Semi discrete discontinuous Galerkin methods and stage-exceeding-order, strong-stability-preserving Runge–Kutta time discretizations

This paper investigates the use of a special class of strong-stability-preserving (SSP) Runge–Kutta time discretization methods in conjunction with discontinuous Galerkin (DG) finite element spatial discretizatons. The class of SSP methods investigated here is defined by the property that the number of stages s is greater than the order k of the method. From analysis, CFL conditions for the linear (L2) stability of the methods defined using the s > k SSP schemes are obtained that are less restrictive than those of the “standard” so-called RKDG methods that use s = k SSP Runge–Kutta schemes. The improvement in the CFL conditions for linear stability of the methods more than offsets the additional work introduced by the increased number of stages. Given that the CFL conditions for linear stability are what must be respected in practice in order to maintain high-order accuracy, the use of the s > k SSP schemes results in RKDG methods that are more efficient than those previously defined. Furthermore, with the application of a slope limiter, the nonlinear stability properties of the forward Euler method and the DG spatial discretization, which have been previously proven, are preserved with these methods under less restrictive CFL conditions than those required for linear stability. Thus, more efficient RKDG methods that possess the same favorable accuracy and stability properties of the “standard” RKDG methods are obtained. Numerical results verify the CFL conditions for stability obtained from analysis and demonstrate the efficiency advantages of these new RKDG methods.