Second-Order Invariant Domain Preserving Approximation of the Euler Equations Using Convex Limiting

Abstract

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.1 γ−1 ρ −1 ) for a polytropic ideal gas. Using the notation s e (ρ, e) := ∂s ∂e (ρ, e) and s ρ (ρ, e) := ∂s ∂ρ (ρ, e), the equation of state then takes the form p := −ρ 2 s ρ s −1 e . To simplify the notation we introduce the Note that (2.9) implies j∈{1: I} c ij = 0. Furthermore, if either ϕ i or ϕ j is zero on ∂D, then c ij = −c ji . In particular we have i∈{1: I} c ij = 0 if ϕ j is zero on ∂D. This property will be used to establish conservation.

High Order ADER Schemes for Continuum Mechanics

“…the resolution of DG in smooth regions and the robustness of FV across discontinuities. Thus, we first evolve the solution everywhere by using our DG scheme; then, we check a posteriori, at the end of each time step, if the obtained DG solution in each cell respects or not some criteria (as density and pressure positivity, a relaxed discrete maximum principle, specific physical bounds, or more elaborate choices as those of [105]), and we mark as troubled those cells where the obtained DG solution is marked as not acceptable. Only for these troubled cells we repeat the time step using, instead of the DG scheme, a second order TVD FV method, which always assures a robust solution.…”

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

“…Entropy stability [8,29,33,34] and preservation of invariant domains [14,16,19] play an important role in the design of numerical methods for nonlinear hyperbolic conservation laws. A failure to comply with these design criteria may result in nonphysical artefacts and/or convergence to wrong weak solutions.…”

Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws

“…Let u(x, t) be a scalar quantity of interest depending on the space location x ∈ R d , d ∈ {1, 2, 3} and time instant t ≥ 0. Consider an initial-boundary value problem of the form [17,33] ∂u ∂t…”

“…As we show in the next section, the bar state form (17) of (15) is also ideally suited for the derivation of high-order extensions that preserve the IDP property using built-in flux limiters.…”

“…This approach, which is described in §5, involves the solution of small sparse linear systems on each macroelement. The IDP property of the corresponding discrete problem is shown using the proof techniques developed in [17,33]. In §6 and §7, we discuss the optional stabilization techniques for the high-order target flux and Hessian-based smoothness indicators that preserve the high-order accuracy near smooth local extrema.…”