Hélène Mathis

Dynamic Model Adaptation for Multiscale Simulation of Hyperbolic Systems with Relaxation

In numerous industrial CFD applications, it is usual to use two (or more) different codes to solve a physical phenomenon: where the flow is a priori assumed to have a simple behavior, a code based on a coarse model is applied, while a code based on a fine model is used elsewhere. This leads to a complex coupling problem with fixed interfaces. The aim of the present work is to provide a numerical indicator to optimize to position of these coupling interfaces. In other words, thanks to this numerical indicator, one could verify if the use of the coarser model and of the resulting coupling does not introduce spurious effects. In order to validate this indicator, we use it in a dynamical multiscale method with moving coupling interfaces. The principle of this method is to use as much as possible a coarse model instead of the fine model in the computational domain, in order to obtain an accuracy which is comparable with the one provided by the fine model. We focus here on general hyperbolic systems with stiff relaxation source terms together with the corresponding hyperbolic equilibrium systems. Using a numerical Chapman–Enskog expansion and the distance to the equilibrium manifold, we construct the numerical indicator. Based on several works on the coupling of different hyperbolic models, an original numerical method of dynamic model adaptation is proposed. We prove that this multiscale method preserves invariant domains and that the entropy of the numerical solution decreases with respect to time. The reliability of the adaptation procedure is assessed on various 1D and 2D test cases coming from two-phase flow modeling.

Model Adaptation for Hyperbolic Systems with Relaxation

We address the numerical coupling of two hyperbolic systems, a relaxation model and the associated equilibrium model, separated by spatial interfaces that automatically evolve in time, the whole being approximated by finite volume schemes. The criterion to choose where each model has to be used results of the Chapman–Enskog expansion of the relaxed model, both on a continuous and a discrete view point. Numerical tests illustrate the good behavior of the algorithm.

Numerical Convergence for a Diffusive Limit of the Goldstein–Taylor System on Bounded Domain

This paper deals with the diffusive limit of the scaled Goldstein–Taylor model and its approximation by an Asymptotic Preserving Finite Volume scheme. The problem is set in some bounded interval with non-homogeneous boundary conditions depending on time. We obtain a uniform estimate in the small parameter \(\varepsilon \) using a relative entropy of the discrete solution with respect to a suitable profile which satisfies the boundary conditions expected to hold as \(\varepsilon \) goes to 0.

Error Estimate for Time-Explicit Finite Volume Approximation of Strong Solutions to Systems of Conservation Laws

We study the finite volume approximation of strong solutions to nonlinear systems of conservation laws. We focus on time-explicit schemes on unstructured meshes, with entropy satisfying numerical fluxes. The numerical entropy dissipation is quantified at each interface of the mesh, which enables to prove a weak–BV estimate for the numerical approximation under a strengthen CFL condition. Then we derive error estimates in the multidimensional case, using the relative entropy between the strong solution and its finite volume approximation. The error terms are carefully studied, leading to a classical

Modeling Phase Transition and Metastable Phases

h^1/4

Relative entropy for the finite volume approximation of strong solutions to systems of conservation laws

estimate in

OSAMOAL: OPTIMIZED SIMULATIONS BY ADAPTED MODELS USING ASYMPTOTIC LIMITS ∗

L^2

Error analysis of a dynamic model adaptation procedure for nonlinear hyperbolic equations

under this strengthen CFL condition.

NUMERICAL CONVERGENCE RATE FOR A DIFFUSIVE LIMIT OF HYPERBOLIC SYSTEMS: p-SYSTEM WITH DAMPING

We propose a model that describes phase transition including metastable phases present in the van der Waals Equation of State (EoS). We introduce a dynamical system that is able to depict the mass transfer between two phases, for which equilibrium states are both metastable and stable states, including mixtures. The dynamical system is then used as a relaxation source term in a isothermal two-phase model. We use a Finite Volume scheme (FV) that treats the convective part and the source term in a fractional step way. Numerical results illustrate the ability of the model to capture phase transition and metastable states.