Filtered RKDG Methods: Fourier Analysis for the Second-Order Case

In this work, we apply a Fourier approach to study the stability and accuracy effects of filtering on Runge–Kutta (RK) discontinuous Galerkin (DG) methods for scalar hyperbolic conservation laws. Both the standard RKDG method and its variant, the RKDG method with compact stencils (cRKDG), will be considered. We focus on the second-order schemes and apply the modal filters immediately after the DG operators at each RK stage. Using a Fourier-type analysis, we quantitatively assess the time step constraints for stability and accuracy of multiple filtered schemes. We show that the application of filters can significantly improve the Courant–Friedrichs–Lewy numbers while maintaining the optimal convergence rate. However, for problems containing sonic points, the convergence order of some filtered schemes may degrade. To address this issue, we apply filters only away from the sonic points and adopt a local time-stepping strategy for the cRKDG method to take advantage of the larger time steps enabled by the filters in these regions.