Stability of Wall Boundary Condition Procedures for Discontinuous Galerkin Spectral Element Approximations of the Compressible Euler Equations

We perform a linear and entropy stability analysis for wall boundary condition procedures for discontinuous Galerkin spectral element approximations of the compressible Euler equations. Two types of boundary procedures are examined. The first defines a special wall boundary flux that incorporates the boundary condition. The other is the commonly used reflection condition where an external state is specified that has an equal and opposite normal velocity. The internal and external states are then combined through an approximate Riemann solver to weakly impose the boundary condition. We show that with the exact upwind and Lax-Friedrichs solvers the approximations are energy dissipative, with the amount of dissipation proportional to the square of the normal Mach number. Standard approximate Riemann solvers, namely Lax-Friedrichs, HLL, HLLC are entropy stable. The Roe flux is entropy stable under certain conditions. An entropy conserving flux with an entropy stable dissipation term (EC-ES) is also presented. The analysis gives insight into why these boundary conditions are robust in that they introduce large amounts of energy or entropy dissipation when the boundary condition is not accurately satisfied, e.g. due to an impulsive start or under resolution.Comment: ICOSAHOM 2018 conference proceedings, 14 pages, 2 Figure

An energy-conserving ultra-weak discontinuous Galerkin method for the generalized Korteweg-De Vries equation

We propose an energy-conserving ultra-weak discontinuous Galerkin (DG) method for the generalized Korteweg-De Vries(KdV) equation in one dimension. Optimal a priori error estimate of order $k + 1$ is obtained for the semi-discrete scheme for the KdV equation without convection term on general nonuniform meshes when polynomials of degree $k\ge 2$ is used. We also numerically observed optimal convergence of the method for the KdV equation with linear or nonlinear convection terms. It is numerically observed for the new method to have a superior performance for long-time simulations over existing DG methods.Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1804.1030

Entropy stable flux correction for scalar hyperbolic conservation laws

It is known that Flux Corrected Transport algorithms can produce entropy-violating solutions of hyperbolic conservation laws. Our purpose is to design flux correction with maximal antidiffusive fluxes to obtain entropy solutions of scalar hyperbolic conservation laws. To do this we consider a hybrid difference scheme that is a linear combination of a monotone scheme and a scheme of high-order accuracy. Flux limiters for the hybrid scheme are calculated from a corresponding optimization problem. Constraints for the optimization problem consist of inequalities that are valid for the monotone scheme and applied to the hybrid scheme. We apply the discrete cell entropy inequality with the proper numerical entropy flux to single out a physically relevant solution of scalar hyperbolic conservation laws. A nontrivial approximate solution of the optimization problem yields expressions to compute the required flux limiters. We present examples that show that not all numerical entropy fluxes guarantee to single out a physically correct solution of scalar hyperbolic conservation laws

Mimetic Properties of Difference Operators: Product and Chain Rules as for Functions of Bounded Variation and Entropy Stability of Second Derivatives

For discretisations of hyperbolic conservation laws, mimicking properties of operators or solutions at the continuous (differential equation) level discretely has resulted in several successful methods. While well-posedness for nonlinear systems in several space dimensions is an open problem, mimetic properties such as summation-by-parts as discrete analogue of integration-by-parts allow a direct transfer of some results and their proofs, e.g. stability for linear systems. In this article, discrete analogues of the generalised product and chain rules that apply to functions of bounded variation are considered. It is shown that such analogues hold for certain second order operators but are not possible for higher order approximations. Furthermore, entropy dissipation by second derivatives with varying coefficients is investigated, showing again the far stronger mimetic properties of second order approximations compared to higher order ones

Entropy conservation property and entropy stabilization of high-order continuous Galerkin approximations to scalar conservation laws

This paper addresses the design of linear and nonlinear stabilization procedures for high-order continuous Galerkin (CG) finite element discretizations of scalar conservation laws. We prove that the standard CG method is entropy conservative for the square entropy. In general, the rate of entropy production/dissipation depends on the residual of the governing equation and on the accuracy of the finite element approximation to the entropy variable. The inclusion of linear high-order stabilization generates an additional source/sink in the entropy budget equation. To balance the amount of entropy production in each cell, we construct entropy-dissipative element contributions using a coercive bilinear form and a parameter-free entropy viscosity coefficient. The entropy stabilization term is high-order consistent, and optimal convergence behavior is achieved in practice. To enforce preservation of local bounds in addition to entropy stability, we use the Bernstein basis representation of the finite element solution and a new subcell flux limiting procedure. The underlying inequality constraints ensure the validity of localized entropy conditions and local maximum principles. The benefits of the proposed modifications are illustrated by numerical results for linear and nonlinear test problems

