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In this work, we present the construction of two distinct finite element approaches to solve the porous medium equation (PME). In the first approach, we transform the PME to a log-density variable formulation and construct a continuous Galerkin method. In the second approach, we introduce additional potential and velocity variables to rewrite the PME into a system of equations, for which we construct a mixed finite element method. Both approaches are first-order accurate, mass conserving, and proved to be unconditionally energy stable for their respective energies. The mixed approach is shown to preserve positivity under a CFL condition, while a much stronger property of unconditional bound preservation is proved for the log-density approach. A novel feature of our schemes is that they can handle compactly supported initial data without the need for any perturbation techniques. Furthermore, the log-density method can handle unstructured grids in any number of dimensions, while the mixed method can handle unstructured grids in two dimensions. We present results from several numerical experiments to demonstrate these properties. [ABSTRACT FROM AUTHOR]

A novel approach for the stabilization of the Discontinuous Galerkin method based on the Dafermos entropy rate crition is presented. First, estimates for the maximal possible entropy dissipation rate of a weak solution are derived. Second, families of conservative Hilbert–Schmidt operators are identified to dissipate entropy. Steering these operators using the bounds on the entropy dissipation results in high-order accurate shock-capturing DG schemes for the one-dimensional Euler equations, satisfying the entropy rate criterion and an entropy inequality. Other testcases include the one-dimensional Buckley–Leverett equation. [ABSTRACT FROM AUTHOR]

We present an entropy stable nodal discontinuous Galerkin spectral element method (DGSEM) for the two-layer shallow water equations on two dimensional curvilinear meshes. We mimic the continuous entropy analysis on the semi-discrete level with the DGSEM constructed on Legendre–Gauss–Lobatto (LGL) nodes. The use of LGL nodes endows the collocated nodal DGSEM with the summation-by-parts property that is key in the discrete analysis. The approximation exploits an equivalent flux differencing formulation for the volume contributions, which generate an entropy conservative split-form of the governing equations. A specific combination of a numerical surface flux and discretization of the nonconservative terms is then applied to obtain a high-order path-conservative scheme that is entropy conservative. Furthermore, we find that this combination yields an analogous discretization for the pressure and nonconservative terms such that the numerical method is well-balanced for discontinuous bathymetry on curvilinear domains. Dissipation is added at the interfaces to create an entropy stable approximation that satisfies the second law of thermodynamics in the discrete case, while maintaining the well-balanced property. We conclude with verification of the theoretical findings through numerical tests and demonstrate results about convergence, entropy stability and well-balancedness of the scheme. [ABSTRACT FROM AUTHOR]

An asymptotic preserving and energy stable scheme for the barotropic Euler system under the low Mach number scaling is designed and analysed. A velocity shift proportional to the pressure gradient is introduced in the convective fluxes, which leads to the dissipation of mechanical energy and the entropy stability at all Mach numbers. The resolution of the semi-implicit in time and upwind finite volume in space fully-discrete scheme involves two steps: the solution of an elliptic problem for the density and an explicit evaluation for the velocity. The proposed scheme possesses several physically relevant attributes, such as the positivity of density, the entropy stability and the consistency with the weak formulation of the continuous Euler system. The AP property of the scheme, i.e. the boundedness of the mesh parameters with respect to the Mach number and its consistency with the incompressible limit system, is shown rigorously. The results of extensive case studies are presented to substantiate the robustness and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

A novel approach for the stabilization of the discontinuous Galerkin method based on the Dafermos entropy rate crition is presented. The approach is centered around the efficient solution of linear or nonlinear optimization problems in every timestep as a correction to the basic discontinuous Galerkin scheme. The thereby enforced Dafermos criterion results in improved stability compared to the basic method while retaining a high order of accuracy in numerical experiments for scalar conservation laws. Further modification of the optimization problem allows also to enforce classical entropy inequalities for the scheme. The proposed stabilization is therefore an alternative to flux-differencing to enforce entropy inequalities. As the shock-capturing abilities of the scheme are also enhanced is the method also an alternative to finite-volume subcells, artificial viscosity, modal filtering, and other shock capturing procedures in one space dimension. Tests are carried out for Burgers’ equation. [ABSTRACT FROM AUTHOR]

