Stable Discontinuous Galerkin Schemes Research Articles

A positivity preserving and oscillation-free entropy stable discontinuous Galerkin scheme for the reactive Euler equations

The reactive Euler equations are a basic model for fluid flows with chemical reactions. In this work, we construct a high order positivity preserving and oscillation-free entropy stable discontinuous Galerkin (DG) scheme for the reactive Euler equations. The main ingredients of the scheme include (i) entropy preserving and entropy stable fluxes to achieve entropy stability, (ii) artificial damping terms to restrain spurious oscillations near the shocks, and (iii) positivity preserving limiters to guarantee the positivity of solutions. These ingredients are compatible with each other so that our scheme simultaneously enjoys the properties of entropy stable, oscillation-free and positivity preserving. Another distinctive feature of our scheme is that it is entropy stable for both the thermodynamic and mathematical entropies. Numerical experiments validate the designed high convergence orders of the scheme and demonstrate its good performances for discontinuous problems.

Entropy Stable Discontinuous Galerkin Schemes for Two-Fluid Relativistic Plasma Flow Equations

Entropy Stable Discontinuous Galerkin Schemes for Two-Fluid Relativistic Plasma Flow Equations

On the Entropy Projection and the Robustness of High Order Entropy Stable Discontinuous Galerkin Schemes for Under-Resolved Flows

High order entropy stable schemes provide improved robustness for computational simulations of fluid flows. However, additional stabilization and positivity preserving limiting can still be required for variable-density flows with under-resolved features. We demonstrate numerically that entropy stable Discontinuous Galerkin (DG) methods which incorporate an “entropy projection” are less likely to require additional limiting to retain positivity for certain types of flows. We conclude by investigating potential explanations for this observed improvement in robustness.

Provably stable flux reconstruction high-order methods on curvilinear elements

Provably stable flux reconstruction (FR) schemes are derived for partial differential equations cast in curvilinear coordinates. Specifically, energy stable flux reconstruction (ESFR) schemes are considered as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, the curvilinear metric split-form for a linear physical flux is examined as it enables the development of energy stability proofs. The first critical step proves, that in curvilinear coordinates, the discontinuous Galerkin (DG) conservative and non-conservative forms are inherently different–even under exact integration and analytically exact metric terms. This analysis demonstrates that the split form is essential to developing provably stable DG schemes on curvilinear coordinates and motivates the construction of metric dependent ESFR correction functions in each element. Furthermore, the provably stable FR schemes differ from schemes in the literature that only apply the ESFR correction functions to surface terms or on the conservative form, and instead incorporate the ESFR correction functions on the full split form of the equations. It is demonstrated that the scheme is divergent when the correction functions are only used for surface reconstruction in curvilinear coordinates. We numerically verify the stability claims for our proposed FR split forms and compare them to ESFR schemes in the literature. Lastly, the newly proposed provably stable FR schemes are shown to obtain optimal orders of convergence. The scheme loses the orders of accuracy at the equivalent correction parameter value c as that of the one-dimensional ESFR scheme.

Entropy stable discontinuous Galerkin schemes for the special relativistic hydrodynamics equations

This article presents entropy stable discontinuous Galerkin numerical schemes for equations of special relativistic hydrodynamics with the ideal equation of state. The numerical schemes use the summation by parts (SBP) property of the Gauss-Lobatto quadrature rules. To achieve entropy stability for the scheme, we use two-point entropy conservative numerical flux inside the cells and a suitable entropy stable numerical flux at the cell interfaces. The resulting semi-discrete scheme is then shown to be entropy stable. Time discretization is performed using SSP Runge-Kutta methods. Several numerical test cases are presented to validate the accuracy and stability of the proposed schemes.