Some remarks about conservation for residual distribution schemes

We are interested in the discretisation of the steady version of hyperbolic problems. We first show that all the known schemes (up to our knowledge) can be rephrased in a common framework. Using this framework, we first show all the known schemes have a flux formulation, with an explicit construction of the flux, and thus are locally conservative. This is well known for the finite volume schemes or the discontinuous Galerkin ones, much less known for the continuous finite element methods. We also show that Tadmor's entropy stability formulation can naturally be rephrased in this framework as an additional conservation relation discretisation, and using this, we show some conenction with the recent papers [1, 2, 3, 4]. This contribution is an enhanced version of [5]

Algebraic entropy fixes and convex limiting for continuous finite element discretizations of scalar hyperbolic conservation laws

In this work, we modify a continuous Galerkin discretization of a scalar hyperbolic conservation law using new algebraic correction procedures. Discrete entropy conditions are used to determine the minimal amount of entropy stabilization and constrain antidiffusive corrections of a property-preserving low-order scheme. The addition of a second-order entropy dissipative component to the antidiffusive part of a nearly entropy conservative numerical flux is generally insufficient to prevent violations of local bounds in shock regions. Our monolithic convex limiting technique adjusts a given target flux in a manner which guarantees preservation of invariant domains, validity of local maximum principles, and entropy stability. The new methodology combines the advantages of modern entropy stable/entropy conservative schemes and their local extremum diminishing counterparts. The process of algebraic flux correction is based on inequality constraints which provably provide the desired properties. No free parameters are involved. The proposed algebraic fixes are readily applicable to unstructured meshes, finite element methods, general time discretizations, and steady-state residuals. Numerical studies of explicit entropy-constrained schemes are performed for linear and nonlinear test problems

Entropy stabilization and property-preserving limiters for discontinuous Galerkin discretizations of nonlinear hyperbolic equations

The methodology proposed in this paper bridges the gap between entropy stable and positivity-preserving discontinuous Galerkin (DG) methods for nonlinear hyperbolic problems. The entropy stability property and, optionally, preservation of local bounds for the cell averages are enforced using flux limiters based on entropy conditions and discrete maximum principles, respectively. Entropy production by the (limited) gradients of the piecewise-linear DG approximation is constrained using Rusanov-type entropy viscosity, as proposed by Abgrall in the context of nodal finite element approximations. We cast his algebraic entropy fix into a form suitable for arbitrary polynomial bases and, in particular, for modal DG approaches. The Taylor basis representation of the entropy stabilization term reveals that it penalizes the solution gradients in a manner similar to slope limiting and requires semi-implicit treatment to achieve the desired effect. The implicit Taylor basis version of the Rusanov entropy fix preserves the sparsity pattern of the element mass matrix. Hence, no linear systems need to be solved if the Taylor basis is orthogonal and an explicit treatment of the remaining terms is adopted. The optional application of a vertex-based slope limiter constrains the piecewise-linear DG solution to be bounded by local maxima and minima of the cell averages. The combination of entropy stabilization with flux and slope limiting leads to constrained approximations that possess all desired properties. Numerical studies of the new limiting techniques and entropy correction procedures are performed for two scalar two-dimensional test problems with nonlinear and nonconvex flux functions

Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability

In the research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series [6], the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly. By applying this technique, the authors demonstrate that a pure continuous Galerkin scheme is indeed linear stable if the boundary conditions are done in the correct way. In this work, we extend this investigation to the non-linear case and focusing on entropy conservation. Switching to entropy variables, we will provide an estimation on the boundary operators also for non-linear problems to guarantee conservation. In numerical simulations, we verify our theoretical analysis.Comment: 21 pages,10 figure

Entropy stable discontinuous Galerkin schemes for the Relativistic Hydrodynamic Equations

In this article, we present 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 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 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