This paper extends a new family of high-order positivity-preserving, entropy stable spectral collocation schemes developed for the one-dimensional compressible Navier–Stokes equations in Upperman and Yamaleev (J Comput Phys 466, 2022; J Comput Phys 466, 2022) to three spatial dimensions. The proposed schemes are constructed by using a flux-limiting technique that combines a positivity-violating entropy stable method of arbitrary order of accuracy and a novel first-order positivity-preserving entropy stable finite volume-type scheme discretized on the same Legendre–Gauss–Lobatto grid points used for constructing the high-order discrete operators. The positivity-preserving and excellent discontinuity-capturing properties are achieved by adding an artificial dissipation in the form of the low- and high-order Brenner–Navier–Stokes diffusion operators. Furthermore, the new schemes are entropy conservative for smooth inviscid flows and freestream preserving. To our knowledge, this is the first family of schemes of arbitrary order of accuracy that provably guarantee both the pointwise positivity of thermodynamic variables and L 2 stability for the 3-D compressible Navier–Stokes equations. Numerical results demonstrating accuracy and positivity-preserving properties of the new schemes are presented for 2-D and 3-D viscous and inviscid flows with strong discontinuities. [ABSTRACT FROM AUTHOR]

Relaxation Runge–Kutta methods reproduce a fully discrete dissipation (or conservation) of entropy for entropy stable semi-discretizations of nonlinear conservation laws. In this paper we derive the discrete adjoint of relaxation Runge–Kutta schemes, which are applicable to discretize-then-optimize approaches for optimal control problems. Furthermore, we prove that the derived discrete relaxation Runge–Kutta adjoint preserves time-symmetry when applied to linear skew-symmetric systems of ODEs. Numerical experiments verify these theoretical results while demonstrating the importance of appropriately treating the relaxation parameter when computing the discrete adjoint. [ABSTRACT FROM AUTHOR]

In this paper, we extend the positivity-preserving, entropy stable first-order scheme developed for the one-dimensional compressible Navier–Stokes equations in Upperman et al. (J Comput Phys 466, 2022) to three spatial dimensions. The new first-order scheme is provably entropy stable, design-order accurate for smooth solutions, and guarantees the pointwise positivity of thermodynamic variables for 3-D compressible viscous flows. Similar to the 1-D counterpart, the proposed scheme for the 3-D Navier–Stokes equations is discretized on Legendre-Gauss-Lobatto grids used for high-order spectral collocation methods. The positivity of density is achieved by adding an artificial dissipation in the form of the first-order Brenner–Navier–Stokes diffusion operator. Another distinctive feature of the proposed scheme is that the Navier–Stokes viscous terms are discretized by high-order spectral collocation summation-by-parts operators. To eliminate time step stiffness caused by the high-order approximation of the viscous terms and the temperature positivity constraint, the velocity and temperature limiters developed for the 1-D compressible Navier–Stokes equations in Upperman et al. (J. Comput. Phys., 466, 2022) are generalized to three spatial dimensions. These limiters bound the magnitude of velocity and temperature gradients and preserve the entropy stability and positivity properties of the baseline scheme. Numerical results are presented to demonstrate design-order accuracy and positivity-preserving properties of the new first-order scheme for 2-D and 3-D inviscid and viscous flows with strong shocks and contact discontinuities. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We develop high-order entropy-conservative semi-discrete schemes for hyperbolic conservation laws applicable to non-conforming curvilinear grids arising from h-, p-, or hp-adaptivity. More precisely, building on previous work with conforming grids by Crean et al. (J Comput Phys 356:410–438, 2018) and Chan et al. (SIAM J Sci Comput 41:A2938–A2966, 2019), we present two schemes: the first couples neighbouring elements in a skew-symmetric method, the second in a pointwise fashion. The key ingredients are degree p diagonal-norm summation-by-parts operators equipped with interface quadrature rules of degree 2p or higher, a skew-symmetric geometric mapping procedure using suitable polynomial functions, and a numerical flux that conserves mathematical entropy. Furthermore, entropy-stable schemes are obtained when augmenting the original schemes with a stabilization term that dissipates mathematical entropy at element interfaces. We provide both theoretical and numerical analysis for the compressible Euler equations demonstrating the element-wise conservation, entropy conservation/dissipation, and accuracy properties of the schemes. While both methods produce comparable results, our studies suggest that the scheme coupling elements in a pointwise manner is more computationally efficient. [ABSTRACT FROM AUTHOR]

