Journal
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Volume 40, Issue 5, Pages A3211-A3239Publisher
SIAM PUBLICATIONS
DOI: 10.1137/17M1149961
Keywords
hyperbolic systems; Riemann problem; invariant domain; entropy inequality; high-order method; exact rarefaction; quasi-convexity; limiting; finite element method
Categories
Funding
- National Science Foundation [DMS-1619892, DMS-1620058]
- Air Force Office of Scientific Research, USAF [FA9550-15-1-0257]
- Army Research Office [W911NF-15-1-0517]
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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 the 2 stands for second-order. The proposed convex limiting is generic and can be applied to other approximation techniques and other hyperbolic systems.
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