Unconditionally energy stable discontinuous Galerkin schemes for the Cahn–Hilliard equation

In this paper, we introduce novel discontinuous Galerkin (DG) schemes for the Cahn–Hilliard equation, which arises in many applications. The method is designed by integrating the mixed DG method for the spatial discretization with the Invariant Energy Quadratization (IEQ) approach for the time discretization. Coupled with a spatial projection, the resulting IEQ–DG schemes are shown to be unconditionally energy dissipative, and can be efficiently solved without resorting to any iteration method. Both one and two dimensional numerical examples are provided to verify the theoretical results, and demonstrate the good performance of IEQ–DG in terms of efficiency, accuracy, and preservation of the desired solution properties.

Numerical Analysis of a Stable Discontinuous Galerkin Scheme for the Hydrostatic Stokes Problem

Abstract We propose a Discontinuous Galerkin (DG) scheme for the Hydrostatic Stokes equations. These equations, related to large-scale PDE models in Oceanography, are characterized by the loss of ellipticity of the vertical momentum equation. This fact provides some interesting challenges, such as the design of stable numerical schemes. The new scheme proposed here is based on the SIP penalty technique, with a particular treatment of the vertical velocity. It is well-known that stability of the mixed formulation of Primitive Equations requires, besides the LBB inf-sup condition, an additional hydrostatic inf-sup restriction relating pressure and vertical velocity. This hydrostatic inf-sup condition invalidates stability of usual Stokes stable continuous finite elements like Taylor-Hood 𝒫2/𝒫1 or bubble 𝒫1b /𝒫1. Here we prove stability for our 𝒫k/𝒫k DG scheme. Some novel numerical tests are provided which are in agreement with the previous analysis.

Numerical analysis of the unconditionally stable discontinuous Galerkin schemes for the nonstationary conduction–convection problem

This work aims at employing the discontinuous Galerkin (DG) methods for solving the nonstationary conduction–convection model, which is motivated by advantages of DG methods. The discontinuous element pair Pk-Pk−1-Pk in the space discretization and backward Euler scheme in time are applied, as well as an explicit upwind scheme based on Picard iteration is introduced for the coupled term. Existence and uniqueness of the approximation solutions are proved, along with preserving the unconditional stability of the numerical schemes. The priori error estimates for velocity, pressure and temperature are also derived. Numerical tests are provided to verify the theoretical analysis and to indicate the efficiency of the DG methods in simulating practical problems.

High-order accurate entropy stable nodal discontinuous Galerkin schemes for the ideal special relativistic magnetohydrodynamics

This paper studies high-order accurate entropy stable nodal discontinuous Galerkin (DG) schemes for the ideal special relativistic magnetohydrodynamics (RMHD). It is built on the modified RMHD equations with a particular source term, which is analogous to the Powell's eight-wave formulation and can be symmetrized so that an “entropy pair” is obtained. We design an affordable “fully consistent” two-point entropy conservative flux, which is not only consistent with the physical flux, but also maintains the zero parallel magnetic component, and then construct high-order accurate semi-discrete entropy stable DG schemes based on the quadrature rules and the entropy conservative and stable fluxes. The Lax-Friedrichs flux is proven to be entropy stable when its viscosity coefficient is chosen as the speed of light, and the other two entropy stable fluxes are also derived by adding the dissipation terms to our proposed entropy conservative flux and compared to the Lax-Friedrichs flux by using numerical experiments in terms of the CPU time and accuracy. All three entropy stable numerical fluxes maintain the zero parallel magnetic component in one dimension. The resulting entropy stable DG schemes satisfy the semi-discrete “entropy inequality” for the given “entropy pair” and are integrated in time by using the high-order explicit strong stability preserving Runge-Kutta schemes to get further the fully-discrete nodal DG schemes. Extensive numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our schemes with the help of the TVB limiter. Moreover, our entropy conservative flux is compared to an existing flux through some numerical tests. The results show that the zero parallel magnetic component in the numerical flux can help to decrease the error in the parallel magnetic component in one-dimensional tests, but two entropy conservative fluxes give similar results since the error in the magnetic field divergence seems dominated in the two-dimensional tests.