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 (Gassner in SIAM J Sci Comput 35(3):A1233–A1253, 2013; Fisher and Carpenter in J Comput Phys 252:518–557, 2013; Carpenter et al. in SIAM J Sci Comput 36(5):B835–B867, 2014; Crean et al. in J Comput Phys 356:410–438, 2018; Chan et al. in Efficient entropy stable Gauss collocation methods, 2018. arXiv:1809.01178) or volume and surface quadrature rules (Chan in J Comput Phys 362:346–374, 2018; Chan and Wilcox in J Comput Phys 378:366–393, 2019) to produce operators which satisfy a summation-by-parts (SBP) property. In this work, we show how to construct "modal" DG formulations which are entropy stable for volume and surface quadratures under which the SBP property in Chan (2018) does not hold. These formulations rely on an alternative skew-symmetric construction of operators which automatically satisfy the SBP property. Entropy stability then follows for choices of volume and surface quadrature which satisfy sufficient accuracy conditions. The accuracy of these new SBP operators depends on a separate set of conditions on quadrature accuracy, with design order accuracy recovered under the usual assumptions of degree 2 N - 1 volume quadratures and degree 2N surface quadratures. We conclude with numerical experiments verifying the accuracy and stability of the proposed formulations, and discuss an application of these formulations for entropy stable DG schemes on mixed quadrilateral-triangle meshes. [ABSTRACT FROM AUTHOR]

We develop a discretely entropy-stable line-based discontinuous Galerkin method for hyperbolic conservation laws based on a flux differencing technique. By using standard entropy-stable and entropy-conservative numerical flux functions, this method guarantees that the discrete integral of the entropy is non-increasing. This nonlinear entropy stability property is important for the robustness of the method, in particular when applied to problems with discontinuous solutions or when the mesh is under-resolved. This line-based method is significantly less computationally expensive than a standard DG method. Numerical results are shown demonstrating the effectiveness of the method on a variety of test cases, including Burgers' equation and the Euler equations, in one, two, and three spatial dimensions. [ABSTRACT FROM AUTHOR]

This work examines the development of an entropy conservative (for smooth solutions) or entropy stable (for discontinuous solutions) space–time discontinuous Galerkin (DG) method for systems of nonlinear hyperbolic conservation laws. The resulting numerical scheme is fully discrete and provides a bound on the mathematical entropy at any time according to its initial condition and boundary conditions. The crux of the method is that discrete derivative approximations in space and time are summation-by-parts (SBP) operators. This allows the discrete method to mimic results from the continuous entropy analysis and ensures that the complete numerical scheme obeys the second law of thermodynamics. Importantly, the novel method described herein does not assume any exactness of quadrature in the variational forms that naturally arise in the context of DG methods. Typically, the development of entropy stable schemes is done on the semidiscrete level ignoring the temporal dependence. In this work, we demonstrate that creating an entropy stable DG method in time is similar to the spatial discrete entropy analysis, but there are important (and subtle) differences. Therefore, we highlight the temporal entropy analysis throughout this work. For the compressible Euler equations, the preservation of kinetic energy is of interest besides entropy stability. The construction of kinetic energy preserving (KEP) schemes is, again, typically done on the semidiscrete level similar to the construction of entropy stable schemes. We present a generalization of the KEP condition from Jameson to the space–time framework and provide the temporal components for both entropy stability and kinetic energy preservation. The properties of the space–time DG method derived herein are validated through numerical tests for the compressible Euler equations. Additionally, we provide, in appendices, how to construct the temporal entropy stable components for the shallow water or ideal magnetohydrodynamic (MHD) equations. [ABSTRACT FROM AUTHOR]

This paper follows up on the author's recent paper "Entropy Production by Explicit Runge–Kutta schemes" (Lozano in J Sci Comput 76(1):521–564, 2018. 10.1007/s10915-017-0627-0), where a formula for the production of entropy by fully discrete schemes with explicit Runge–Kutta time integrators was presented. In this paper, the focus is on implicit Runge–Kutta schemes, for which the fully discrete numerical entropy evolution scheme is derived and tested. [ABSTRACT FROM AUTHOR]

This work reviews the entropy conservative (EC) schemes introduced by Tadmor and examines their use in the receding flow problem extensively studied by Liou. This is motivated by Liou's recent findings that an abnormal spike in temperature observed with finite-volume schemes is linked to a spurious entropy rise that can be prevented in principle by conserving entropy. While a semi-discrete analysis suggests EC schemes are a good fit, a fully discrete analysis based on Tadmor's framework shows the non-negligible impact of time-integration on the solution behavior. In addition, an EC time-integration scheme is developed to show that enforcing conservation of entropy at the fully discrete level does not necessarily prevent the temperature spike. [ABSTRACT FROM AUTHOR]

