Entropy-satisfying relaxation method with large time-steps for Euler IBVPs

Abstract

Abstract. This paper could have been given the title: "How to positively and implicitly solve Euler equations using only linear scalar advections." The new relaxation method we propose is able to solve Euler-like systems-as well as initial and boundary value problems-with real state laws at very low cost, using a hybrid explicit-implicit time integration associated with the Arbitrary Lagrangian-Eulerian formalism. Furthermore, it possesses many attractive properties, such as: (i) the preservation of positivity for densities; (ii) the guarantee of min-max principle for mass fractions; (iii) the satisfaction of entropy inequality, under an expressible bound on the CFL ratio. The main feature that will be emphasized is the design of this optimal time-step, which takes into account data not only from the inner domain but also from the boundary conditions.

Large Time-Step Numerical Scheme for the Seven-Equation Model of Compressible Two-Phase Flows

“…This transformation aims at proposing in the next section an implicit-explicit Lagrange-Projection scheme similar to [6], and at treating separately the nonconservative products. Note from now on that the overall algorithm will be conservative on the partial mass α k ̺ k , total momentum α1̺1u1 + α2̺2u2 and on the total energy α1̺1e1 + α2̺2e2, as it is expected from the original form (1) of the model.…”

“…As already stated, our aim is to propose an implicit treatment of the fast waves u k ± c k , and an explicit treatment of u k . With this in mind, we follow [6] and adopt a Lagrange-Projection scheme, coupled with a pressure relaxation strategy that is well adapted to this purpose. A Lagrange-Projection splitting strategy applied to (4) amounts to introducing the Lagrangian system…”

“…The idea is then to propose a timeimplicit treatment of the (fast) acoustic waves, in order to get rid of a too restrictive CFL condition, together with an explicit treatment of the (slow) contact waves in order to preserve accuracy. This was recently proposed in Coquel et al [6] in the context of Euler equations, using a Lagrange-Projection approach. This approach is well-adapted as it naturally decouples the fast and slow waves in the Lagrange and Projection steps respectively.…”

“…The idea is to operate a relevant operator splitting between the conservative and nonconservative parts of the original model, in order to make Euler systems for each phase appear. The latter parts are treated as in [6]. The nonconservative products are then discretized so as to maintain conservativity properties of the model on each partial mass, on the total momentum and total energy.…”

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AbstractWe consider the seven-equation model for compressible two-phase flows and propose a large time-step numerical scheme based on a time implicit-explicit Lagrange-Projection strategy introduced in Coquel et al [6] for Euler equations. The main objective is to get a Courant-Friedrichs-Lewy (CFL) condition driven by (slow) contact waves instead of (fast) acoustic waves.

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