A Space-time Approach to Entropy-Stable Discontinuous Galerkin and Flux Reconstruction

Abstract: We present a high-order space-time discretization equipped with fully-discrete entropy stability properties for general choices of volume and surface quadrature rules. The formulation uses flux reconstruction (FR) in the spatial dimension paired with a discontinuous Galerkin (DG) method in the temporal dimension. The result is a fully-implicit system using polynomial bases in space and time. An energy-stable discretization is applied to the linear advection equation, yielding optimal $p+1$ convergence for small FR correction parameters and $p$ convergence at the same filter strength as method-of-lines implementations. We can thus recover the space-time equivalent to schemes such as DG, Huynh's FR, or spectral difference through a single parameter $c$. We follow with a similar space-time nonlinearly-stable flux reconstruction (ST-NSFR) scheme, which uses skew-symmetric stiffness operators in both space and time. The ST-NSFR scheme is fully-discretely entropy preserving using the $c_{DG}$ parameter or entropy-stable for small $c$. Numerical experiments using the linear advection and Euler equations confirm convergence orders and stability properties. The advantage of FR in a space-time context is demonstrated by a reduction in computational cost up to about $70\%$ as $c$ is increased.