This work presents an entropy stable discontinuous Galerkin (DG) spectral element approximation for systems of non-linear conservation laws with general geometric (h) and polynomial order (p) non-conforming rectangular meshes. The crux of the proofs presented is that the nodal DG method is constructed with the collocated Legendre-Gauss-Lobatto nodes. This choice ensures that the derivative/mass matrix pair is a summation-by-parts (SBP) operator such that entropy stability proofs from the continuous analysis are discretely mimicked. Special attention is given to the coupling between non-conforming elements as we demonstrate that the standard mortar approach for DG methods does not guarantee entropy stability for non-linear problems, which can lead to instabilities. As such, we describe a precise procedure and modify the mortar method to guarantee entropy stability for general non-linear hyperbolic systems on h / p non-conforming meshes. We verify the high-order accuracy and the entropy conservation/stability of fully non-conforming approximation with numerical examples. [ABSTRACT FROM AUTHOR]

In this work we prove that the original (Bassi and Rebay in J Comput Phys 131:267-279, 1997) scheme (BR1) for the discretization of second order viscous terms within the discontinuous Galerkin collocation spectral element method (DGSEM) with Gauss Lobatto nodes is stable. More precisely, we prove in the first part that the BR1 scheme preserves energy stability of the skew-symmetric advection term DGSEM discretization for the linearized compressible Navier-Stokes equations (NSE). In the second part, we prove that the BR1 scheme preserves the entropy stability of the recently developed entropy stable compressible Euler DGSEM discretization of Carpenter et al. (SIAM J Sci Comput 36:B835-B867, 2014) for the non-linear compressible NSE, provided that the auxiliary gradient equations use the entropy variables. Both parts are presented for fully three-dimensional, unstructured curvilinear hexahedral grids. Although the focus of this work is on the BR1 scheme, we show that the proof naturally includes the Local DG scheme of Cockburn and Shu. [ABSTRACT FROM AUTHOR]

Entropy conservation and stability of numerical methods in gas dynamics have received much interest. Entropy conservative numerical fluxes can be used as ingredients in two kinds of schemes: firstly, as building blocks in the subcell flux differencing form of Fisher and Carpenter (Technical Report NASA/TM-2013-217971, NASA, 2013; J Comput Phys 252:518-557, 2013) and secondly (enhanced by dissipation) as numerical surface fluxes in finite volume like schemes. The purpose of this article is threefold. Firstly, the flux differencing theory is extended, guaranteeing high-order for general symmetric and consistent numerical fluxes and investigating entropy stability in a generalised framework of summation-by-parts operators applicable to multiple dimensions and simplex elements. Secondly, a general procedure to construct affordable entropy conservative fluxes is described explicitly and used to derive several new fluxes. Finally, robustness properties of entropy stable numerical fluxes are investigated and positivity preservation is proven for several entropy conservative fluxes enhanced with local Lax-Friedrichs type dissipation operators. All these theoretical investigations are supplemented with numerical experiments. [ABSTRACT FROM AUTHOR]

In this article, we propose high order, semi-discrete, entropy stable, finite difference schemes for Ten-Moment Gaussian Closure equations. The crucial components of these schemes are an entropy conservative flux and suitable high order entropy dissipative operators to ensure entropy stability. We design two numerical fluxes, one is approximately entropy conservative, and another is entropy conservative flux. For the construction of appropriate entropy dissipative operators, we also derive entropy scaled right eigenvectors. This is used for sign preserving reconstruction of scaled entropy variables, which results in second and third order entropy stable schemes. We also extend these schemes to a plasma flow model with source term. Several numerical results are presented for homogeneous and non-homogeneous cases to demonstrate stability and performance of these schemes. [ABSTRACT FROM AUTHOR]

We propose a third-order WENO reconstruction which satisfies the sign property, required for constructing high resolution entropy stable finite difference scheme for conservation laws. The reconstruction technique, which is termed as SP-WENO, is endowed with additional properties making it a more robust option compared to ENO schemes of the same order. The performance of the proposed reconstruction is demonstrated via a series of numerical experiments for linear and nonlinear scalar conservation laws. The scheme is easily extended to multi-dimensional conservation laws. [ABSTRACT FROM AUTHOR]

In this work, we design an entropy stable, finite volume approximation for the shallow water magnetohydrodynamics (SWMHD) equations. The method is novel as we design an affordable analytical expression of the numerical interface flux function that exactly preserves the entropy, which is also the total energy for the SWMHD equations. To guarantee the discrete conservation of entropy requires a special treatment of a consistent source term for the SWMHD equations. With the goal of solving problems that may develop shocks, we determine a dissipation term to guarantee entropy stability for the numerical scheme. Numerical tests are performed to demonstrate the theoretical findings of entropy conservation and robustness. [ABSTRACT FROM AUTHOR]