for inner RK stages and the usual DG spatial discretization used for the final stage only. Limiters are applied only at the final stage for the control of spurious oscillations. Their connections with the Lax-Wendroff DG schemes and the ADER DG schemes are also investigated. Numerical examples are given to confirm that the new RKDG schemes are as accurate as standard RKDG methods, while being more compact and cost-effective, for certain problems including two-dimensional Euler systems of compressible gas dynamics.
