Adaptive Modal Filters Based on Artificial and Spectral Viscosity Techniques

In this work, we adapt techniques of artificial and spectral viscosities [especially those presented in Klockner et al. (Math Model Nat Phenom 6(03):57-83, 2011) and Tadmor and Waagan (SIAM J Sci Comput 34(2):A993-A1009, 2012)] to develop adaptive modal filters for Legendre polynomial bases and demonstrate their application in a shock-capturing discontinuous Galerkin method. While traditional artificial viscosities can impose additional stability restrictions on the time step size, we show that our filtering methods can achieve a similar shock-capturing capability without additional restriction of the time step size. We further demonstrate that the accuracy of the artificial viscosity filter can be significantly improved upon with a hybrid spectral viscosity-artifical viscosity filter, which we also develop and demonstrate. We show that we can achieve computational efficiency with an analytical time integration technique similar to one presented in Moura et al. (Diffusion-based limiters for discontinuous galerkin methods-part I: one-dimensional equations, 2013). We consider several 1D numerical examples for the linear advection, scalar wave, Burgers', and Euler equations to demonstrate the accuracy and uniform stability of our methods.