An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: Theory and numerical verification

We develop an entropy stable two-phase incompressible Navier-Stokes/Cahn-Hilliard discontinuous Galerkin (DG) flow solver method. The model poses the Cahn-Hilliard equation as the phase field method, a skew-symmetric form of the momentum equation, and an artificial compressibility method to compute the pressure. We design the model so that it satisfies an entropy law, including free-and no-slip wall boundary conditions with non-zero wall contact angle. We then construct a high-order DG approximation of the model that satisfies the SBP-SAT property. With the help of a discrete stability analysis, the scheme has two modes: an entropy conserving approximation with central advective fluxes and the Bassi-Rebay 1 (BR1) method for diffusion, and an entropy stable approximation with an exact Riemann solver for advection and interface stabilization added to the BR1 method. The scheme is applicable to, and the stability proofs hold for, three-dimensional unstructured meshes with curvilinear hexahedral elements. We test the convergence of the schemes on a manufactured solution, and their robustness by solving a flow initialized from random numbers. In the latter, we find that a similar scheme that does not satisfy an entropy inequality had 30% probability to fail, while the entropy stable scheme never does. We also solve the static and rising bubble test problems, and to challenge the solver capabilities we compute a three-dimensional pipe flow in the annular regime.

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“…With these results, we confirm that the iNS/CH system, (20), with entropy, (37), satisfies the conservation law…”

Section: Property 1 If Inviscid Fluxessupporting

confidence: 75%

Entropy–stable discontinuous Galerkin approximation with summation–by–parts property for the incompressible Navier–Stokes/Cahn–Hilliard system

Manzanero

Rubio

Kopriva

et al. 2020

Journal of Computational Physics

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“…is methodology has been used in the construction of high-order entropy stable DGSEM on quadrilateral/hexahedral elements, e.g. [3,26,60]. In this section, it will be shown that similar ideas can be used to construct high-order entropy stable moving mesh DGSEM, when there are two-point ux functions with the property (2.19) available for the low-order FV method (2.16).…”

Section: Entropy Stable Dgsem On Moving Meshesmentioning

confidence: 99%

“…is methodology has been used in the construction of high-order entropy stable DGSEM on quadrilateral/hexahedral elements, e.g. [3,26,60], or on triangular/tetrahedral elements, e.g. [6,11,14].…”

Section: Introductionmentioning

confidence: 99%

Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes for Hyperbolic Conservation Laws

et al. 2020

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is work is focused on the entropy analysis of a semi-discrete nodal discontinuous Galerkin spectral element method (DGSEM) on moving meshes for hyperbolic conservation laws. e DGSEM is constructed with a local tensor-product Lagrange-polynomial basis computed from Legendre-Gauss-Loba o (LGL) points. Furthermore, the collocation of interpolation and quadrature nodes is used in the spatial discretization. is approach leads to discrete derivative approximations in space that are summation-by-parts (SBP) operators. On a static mesh, the SBP property and suitable two-point ux functions, which satisfy the entropy condition from Tadmor, allow to mimic results from the continuous entropy analysis on the discrete level. In this paper, Tadmor's condition is extended to the moving mesh framework. Based on the moving mesh entropy condition, entropy conservative two-point ux functions for the homogeneous shallow water equations and the compressible Euler equations are constructed. Furthermore, it will be proven that the semi-discrete moving mesh DGSEM is an entropy conservative scheme when a two-point ux function, which satis es the moving mesh entropy condition, is applied in the split form DG framework. is proof does not require any exactness of quadrature in the spatial integrals of the variational form. Nevertheless, entropy conservation is not su cient to tame discontinuities in the numerical solution and thus the entropy conservative moving mesh DGSEM is modi ed by adding numerical dissipation matrices to the entropy conservative uxes. en, the method becomes entropy stable such that the discrete mathematical entropy is bounded at any time by its initial and boundary data when the boundary conditions are speci ed appropriately.Besides the entropy stability, the time discretization of the moving mesh DGSEM will be investigated and it will be proven that the moving mesh DGSEM satis es the free stream preservation property for an arbitrary s-stage Runge-Ku a method. e theoretical properties of the moving mesh DGSEM will be validated by numerical experiments for the compressible Euler equations. Entropy Stable DG Schemes on Moving Meshes where ν is the grid velocity. e chain rule and (2.2) provide J du dt = J ∂u ∂t + ν ∂u ∂ξ ⇔ J ∂u ∂t = J du dt − ν ∂u ∂ξ . (2.3) Hence, by applying the chain rule and rearranging terms the conservation law (2.1) becomes J du dt + ∂f ∂ξ = ν ∂u ∂ξ (2.4) on the reference element. e product rule allows to write the equation (2.4) as J du dt + ∂g ∂ξ = − ∂ν ∂ξ u, (2.5) d dt 1 2

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“…The state vector of conserved quantities is u and the Cartesian fluxes are denoted by f 1 , f 2 , f 3 . As in [2,23], we define block vector notation with a double arrow…”

Section: Polytropic Euler Equationsmentioning

confidence: 99%

Entropy stable numerical approximations for the isothermal and polytropic Euler equations

et al. 2019

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In this work we analyze the entropic properties of the Euler equations when the system is closed with the assumption of a polytropic gas. In this case, the pressure solely depends upon the density of the fluid and the energy equation is not necessary anymore as the mass conservation and momentum conservation then form a closed system. Further, the total energy acts as a convex mathematical entropy function for the polytropic Euler equations. The polytropic equation of state gives the pressure as a scaled power law of the density in terms of the adiabatic index γ. As such, there are important limiting cases contained within the polytropic model like the isothermal Euler equations (γ = 1) and the shallow water equations (γ = 2). We first mimic the continuous entropy analysis on the discrete level in a finite volume context to get special numerical flux functions. Next, these numerical fluxes are incorporated into a particular discontinuous Galerkin (DG) spectral element framework where derivatives are approximated with summation-by-parts operators. This guarantees a high-order accurate DG numerical approximation to the polytropic Euler equations that is also consistent to its auxiliary total energy behavior. Numerical examples are provided to verify the theoretical derivations, i.e., the entropic properties of the high order DG scheme.the internal energy e ∈ R, and the pressure p ∈ R + . Thus, in order to close the system, an equation of state is necessary to relate thermodynamic state variables like pressure, density, and internal energy. Depending on the fluid and physical processes we wish to model the equation of state changes.

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An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: Theory and numerical verification

Cited by 53 publications

References 53 publications

Entropy–stable discontinuous Galerkin approximation with summation–by–parts property for the incompressible Navier–Stokes/Cahn–Hilliard system

Entropy–stable discontinuous Galerkin approximation with summation–by–parts property for the incompressible Navier–Stokes/Cahn–Hilliard system

Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes for Hyperbolic Conservation Laws

Entropy stable numerical approximations for the isothermal and polytropic Euler equations

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