A conservative Lagrangian-Eulerian finite volume approximation method for balance law problems

Abstract

We present a simple numerical method based on a Lagrangian-Eulerian framework for approximate solutions of nonlinear balance law problems. This framework has been used for numerically solving partial differential equations of several types, such as hyperbolic conservation laws [3,8], balance laws problems [4]. As in [3,5] the mass conservation takes place in the space-time volume D n j , and this region in the form of [3] is used to define naturally a balance law. This balance law is the central idea to build a efficient numerical method to approximate solution to balance law problems. Verification of the technique is also made by comparison with analytical solutions when they are available.

On the Conservation Properties in Multiple Scale Coupling and Simulation for Darcy Flow with Hyperbolic-Transport in Complex Flows

“…Conservation properties of the no flow surface region for hyperbolic conservation laws. The aim of this section is to present an extension of the Lagragian-Eulerian scheme (see [17,18,19,20,21,22]) for hyperbolic conservation laws in two-space dimensions with some initial condition coming from abstract nonlinear problems of hyperbolic conservation laws. We can also consider problems of physical interest in fluid mechanics such as multiscale flow in porous media scalar and systems treated in this work.…”

“…It turn out that our monotone Lagrangian-Eulerian is a building block for construction of a novel class of Lagrangian-Eulerian shock-capturing schemes for first-order hyperbolic problems. The early monotone versions of the Lagrangian-Eulerian approach has been employed successfully in a number of very non-trivial problems and also developed theoretically [17,20,18,19,21,22] linked to several transport models such as the Burgers' equation with Greenberg-LeRoux's and Riccati's source terms, the shallow-water system, Broadwell's rarefied gas dynamics, Baer-Nunziato's system linear, non-linear convex and non-linear non-convex 2D scalar conservation laws (see [18,17]). It is worth mentioning that the Lagrangian-Eulerian framework is able to compute qualitatively correct (entropy) solutions involving intricate non-linear wave interactions of rarefaction and shock waves.…”

“…It is worth mentioning that the Lagrangian-Eulerian framework is able to compute qualitatively correct (entropy) solutions involving intricate non-linear wave interactions of rarefaction and shock waves. It is of significance to mention that the scheme is able to handle resonance effect associated to non classical transitional shock in a 2×2 three-phase flow water-oil-gas system [18,19] and an intricate shock structure linked to a 5 × 5 isentropic Baer-Nunziato model (see [17]). The Lagrangian-Eulerian scheme does handle properly the sonic rarefaction linked to Burgers' equation, namely, a typical small (and unphysical) discontinuity jump within the rarefaction structure; such discontinuation in the solution is unphysical, and thus with no mathematical relation with an entropy violating shock.…”

“…We assume there is not flow through the surface S n i,j (that is, S n i,j is impervious; this is natural in many applications [17,18,19,20,22]). Therefore surface integral of S n i,j is zero, i.e., (3.8)…”

“…In particular is also applicable to the Generalized Multiscale Finite Element Method (GMsFEM); see [16,74,119,75,118] and references therein. On the other hand, the new Lagrangian-Eulerian method seems to be a promissing general approach to capture nonlinear wave interactions linked to multiscale behavior of interactions of waves in a wide range of models and, in particular, for systems (see [17,18,19,20,21,22]). Numerical results show that our combined multiscale approach provides an accurate and robust procedure to study phenomena which couple distinct length or time scales.…”