An asymptotic-preserving method for a relaxation of the Navier–Stokes–Korteweg equations

Chertock, Alina; Degond, Pierre; Neusser, Jochen

doi:10.1016/j.jcp.2017.01.030

Cited by 12 publications

(8 citation statements)

References 39 publications

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“…(because the pressure stays below p * = max(p * 1 , p * 2 ), see also (13)). During this period the GFM and the HSM are not synchronized.…”

Section: Discussionmentioning

confidence: 99%

“…satisfy assumptions (12) and (13). Suppose the limits of the densities n 1 , n 2 , of the pressure p and of q m as → 0, m → +∞ and α → 0 exist and are denoted by n ∞ 1 , n ∞ 2 , p ∞ and q ∞ .…”

Section: Formal Limitmentioning

confidence: 99%

“…The fourth order terms in ( 8) -( 11) are now replaced by second order terms supplemented with Poisson equations which define c 1 and c 2 . This type of relaxation has been used for examples, for the numerical approximation of the Navier-Stokes-Korteweg equations [13,41]. The Relaxation Gradient Flow Model (RGFM) formally converges toward the GFM when ν → 0.…”

Section: Relaxed Modelmentioning

confidence: 99%

See 2 more Smart Citations

Incompressible limit of a continuum model of tissue growth with segregation for two cell populations

Chertock

Degond

Hecht

et al. 2019

Mathematical Biosciences and Engineering

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This paper proposes a model for the growth two interacting populations of cells that do not mix. The dynamics is driven by pressure and cohesion forces on the one hand and proliferation on the other hand. Following earlier works on the single population case, we show that the model approximates a free boundary Hele Shaw type model that we characterise using both analytical and numerical arguments.

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“…(because the pressure stays below p * = max(p * 1 , p * 2 ), see also (13)). During this period the GFM and the HSM are not synchronized.…”

Section: Discussionmentioning

confidence: 99%

Section: Formal Limitmentioning

confidence: 99%

Section: Relaxed Modelmentioning

confidence: 99%

See 1 more Smart Citation

Incompressible limit of a continuum model of tissue growth with segregation for two cell populations

Chertock

Degond

Hecht

et al. 2019

Mathematical Biosciences and Engineering

Self Cite

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show abstract

“…First, one can substitute the Laplacian in the momentum balance of Eq. ( 20) by a convolution term (see [8,28]) that relies on analytical arguments from [9]. This approach necessitates the solution of an extra elliptic equation.…”

Section: Navier-stokes-korteweg (Nsk) Equations For Two-phase Flowmentioning

confidence: 99%

Analysis and Numerics of Sharp and Diffuse Interface Models for Droplet Dynamics

Magiera

Rohde

2022

Fluid Mechanics and Its Applications

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The modelling of liquid–vapour flow with phase transition poses many challenges, both on the theoretical level, as well as on the level of discretisation methods. Therefore, accurate mathematical models and efficient numerical methods are required. In that, we focus on two modelling approaches: the sharp-interface (SI) approach and the diffuse-interface (DI) approach. For the SI-approach, representing the phase boundary as a co-dimension-1 manifold, we develop and validate analytical Riemann solvers for basic isothermal two-phase flow scenarios. This ansatz becomes cumbersome for increasingly complex thermodynamical settings. A more versatile multiscale interface solver, that is based on molecular dynamics simulations, is able to accurately describe the evolution of phase boundaries in the temperature-dependent case. It is shown to be even applicable to two-phase flow of multiple components. Despite the successful developments for the SI approach, these models fail if the interface undergoes topological changes. To understand merging and splitting phenomena for droplet ensembles, we consider DI models of second gradient type. For these Navier–Stokes–Korteweg systems, that can be seen as a third order extension of the Navier–Stokes equations, we propose variants that are more accessible to standard numerical schemes. More precisely, we reformulate the capillarity operator to restore the hyperbolicity of the Euler operator in the full system.

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“…Since then, AP schemes have been developed for a wide variety of PDE systems including hyperbolic conservation laws, [13,12,20,17]. Moreover, an increasing number of authors has focused on the development of AP schemes for Euler and Navier-Stokes equations, see [128,123,61,68,16,56,22,21,1,2,139] and references therein. This research has set the basis for novel developments of AP schemes for the shallow water equations.…”

Section: Introductionmentioning

confidence: 99%

A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers

Busto,

Dumbser

2023

Preprint

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We present a novel staggered semi-implicit hybrid finite volume / finite element method for the numerical solution of the shallow water equations at all Froude numbers on unstructured meshes. A semi-discretization in time of the conservative Saint-Venant equations with bottom friction terms leads to its decomposition into a first order hyperbolic subsystem containing the nonlinear convective term and a second order wave equation for the pressure. For the spatial discretization of the free surface elevation an unstructured mesh composed of triangular simplex elements is considered, whereas a dual grid of the edge-type is employed for the computation of the depth-averaged momentum vector. The first stage of the proposed algorithm consists in the solution of the nonlinear convective subsystem using an explicit Godunov-type finite volume method on the staggered dual grid. Next, a classical continuous finite element scheme provides the free surface elevation at the vertices of the primal simplex mesh. The semi-implicit strategy followed in this paper circumvents the contribution of the surface wave celerity to the CFL-type time step restriction, hence making the proposed algorithm well-suited for the solution of low Froude number flows. It can be shown that in the low Froude number limit the proposed algorithm reduces to a semi-implicit hybrid FV/FE projection method for the incompressible Navier-Stokes equations. At the same time, the conservative formulation of the governing equations also allows the discretization of high Froude number flows with shock waves. As such, the new hybrid FV/FE scheme can be considered an all-Froude number solver, able to deal simultaneously with both, subcritical as well as supercritical flows. Besides, the algorithm is well balanced by construction. The accuracy of the overall methodology is studied numerically and the C-property is proven theoretically and is also validated via numerical experiments. The numerical solution of several Riemann problems, including flat bottom and a bottom with jumps, attests the robustness of the new method to deal also with flows containing bores and discontinuities. Finally, a 3D dam break problem over a dry bottom is studied and our numerical results are successfully compared with numerical reference solutions obtained for the 3D free surface Navier-Stokes equations and with available experimental reference data.

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An asymptotic-preserving method for a relaxation of the Navier–Stokes–Korteweg equations

Cited by 12 publications

References 39 publications

Incompressible limit of a continuum model of tissue growth with segregation for two cell populations

Incompressible limit of a continuum model of tissue growth with segregation for two cell populations

Analysis and Numerics of Sharp and Diffuse Interface Models for Droplet Dynamics

A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers

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