Entropy viscosity method for nonlinear conservation laws

Guermond, Jean‐Luc; Pasquetti, Richard; Popov, Bojan

doi:10.1016/j.jcp.2010.11.043

Cited by 324 publications

(324 citation statements)

References 31 publications

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“…Several directions exist to cure this problem. For instance, the artificial viscosity necessary for the scheme to converge could be monitored by a posteriori indicators, following the ideas developed in [12,13]. Another route, especially for the explicit variant of the scheme, is to implement MUSCL techniques; this work is underway.…”

Section: Resultsmentioning

confidence: 99%

“…Since combining a centred discretization of the momentum balance equation with the addition of an artificial viscosity may seem to be an appealing technique to avoid an excessive numerical dissipation (for instance, associated to an adjustment of µ as a function of the regularity of the solution, in the spirit of [12,13]), we now investigate the influence of the value of µ on the accuracy of the centred scheme. We observe in Figure 7 and Figure 8 that taking a large viscosity yields inaccurate results, which is easily explained by the fact that the solved problem is too far from the original one.…”

Section: Sod Shock Tube Problemmentioning

confidence: 99%

“…To this purpose, we use in this section a "barotropic one-phase version", which we then extend to obtain a two-phase flow problem, of a test which is classical for (non-barotropic one-phase) Euler equations [29,13], and is often referred to as "the Mach 3 wind tunnel with step".…”

Section: Wind Tunnel With a Stepmentioning

confidence: 99%

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Pressure correction staggered schemes for barotropic one-phase and two-phase flows

Kheriji¹,

Herbin²,

Latché³

2013

Computers & Fluids

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SummaryWe assess in this paper the capability of a pressure correction scheme to compute shock solutions of the homogeneous model for barotropic two-phase flows. This scheme is designed to inherit the stability properties of the continuous problem: the unknowns (in particular the density and the dispersed phase mass fraction y) are kept within their physical bounds, and the entropy of the system is conserved, thus providing an unconditional stability property. In addition, the scheme keeps the velocity and pressure constant through contact discontinuities. These properties are obtained by coupling the mass balance and the transport equation for y in an original pressure correction step. The space discretization is staggered; the numerical schemes which are considered are the Marker-And Cell (MAC) finite volume scheme and the nonconforming low-order Rannacher-Turek and Crouzeix-Raviart finite element approximation. In either case, a finite volume technique is used for all convection terms. Numerical experiments performed here show that, provided that a sufficient dissipation is introduced in the scheme, it converges to the (weak) solution of the continuous hyperbolic system. Observed orders of convergence for 1D Riemann problems as a function of the mesh and time step at constant CFL number vary with the studied case, and the CFL number and on the regularity of the solution. They range from 0.5 to greater than 1 for the velocity and the pressure; in most cases, the density and mass fraction converge with a 0.5 order. Finally, the scheme shows a satisfactory behaviour up to large CFL numbers.

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Section: Resultsmentioning

confidence: 99%

Section: Sod Shock Tube Problemmentioning

confidence: 99%

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Pressure correction staggered schemes for barotropic one-phase and two-phase flows

Kheriji¹,

Herbin²,

Latché³

2013

Computers & Fluids

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“…Unfortunately, this does not occur efficiently on contact discontinuities where only numerical diffusion is active. Nonetheless, the work by Guermond et al [10] provides a remarkable and simple way to solve this problem by making the numerical diffusion to be proportional to the numerical density of entropy production (i.e. adding large numerical dissipation in the shock regions and almost no dissipation in the regions where the solution remains smooth).…”

Section: Introductionmentioning

confidence: 99%

Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

Ersoy¹,

Golay²,

Yushchenko³

2013

Open Mathematics

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We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). This density is used as an a posteriori error which provides information if the mesh should be refined in the regions where discontinuities occur or coarsened in the regions where the solution remains smooth. As due to the Courant-Friedrich-Levy stability condition the time step is restricted and leads to time consuming simulations, we propose a local time stepping algorithm. We also use high order time extensions applying the Adams-Bashforth time integration technique as well as the second order linear reconstruction in space. We numerically investigate the efficiency of the scheme through several test cases: Sod's shock tube problem, Lax's shock tube problem and the Shu-Osher test problem. MSC:

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“…We leave as a subject of future studies the implementation of more recent and more sophisticated damping methods, e.g. entropy viscosity method as described in Guermond et al (2011).…”

Section: Nonlinearities and Their Stabilizationmentioning

confidence: 99%

Flexible parallel implicit modelling of coupled thermal–hydraulic–mechanical processes in fractured rocks

Cacace

Jacquey

2017

Solid Earth

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Abstract. Theory and numerical implementation describing groundwater flow and the transport of heat and solute mass in fully saturated fractured rocks with elasto-plastic mechanical feedbacks are developed. In our formulation, fractures are considered as being of lower dimension than the hosting deformable porous rock and we consider their hydraulic and mechanical apertures as scaling parameters to ensure continuous exchange of fluid mass and energy within the fracturesolid matrix system. The coupled system of equations is implemented in a new simulator code that makes use of a Galerkin finite-element technique. The code builds on a flexible, object-oriented numerical framework (MOOSE, Multiphysics Object Oriented Simulation Environment) which provides an extensive scalable parallel and implicit coupling to solve for the multiphysics problem. The governing equations of groundwater flow, heat and mass transport, and rock deformation are solved in a weak sense (either by classical Newton-Raphson or by free Jacobian inexact NewtonKrylow schemes) on an underlying unstructured mesh. Nonlinear feedbacks among the active processes are enforced by considering evolving fluid and rock properties depending on the thermo-hydro-mechanical state of the system and the local structure, i.e. degree of connectivity, of the fracture system. A suite of applications is presented to illustrate the flexibility and capability of the new simulator to address problems of increasing complexity and occurring at different spatial (from centimetres to tens of kilometres) and temporal scales (from minutes to hundreds of years).

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Entropy viscosity method for nonlinear conservation laws

Cited by 324 publications

References 31 publications

Pressure correction staggered schemes for barotropic one-phase and two-phase flows

Pressure correction staggered schemes for barotropic one-phase and two-phase flows

Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

Flexible parallel implicit modelling of coupled thermal–hydraulic–mechanical processes in fractured rocks

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