A Space-time Approach to Entropy-Stable Discontinuous Galerkin and Flux Reconstruction
This work addresses computational efficiency and stability in numerical simulations for fluid dynamics, though it appears incremental as it builds on existing flux reconstruction and discontinuous Galerkin methods.
The authors tackled the problem of achieving fully-discrete entropy stability in high-order space-time discretizations for partial differential equations, resulting in a method that reduces computational cost by up to 70% while maintaining optimal convergence rates for linear advection and Euler equations.
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.