Second-order invariant domain preserving approximation of the Euler equations using convex limiting
For computational fluid dynamics, this provides a provably invariant domain preserving second-order scheme, addressing a key challenge in hyperbolic conservation laws.
A second-order method (GMS-GV2) for the compressible Euler equations preserves all known invariant domains (positivity of density, internal energy, and entropy minimum principle) by combining a first-order invariant domain preserving method with convex limiting. Numerical tests confirm second-order accuracy in the maximum norm.
A new second-order method for approximating the compressible Euler equations is introduced. The method preserves all the known invariant domains of the Euler system: positivity of the density, positivity of the internal energy and the local minimum principle on the specific entropy. The technique combines a first-order, invariant domain preserving, Guaranteed Maximum Speed method using a Graph Viscosity (GMS-GV1) with an invariant domain violating, but entropy consistent, high-order method. Invariant domain preserving auxiliary states, naturally produced by the GMS-GV1 method, are used to define local bounds for the high-order method which is then made invariant domain preserving via a convex limiting process. Numerical tests confirm the second-order accuracy of the new GMS-GV2 method in the maximum norm, where 2 stands for second-order. The proposed convex limiting is generic and can be applied to other approximation techniques and other hyperbolic systems.