A structure-preserving staggered semi-implicit finite volume scheme for continuum mechanics

Boscheri, Walter; Dumbser, Michael; Ioriatti, Matteo; Peshkov, Ilya; Romenski, Evgeniy

doi:10.1016/j.jcp.2020.109866

Cited by 45 publications

(59 citation statements)

References 136 publications

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“…In the future we plan to extend the new family of thermodynamically compatible schemes to the equations of nonlinear hyperelasticity [14,67,75,77,87,102] and to the unified hyperbolic model of continuum mechanics [13,17,47,93,102], as well as to hyperbolic reformulations of dispersive systems [7,18,37,58]. Further work will also concern the extension of the discrete Godunov formalism presented in this paper to higher order semi-discrete discontinuous Galerkin finite element schemes, see e.g.…”

Section: Discussionmentioning

confidence: 99%

“…Another open challenge remains the development of thermodynamically compatible schemes like those presented in this paper that also maintain curl and divergence involution constraints exactly at the semi-discrete level, similar to the structure-preserving semi-implicit method recently proposed in [13], but which was not thermodynamically compatible. 2h .…”

Section: Discussionmentioning

confidence: 99%

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On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

et al. 2021

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In this paper we propose a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows recently proposed in Gavrilyuk et al. (J Comput Phys 366:252–280, 2018). The novelty of the formulation forwarded here is the use of a new evolution variable that guarantees the trace of the discrete Reynolds stress tensor to be always non-negative. The mathematical model is particularly challenging because one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. Therefore, in this paper we first consider a thermodynamically compatible viscous extension of the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamically compatible numerical scheme. We then introduce two different, but related, families of numerical methods for its solution. The first scheme is a provably thermodynamically compatible semi-discrete finite volume scheme that makes direct use of the Godunov form of the equations and can therefore be called a discrete Godunov formalism. The new method mimics the underlying continuous viscous system exactly at the semi-discrete level and is thus consistent with the conservation of total energy, with the entropy inequality and with the vanishing viscosity limit of the model. The second scheme is a general purpose high order path-conservative ADER discontinuous Galerkin finite element method with a posteriori subcell finite volume limiter that can be applied to the inviscid as well as to the viscous form of the model. Both schemes have in common that they make use of path integrals to define the jump terms at the element interfaces. The different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in Gavrilyuk et al. (2018) on the example of three Riemann problems. Moreover, we make the comparison with a fully resolved solution of the underlying viscous system with small viscosity parameter (vanishing viscosity limit). In all cases an excellent agreement between the different schemes is achieved. We furthermore show numerical convergence rates of ADER-DG schemes up to sixth order in space and time and also present two challenging test problems for the model where we also compare with available experimental data.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

et al. 2021

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“…In the future, we plan to apply the numerical method developed in this paper also to other nonlinear dispersive systems, such as the Navier-Stokes-Korteweg system, for which a hyperbolic reformulation is possible by combining the GPR model of continuum mechanics [17,47,88] with the hyperbolic reformulation of the nonlinear Schrödinger equation introduced in [38] and studied in the present paper. As further research, we also plan to develop novel structure-preserving semi-implicit schemes on staggered meshes to solve the systems under investigation in this paper, following the ideas outlined in [9,11,18,19], with the aim to preserve the curl constraint exactly at the discrete level.…”

Section: Discussionmentioning

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%

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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“…Regarding the implicit treatment of acoustic waves, pressure based semi‐implicit schemes on staggered grids date back to Harlow and Welch, 14 and became soon widely used, especially in the last decades, see for example, References 15‐31 to mention a few applications to incompressible fluid‐dynamics. In so‐called pressure‐based methods, the pressure field is obtained implicitly by solving a simple system of linear algebraic equations.…”

Section: Introductionmentioning

confidence: 99%

A novel structure preserving semi‐implicit finite volume method for viscous and resistive magnetohydrodynamics

Fambri

2021

Numerical Methods in Fluids

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In this work we introduce a novel semi‐implicit structure‐preserving finite‐volume/finite‐difference scheme for the viscous and resistive equations of magneto‐hydrodynamics (VRMHD) based on an appropriate 3‐split of the governing PDE system, which is decomposed into a first convective subsystem, a second subsystem involving the coupling of the velocity field with the magnetic field, and a third subsystem involving the pressure‐velocity coupling. The nonlinear convective terms are discretized explicitly, while the remaining two subsystems accounting for the Alfvén waves and the magneto‐acoustic waves are treated implicitly. Thanks to this, the final algorithm is at least formally constrained only by a mild CFL stability condition depending on the velocity field of the pure hydrodynamic convection. To preserve the divergence‐free constraint of the magnetic field exactly at the discrete level, a proper set of overlapping dual (staggered) meshes is employed. The resulting linear algebraic systems are shown to be symmetric and therefore can be solved by means of an efficient standard matrix‐free conjugate gradient algorithm. The final scheme can be regarded as a novel shock‐capturing, conservative, and structure preserving semi‐implicit scheme for VRMHD. Several numerical tests are presented to show the main features of our novel divergence‐free semi‐implicit FV/FD solver: linear‐stability in the sense of Lyapunov is verified at a prescribed constant equilibrium solution; a second‐order of convergence is obtained for a smooth time‐dependent solution; shock‐capturing capabilities are proven against a standard set of stringent MHD shock‐problems; accuracy and robustness are verified against a non‐trivial set of two‐ and three‐dimensional MHD problems.

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A structure-preserving staggered semi-implicit finite volume scheme for continuum mechanics

Cited by 45 publications

References 136 publications

On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

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

A novel structure preserving semi‐implicit finite volume method for viscous and resistive magnetohydrodynamics

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