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In this paper, we develop a new type of Runge--Kutta (RK) discontinuousGalerkin (DG) method for solving hyperbolic conservation laws. Compared withthe original RKDG method, the new method features improved compactness andallows simple boundary treatment. The key idea is to hybridize two differentspatial operators in an explicit RK scheme, utilizing local projectedderivatives for inner RK stages and the usual DG spatial discretization for thefinal stage only. Limiters are applied only at the final stage for the controlof spurious oscillations. We also explore the connections between our methodand Lax--Wendroff DG schemes and ADER-DG schemes. Numerical examples are givento confirm that the new RKDG method is as accurate as the original RKDG method,while being more compact, for problems including two-dimensional Eulerequations for compressible gas dynamics.

The Runge--Kutta discontinuous Galerkin method with compact stencils for hyperbolic conservation laws