On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows
type: Item
instance of: scholarly article
title: On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows
zbMATH Open document ID: 1501.65047
author: Saray Busto
author: Michael Dumbser
author: Kseniya Ivanova
author: S. L. Gavrilyuk
published in: Journal of Scientific Computing
publication date: 2021-08-24T00:00:00Z
review text: The authors introduce a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows introduced and studied in some previous works, e.g. [\textit{S. Gavrilyuk} et al., J. Comput. Phys. 366, 252--280 (2018; Zbl 1406.65068)]. The main key of the model reformulation proposed is the decomposition of the specific Reynolds stress tensor \({\textbf{P}}\) at the aid of a new object \({\textbf{Q}}\) such that \({\textbf{P}}={\textbf{Q}}{\textbf{Q}}^T\). This yields \(\mathrm{tr} {\textbf{P}}\geq 0\). The mathematical model is interesting since one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. A thermodynamically compatible \textit{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 is considered. Two families of numerical methods are introduced. The first scheme is a provably \textit{thermodynamically compatible} semi-discrete finite volume scheme that makes direct use of the \textit{Godunov form} of the equations and can therefore be called a \textit{discrete Godunov formalism}. The new method mimics the underlying continuous viscous system \textit{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 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 use path integrals to define the jump terms at the element interfaces. Some different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in [the th…
reviewed by: Abdallah Bradji
Mathematics Subject Classification ID: 65M08
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65N08
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 35L02
Mathematics Subject Classification ID: 35L45
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 76F50
Mathematics Subject Classification ID: 76N10
Mathematics Subject Classification ID: 76M10
Mathematics Subject Classification ID: 76M12
Mathematics Subject Classification ID: 35Q31
zbMATH DE Number: 7384714
zbMATH Keywords: Godunov form of hyperbolic equations
zbMATH Keywords: discrete Godunov formalism
zbMATH Keywords: thermodynamically compatible finite volume schemes
zbMATH Keywords: vanishing viscosity limit
zbMATH Keywords: path-conservative ADER discontinuous Galerkin schemes
zbMATH Keywords: unsteady turbulent shallow water flows
zbMATH Keywords: realizable hyperbolic turbulence model
describes a project that uses: MOOD
MaRDI profile type: Publication
full work available at URL: s10915-021-01521-z
OpenAlex ID: W3166647868
cites work: An Asymptotic-Preserving All-Speed Scheme for Fluid Dynamics and Nonlinear Elasticity
cites work: A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes
cites work: A high-order nonconservative approach for hyperbolic equations in fluid dynamics
cites work: Computations of compressible multifluids.
cites work: A comment on the computation of non-conservative products
cites work: Exact and approximate solutions of Riemann problems in nonlinear elasticity
cites work: A hyperbolic reformulation of the Serre-Green-Naghdi model for general bottom topographies
cites work: High order ADER-DG schemes for the simulation of linear seismic waves induced by nonlinear dispersive free-surface water waves
cites work: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations
cites work: Q4952986
cites work: Fluctuation splitting Riemann solver for a non-conservative modeling of shear shallow water flow
cites work: Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes
cites work: A structure-preserving staggered semi-implicit finite volume scheme for continuum mechanics
cites work: A Cartesian Scheme for Compressible Multimaterial Hyperelastic Models with Plasticity
cites work: On high order ADER discontinuous Galerkin schemes for first order hyperbolic reformulations of nonlinear dispersive systems
cites work: Efficient high order accurate staggered semi-implicit discontinuous Galerkin methods for natural convection problems
cites work: Design and analysis of ADER-type schemes for model advection-diffusion-reaction equations
cites work: Well-Balanced High Order Extensions of Godunov's Method for Semilinear Balance Laws
cites work: High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems
cites work: Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes
cites work: High order extensions of roe schemes for two-dimensional nonconservative hyperbolic systems
cites work: A path conservative finite volume method for a shear shallow water model
cites work: Convergence of second-order, entropy stable methods for multi-dimensional conservation laws
cites work: The local projection $P^0-P^1$-discontinuous-Galerkin finite element method for scalar conservation laws
cites work: Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws
cites work: A high-order finite volume method for systems of conservation laws-multi-dimensional optimal order detection (MOOD)
cites work: The Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws. IV: The Multidimensional Case
cites work: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III: One-dimensional systems
cites work: TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws II: General Framework
cites work: The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws
cites work: The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
cites work: Runge--Kutta discontinuous Galerkin methods for convection-dominated problems
cites work: Ideal GLM-MHD: about the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations
cites work: Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation
cites work: Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials
cites work: The Multidimensional Optimal Order Detection method in the three‐dimensional case: very high‐order finite volume method for hyperbolic systems
cites work: Arbitrary high order \(P_{N}P_{M}\) schemes on unstructured meshes for the compressible Navier-Stokes equations
cites work: A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes
cites work: ADER schemes on unstructured meshes for nonconservative hyperbolic systems: applications to geophysical flows
cites work: Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws
cites work: A space-time discontinuous Galerkin method for Boussinesq-type equations
cites work: FORCE schemes on unstructured meshes. II: Non-conservative hyperbolic systems
cites work: A simple robust and accurate \textit{a posteriori} sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes
cites work: High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: \textit{A posteriori} subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Spectral/\(hp\) discontinuous Galerkin methods for modelling 2D Boussinesq equations

