A hyperbolic model for viscous Newtonian flows

Peshkov, Ilya; Romenski, Evgeniy

doi:10.1007/s00161-014-0401-6

Cited by 141 publications

(227 citation statements)

References 61 publications

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“…[106][107][108]116,117 In the near future we also plan an extension of this new family of efficient semi-implicit finite volume schemes to the unified Godunov-Peshkov-Romenski model of continuum mechanics 99,115,118 and to the Baer-Nunziato model of compressible multiphase flows, [119][120][121] where low Mach number problems are particularly important due to the simultaneous presence of two different phases. [106][107][108]116,117 In the near future we also plan an extension of this new family of efficient semi-implicit finite volume schemes to the unified Godunov-Peshkov-Romenski model of continuum mechanics 99,115,118 and to the Baer-Nunziato model of compressible multiphase flows, [119][120][121] where low Mach number problems are particularly important due to the simultaneous presence of two different phases.…”

Section: Discussionmentioning

confidence: 99%

“…Future work will consist in an extension of the present approach to general unstructured meshes in multiple space dimensions and to higher order of accuracy at the aid of staggered semi-implicit DG finite element schemes, following the ideas outlined in other works. [106][107][108]116,117 In the near future we also plan an extension of this new family of efficient semi-implicit finite volume schemes to the unified Godunov-Peshkov-Romenski model of continuum mechanics 99,115,118 and to the Baer-Nunziato model of compressible multiphase flows, [119][120][121] where low Mach number problems are particularly important due to the simultaneous presence of two different phases.…”

Section: Discussionmentioning

confidence: 99%

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A divergence‐free semi‐implicit finite volume scheme for ideal, viscous, and resistive magnetohydrodynamics

Dumbser

Balsara

Tavelli

et al. 2018

Numerical Methods in Fluids

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Summary In this paper, we present a novel pressure‐based semi‐implicit finite volume solver for the equations of compressible ideal, viscous, and resistive magnetohydrodynamics (MHD). The new method is conservative for mass, momentum, and total energy, and in multiple space dimensions, it is constructed in such a way as to respect the divergence‐free condition of the magnetic field exactly, also in the presence of resistive effects. This is possible via the use of multidimensional Riemann solvers on an appropriately staggered grid for the time evolution of the magnetic field and a double curl formulation of the resistive terms. The new semi‐implicit method for the MHD equations proposed here discretizes the nonlinear convective terms as well as the time evolution of the magnetic field explicitly, whereas all terms related to the pressure in the momentum equation and the total energy equation are discretized implicitly, making again use of a properly staggered grid for pressure and velocity. Inserting the discrete momentum equation into the discrete energy equation then yields a mildly nonlinear symmetric and positive definite algebraic system for the pressure as the only unknown, which can be efficiently solved with the (nested) Newton method of Casulli et al. The pressure system becomes linear when the specific internal energy is a linear function of the pressure. The time step of the scheme is restricted by a CFL condition based only on the fluid velocity and the Alfvén wave speed and is not based on the speed of the magnetosonic waves. Being a semi‐implicit pressure‐based scheme, our new method is therefore particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, for which it is well known that explicit density‐based Godunov‐type finite volume solvers become increasingly inefficient and inaccurate because of the more and more stringent CFL condition and the wrong scaling of the numerical viscosity in the incompressible limit. We show a relevant MHD test problem in the low Mach number regime where the new semi‐implicit algorithm is a factor of 50 faster than a traditional explicit finite volume method, which is a very significant gain in terms of computational efficiency. However, our numerical results confirm that our new method performs well also for classical MHD test cases with strong shocks. In this sense, our new scheme is a true all Mach number flow solver.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

A divergence‐free semi‐implicit finite volume scheme for ideal, viscous, and resistive magnetohydrodynamics

Dumbser

Balsara

Tavelli

et al. 2018

Numerical Methods in Fluids

View full text Add to dashboard Cite

show abstract

“…Good progress is being made at the moment on a steady-state version, and on limiting methods. The principle challenge ahead is the extension to Navier-Stokes, and the intention is to adopt the hyperbolic versions of Navier-Stokes recently put forward by Nishikawa [25] and by Peshkov and Romenski [28]. This will involve considering, as in solid mechanics, both compressive and torsional waves, and it is encouraging that these will also commute, and may be treated independently.…”

Section: Discussionmentioning

confidence: 99%

Is Discontinuous Reconstruction Really a Good Idea?

Roe

2017

J Sci Comput

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It has been almost automatically assumed for a quarter century that the numerical solution of hyperbolic conservation laws is best accomplished by making a reconstruction of the initial data that is only piecewise continuous. The effect of the discontinuities is taken into account by means of Riemann solvers. This strategy has enjoyed great practical success but introduces only one-dimensional physics as a guide to the discretization of multidimensional problems. This article points out some of the resulting defects and proposes an alternative viewpoint. The chief novelty of the new "Active Flux" method, apart from the elimination of discontinuities, is the division into advective and acoustic disturbances, with acoustics being handled by exploiting classical solutions to the scalar wave equation. Results for a standard test problem for the compressible Euler equations indicate that an order of magnitude increase in cost effectiveness may be possible by following this approach, compared to the traditional one.

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“…That is why we consider distortion A as a measure of deformation of the whole medium and do not consider deformation of the skeleton separately. In this context, we recall that our unified continuum model for fluids and solids [29,8,7] also relies on a deformation-based rather than a strain-rate-based description of fluid flows.…”

Section: Comparison Of Shtc and Biot Models 41 Theoretical Comparisonmentioning

confidence: 99%

Modeling of Multiphase Flow in Elastic Porous Media Based on Thermodynamically Compatible Systems Theory

Romenskiy¹,

Perepechko²,

Reshetova³

2014

Proceedings

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A multiphase model and its application to wavefields numerical simulation are discussed in the context of modeling of compressible fluid flows in elastic porous media. The derivation of the model is based on a theory of thermodynamically compatible systems and on a model of nonlinear elastoplasticity combined with a multiphase compressible fluid flow model. The governing equations of the model include phase mass conservation laws, a total momentum conservation law, an equation for the relative velocities of the phases, an equation for mixture distortion, and a balance equation for porosity. They form a hyperbolic system of conservation equations that satisfy the fundamental laws of thermodynamics. Two types of phase interaction are introduced in the model: phase pressure relaxation to a common value and interfacial friction. Inelastic deformations also can be accounted for by source terms in the equation for distortion. The thus formulated model can be used for studying general compressible fluid flows in a deformable elastoplastic porous medium, and for modeling wave propagation in a saturated porous medium. Governing equations for small-amplitude wave propagation in a uniform porous medium saturated with a single fluid are derived. They form a first-order hyperbolic PDE system written in terms of stress and velocities and, like in Biot's model, predict three type of waves existing in real fluid-saturated porous media: fast and slow longitudinal waves and a shear wave. For the numerical solution of these equations, an efficient numerical method based on a staggered-grid finite difference scheme is used. The results of solving some numerical test problems are presented and discussed.

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A hyperbolic model for viscous Newtonian flows

Cited by 141 publications

References 61 publications

A divergence‐free semi‐implicit finite volume scheme for ideal, viscous, and resistive magnetohydrodynamics

A divergence‐free semi‐implicit finite volume scheme for ideal, viscous, and resistive magnetohydrodynamics

Is Discontinuous Reconstruction Really a Good Idea?

Modeling of Multiphase Flow in Elastic Porous Media Based on Thermodynamically Compatible Systems Theory

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