High order ADER schemes and GLM curl cleaning for a first order hyperbolic formulation of compressible flow with surface tension

Chiocchetti, Simone; Peshkov, Ilya; Gavrilyuk, Sergey; Dumbser, Michael

doi:10.1016/j.jcp.2020.109898

Cited by 33 publications

(25 citation statements)

References 141 publications

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“…has been added in order to restore the Galilean invariance of the system, see [27,47,60,92]. We note that the mathematical structure of system ( 15) is very similar to the one of (1)-( 5), apart from the additional field p k that was not present in the hyperbolic reformulation of the Serre-Green-Naghdi model.…”

Section: Hyperbolic Reformulation Of the Nonlinear Schrödinger Equationmentioning

confidence: 93%

“…In order to fix this issue, in Sect. 2.2.3 we will introduce an augmented system that accounts for the curl involution on p via a hyperbolic generalized Lagrangian multiplied (GLM) curl cleaning approach, which goes back to the ideas forwarded by Munz et al [35,83] for divergence-type constraints and in [27,46] for curl-type constraints.…”

Section: Eigenstructure Of the Original Weakly Hyperbolic Systemmentioning

confidence: 99%

“…The hyperbolic reformulation of the nonlinear Schrödinger equation [38] has the additional difficulties that it is only weakly hyperbolic in multiple space dimensions and that it is endowed with a curl involution constraint that needs to be properly accounted for. In this paper we will make use of the hyperbolic generalized Lagrangian multiplier (GLM) curl cleaning approach of [27,46] that goes back to the hyperbolic GLM divergence cleaning technique of Munz et al for Maxwell and MHD equations, see [35,83]. In alternative to the GLM method proposed here, also exactly curl-preserving schemes could be used, see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…In alternative to the GLM method proposed here, also exactly curl-preserving schemes could be used, see e.g. [1,11,27,66,67,71], but they require an appropriately staggered mesh and are therefore not as easy to implement in an existing general purpose DG solver as the simple GLM method, which only requires the solution of additional PDEs for the cleaning quantities. The main novelty proposed in the present paper is a new GLM curl cleaning that is also thermodynamically compatible with the conservation of total energy.…”

Section: Introductionmentioning

confidence: 99%

“…Compared to the work on the hyperbolic reformulation of the Schrödinger equation [38] in this paper an additional term is added to the system in order to restore Galilean invariance and also the curl involution constraint of the system is explicitly taken into account, which is necessary for the multi-dimensional case. As already mentioned previously, in this paper we will make use of a hyperbolic generalized Lagrangian multiplier (GLM) approach, see [27,35,46,83], which also restores strong hyperbolicity of the system.…”

Section: Introductionmentioning

confidence: 99%

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On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems

et al. 2021

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This paper is on arbitrary high order fully discrete one-step ADER discontinuous Galerkin schemes with subcell finite volume limiters applied to a new class of first order hyperbolic reformulations of nonlinear dispersive systems based on an extended Lagrangian approach introduced by Dhaouadi et al. (Stud Appl Math 207:1–20, 2018), Favrie and Gavrilyuk (Nonlinearity 30:2718–2736, 2017). We consider the hyperbolic reformulations of two different nonlinear dispersive systems, namely the Serre–Green–Naghdi model of dispersive water waves and the defocusing nonlinear Schrödinger equation. The first order hyperbolic reformulation of the Schrödinger equation is endowed with a curl involution constraint that needs to be properly accounted for in multiple space dimensions. We show that the original model proposed in Dhaouadi et al. (2018) is only weakly hyperbolic in the multi-dimensional case and that strong hyperbolicity can be restored at the aid of a novel thermodynamically compatible GLM curl cleaning approach that accounts for the curl involution constraint in the PDE system. We show one and two-dimensional numerical results applied to both systems and compare them with available exact, numerical and experimental reference solutions whenever possible.

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Section: Hyperbolic Reformulation Of the Nonlinear Schrödinger Equationmentioning

confidence: 93%

Section: Eigenstructure Of the Original Weakly Hyperbolic Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems

et al. 2021

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Modelling interactions between waves and diffused interfaces

Schmidmayer

Cazé

Petitpas

et al. 2022

Numerical Methods in Fluids

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When simulating multiphase compressible flows using the diffuse-interface methods, the test cases presented in the literature to validate the modellings with regard to interface problems are always textbook cases: interfaces are sharp and the simulations therefore easily converge to the exact solutions. In real problems, it is rather different because the waves encounter moving interfaces which consequently have already undergone the effects of numerical diffusion. Numerical solutions resulting from the interactions of waves with diffused interfaces have never been precisely investigated and for good reasons, the results obtained are extremely dependent on the model used. Precisely, well-posed models present similar and important issues when such an interaction occurs, coming from the appearance of a wave-trapping phenomenon.To circumvent those issues, we propose to use a thermodynamically-consistent pressure-disequilibrium model with finite, instead of infinite, pressure-relaxation rate to overcome the difficulties inherent in the computation of these interactions. Because the original method to solve this model only enables infinite relaxation, we propose a new numerical method allowing infinite as well as finite relaxation rates. Solutions of the new modelling are examined and compared to literature, in particular we propose the study of a shock on a water-air interface, but also for problems of helium-air and water-air shock tubes, spherical and non-spherical bubble collapses.

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On Numerical Methods for Hyperbolic PDE with Curl Involutions

Dumbser

Chiocchetti

Peshkov

2020

Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov's Legacy

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In this paper we present three different numerical approaches to account for curl-type involution constraints in hyperbolic partial differential equations for continuum physics. All approaches have a direct analogy to existing and wellknown divergence-preserving schemes for the Maxwell and MHD equations. The first method consists in a generalization of the Godunov-Powell terms, which means adding suitable multiples of the involution constraints to the PDE system in order to achieve the symmetric Godunov form. The second method is an extension of the generalized Lagrangian multiplier (GLM) approach of Munz et al., where the numerical errors in the involution constraint are propagated away via an augmented PDE system. The last method is an exactly involution preserving discretization, similar to the exactly divergence-free schemes for the Maxwell and MHD equations, making use of appropriately staggered meshes. We present some numerical results that allow to compare all three approaches with each other.

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High order ADER schemes and GLM curl cleaning for a first order hyperbolic formulation of compressible flow with surface tension

Cited by 33 publications

References 141 publications

On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems

On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems

Modelling interactions between waves and diffused interfaces

On Numerical Methods for Hyperbolic PDE with Curl Involutions

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