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Diffuse interface model for compressible fluid -- compressible elastic-plastic solid interaction

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Solid-fluid diffuse interface model in cases of extreme deformations

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Accurate numerical discretizations of non-conservative hyperbolic systems

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Symmetric positive linear differential equations

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Systems of Conservation Equations with a Convex Extension

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Modelling wave dynamics of compressible elastic materials

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Geometric evolution of the Reynolds stress tensor

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Multi-dimensional shear shallow water flows: problems and solutions

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Q5329330

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Thermodynamically consistent nonlinear model of elastoplastic Maxwell medium

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Q4331672

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Q4425058

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Thermodynamic formalization of the fluid dynamics equations for a charged dielectric in an electromagnetic field

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Structure of the hydraulic jump in convergent radial flows

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: A unified Eulerian framework for multimaterial continuum mechanics

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Space-time discontinuous Galerkin method for the compressible Navier--Stokes equations

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Local discontinuous Galerkin methods for nonlinear dispersive equations

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Definition and weak stability of nonconservative products

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Godunov method for nonconservative hyperbolic systems

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Criterion of hyperbolicity in hyperelasticity in the case of the stored energy in separable form

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Multi-solid and multi-fluid diffuse interface model: applications to dynamic fracture and fragmentation

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Numerical methods for nonconservative hyperbolic systems: a theoretical framework.

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: On the well-balance property of Roe's method for nonconservative hyperbolic systems. applications to shallow-water systems

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Theoretical and numerical comparison of hyperelastic and hypoelastic formulations for Eulerian non-linear elastoplasticity

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: A hyperbolic model for viscous Newtonian flows

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Fully discrete explicit locally entropy-stable schemes for the compressible Euler and Navier-Stokes equations

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: A space-time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: A space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: A new model of roll waves: comparison with Brock’s experiments

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: The classical hydraulic jump in a model of shear shallow-water flows

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics.

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Conservative formulation for compressible multiphase flows

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Conservative models and numerical methods for compressible two-phase flow

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: A new continuum model for general relativistic viscous heat-conducting media

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Conservative hyperbolic formulation for compressible two-phase flow with different phase pressures and temperatures

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: The Numerical Viscosity of Entropy Stable Schemes for Systems of Conservation Laws. I

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equations at all Mach numbers

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Arbitrary high order accurate space-time discontinuous Galerkin finite element schemes on staggered unstructured meshes for linear elasticity

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Gas-dynamic analogy for vortex free-boundary flows

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: ADER: Arbitrary high-order Godunov approach

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: ADER schemes for three-dimensional non-linear hyperbolic systems

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Q4255380

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Shock-capturing methods for free-surface shallow flows

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Towards very high order Godunov schemes

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Solution of the generalized Riemann problem for advection–reaction equations

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Derivative Riemann solvers for systems of conservation laws and ader methods

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows. I: General formulation.

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows. II: Efficient flux quadrature.

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: A Local Discontinuous Galerkin Method for KdV Type Equations

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Local discontinuous Galerkin methods for partial differential equations with higher order derivatives

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (cites work)
cites work: Space-time adaptive ADER discontinuous Galerkin finite element schemes with \textit{a posteriori} sub-cell finite volume limiting

On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows (DOI)
DOI: 10.1007/S10915-021-01521-Z