A finite volume method for undercompressive shock waves in two space dimensions

Chalons, Christophe; Rohde, Christian; Wiebe, Maria

doi:10.1051/m2an/2017027

Cited by 15 publications

(23 citation statements)

References 38 publications

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“…Consequently, it is important to resolve it accurately in numerical simulations. This can be achieved by using front-tracking algorithms such as the ghost-uid method [11,10], or front-capturing algorithms such as moving mesh methods [6]. For each of these numerical algorithms it is essential to describe the dynamics of the wave of interest precisely.…”

Section: Problem Descriptionmentioning

confidence: 99%

“…However, (6) might not hold if the solution of the Riemann problem (3) is computed from dissipative approximations, such as the Navier-Stokes equations or particle simulations [26]. Especially in the former case, the solution might be corrupted by noise, due to uctuations in the particle distribution, and therefore (6) might not hold true.…”

Section: Problem Descriptionmentioning

confidence: 99%

“…As (19) is a scalar law, we can resolve the constraint (23) by computing the correct wave speed s for a jump from u * − to u * + from (6). The resolving function Ψ (see Section 4.1) is therefore given by Figure 5: Example of a Riemann solution consisting of a rarefaction wave followed by an undercompressive shock wave.…”

Section: Cubic Flux Model: Undercompressive Wavementioning

confidence: 99%

“…To implement the numerical discretization of conservation laws while resolving discontinuous wave fronts within the mesh, we apply the one-dimensional version of the front-capturing scheme described in [6]. It is a nite volume scheme with moving edges, such that the position x Γ (t ) of a discontinuous wave is always be fully resolved within the mesh.…”

Section: Finite Volume Moving Mesh Front Capturing Schemementioning

confidence: 99%

“…In our case, we seek to keep track of discontinuous waves that connect regions with values in − and regions with values in + as illustrated in Figure 9. In the following we present the most relevant parts of the algorithm -for more details we refer to [6].…”

Section: Finite Volume Moving Mesh Front Capturing Schemementioning

confidence: 99%

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Constraint-aware neural networks for Riemann problems

Magiera

Ray

Hesthaven

et al. 2020

Journal of Computational Physics

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Neural networks are increasingly used in complex (data-driven) simulations as surrogates or for accelerating the computation of classical surrogates. In many applications physical constraints, such as mass or energy conservation, must be satis ed to obtain reliable results. However, standard machine learning algorithms are generally not tailored to respect such constraints. We propose two di erent strategies to generate constraint-aware neural networks. We test their performance in the context of front-capturing schemes for strongly nonlinear wave motion in compressible uid ow. Precisely, in this context so-called Riemann problems have to be solved as surrogates. Their solution describes the local dynamics of the captured wave front in numerical simulations. Three model problems are considered: a cubic ux model problem, an isothermal two-phase ow model, and the Euler equations. We demonstrate that a decrease in the constraint deviation correlates with low discretization errors for all model problems, in addition to the structural advantage of ful lling the constraint.

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Section: Problem Descriptionmentioning

confidence: 99%

Section: Problem Descriptionmentioning

confidence: 99%

Section: Cubic Flux Model: Undercompressive Wavementioning

confidence: 99%

Section: Finite Volume Moving Mesh Front Capturing Schemementioning

confidence: 99%

Section: Finite Volume Moving Mesh Front Capturing Schemementioning

confidence: 99%

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Constraint-aware neural networks for Riemann problems

Magiera

Ray

Hesthaven

et al. 2020

Journal of Computational Physics

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A Fully Conforming Finite Volume Approach to Two-Phase Flow in Fractured Porous Media

Burbulla

Rohde

2020

Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples

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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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A finite volume method for undercompressive shock waves in two space dimensions

Cited by 15 publications

References 38 publications

Constraint-aware neural networks for Riemann problems

Constraint-aware neural networks for Riemann problems

A Fully Conforming Finite Volume Approach to Two-Phase Flow in Fractured Porous Media

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

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