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

This 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. The DGSEM is constructed with a local tensor-product Lagrange-polynomial basis computed from Legendre–Gauss–Lobatto points. Furthermore, the collocation of interpolation and quadrature nodes is used in the spatial discretization. This 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 flux functions, which satisfy the entropy condition from Tadmor, allow to mimic results from the continuous entropy analysis, if it is ensured that properties such as positivity preservation (of the water height, density or pressure) are satisfied on the discrete level. In this paper, Tadmor’s condition is extended to the moving mesh framework. We show that the volume terms in the semi-discrete moving mesh DGSEM do not contribute to the discrete entropy evolution when a two-point flux function that satisfies the moving mesh entropy condition is applied in the split form DG framework. The discrete entropy behavior then depends solely on the interface contributions and on the domain boundary contribution. The interface contributions are directly controlled by proper choice of the numerical element interface fluxes. If an entropy conserving two-point flux is chosen, the interface contributions vanish. To increase the robustness of the discretization we use so-called entropy stable two-point fluxes at the interfaces that are guaranteed entropy dissipative and thus give a bound on the interface contributions in the discrete entropy balance. The remaining boundary condition contributions depend on the type of the considered boundary condition. E.g. for periodic boundary conditions that are of entropy conserving type, our methodology with the entropy conserving interface fluxes is fully entropy conservative and with the entropy stable interface fluxes is guaranteed entropy stable. The presented proof does not require any exactness of quadrature in the spatial integrals of the variational forms. As it is the case for static meshes, these results rely on the assumption that additional properties like positivity preservation are satisfied on the discrete level. 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 satisfies the free stream preservation property for an arbitrary s-stage Runge–Kutta method, when periodic boundary conditions are used. The theoretical properties of the moving mesh DGSEM will be validated by numerical experiments for the compressible Euler equations with periodic boundary conditions.

Skew-Symmetric Entropy Stable Modal Discontinuous Galerkin Formulations

High order entropy stable discontinuous Galerkin (DG) methods for nonlinear conservation laws satisfy an inherent discrete entropy inequality. The construction of such schemes has relied on the use of carefully chosen nodal points or volume and surface quadrature rules to produce operators which satisfy a summation-by-parts (SBP) property. In this work, we show how to construct "modal" skew-symmetric DG formulations which are entropy stable for volume and surface quadratures under which a traditional SBP property does not hold. These skew-symmetric formulations avoid the use of a "strong" matrix-based SBP property, and instead rely on the assumption that discrete operators exactly differentiate constants and satisfy a discrete form of the fundamental theorem of calculus. We conclude with numerical experiments verifying the accuracy and stability of the proposed formulations, and discuss an application of skew-symmetric formulations for entropy stable DG schemes on mixed quadrilateral-triangle meshes.

On discretely entropy stable weight-adjusted discontinuous Galerkin methods: curvilinear meshes

We construct entropy conservative and entropy stable discontinuous Galerkin (DG) discretizations for time-dependent nonlinear hyperbolic conservation laws on curvilinear meshes. The resulting schemes preserve a semi-discrete quadrature approximation of a continuous global entropy inequality. The proof requires the satisfaction of a discrete geometric conservation law, which we enforce through an appropriate polynomial approximation. We extend the construction of entropy conservative and entropy stable DG schemes to the case when high order curvilinear mass matrices are approximated using low-storage weight-adjusted approximations, and describe how to retain global conservation properties under such an approximation. For certain types of curvilinear meshes, these weight-adjusted approximations deliver optimal rates of convergence. Finally, the high order accuracy, local conservation, and discrete conservation or dissipation of entropy for under-resolved solutions are verified through numerical experiments for the compressible Euler equations on triangular and tetrahedral meshes.

Correction to: The BR1 Scheme is Stable for the Compressible Navier–Stokes Equations

An open-source code that implements the entropy stable discontinuous Galerkin scheme with Legendere–Gauss–Lobatto collocation (DGSEM) on curved unstructured hexahedral grids for compressible Navier–Stokes equations (NSE) is available at github.com/project-fluxo.