FullyDiscrete Explicit Locally EntropyStable Schemes for the Compressible Euler and NavierStokes Equations
Abstract
Recently, relaxation methods have been developed to guarantee the preservation of a single global functional of the solution of an ordinary differential equation. Here, we generalize this approach to guarantee local entropy inequalities for finitely many convex functionals (entropies) and apply the resulting methods to the compressible Euler and NavierStokes equations. Based on the unstructured $hp$adaptive SSDC framework of entropy conservative or dissipative semidiscretizations using summationbyparts and simultaneousapproximationterm operators, we develop the first discretizations for compressible computational fluid dynamics that are primary conservative, locally entropy stable in the fully discrete sense under a usual CFL condition, explicit except for the parallelizable solution of a single scalar equation per element, and arbitrarily highorder accurate in space and time. We demonstrate the accuracy and the robustness of the fullydiscrete explicit locally entropystable solver for a set of test cases of increasing complexity.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 arXiv:
 arXiv:2003.08831
 Bibcode:
 2020arXiv200308831R
 Keywords:

 Mathematics  Numerical Analysis;
 Physics  Computational Physics;
 Physics  Fluid Dynamics;
 65M12;
 65M70;
 65L06;
 65L20;
 65P10;
 76M10;
 76M20;
 76M22;
 76N99
 EPrint:
 Computers and Mathematics with Applications, 2020