Skew-Symmetric Entropy Stable Modal Discontinuous Galerkin Formulations

Abstract

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 [1][2][3][4][5] or volume and surface quadrature rules [6,7] 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 [6] 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 2N −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. IntroductionHigh order methods for the simulation of time-dependent compressible flow have the potential to achieve higher levels of accuracy at lower costs compared to current low order schemes [8]. In addition to superior accuracy, the low numerical dispersion and dissipation of high order methods [9] enables the accurate propagation of waves over long distances and time scales. The same properties also make high order methods attractive for unsteady phenomena such as vorticular and turbulent flows, which are sensitive to numerical dissipation [8,10].

A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equations

“…where x 0 = 0.33, γ = 5 3 . We set the exterior values at endpoints of the domain x = 0, 1 to be u L , u R to enforce inhomogeneous Dirichlet boundary condition.…”

“…However, high-order DG methods typically suffer from stability issues in the presence of discontinuous solutions, due to the lack of numerical dissipation. High-order entropy stable DG (ESDG) discretizations are high order accurate and satisfy entropy balances at the discrete level for the compressible Euler and Navier-Stokes equations, so they remain robust even on the absence of the chain rule and exact integration at the discrete level [4,5,6,7,8,9,10]. However, a discrete entropy balance is not sufficient to guarantee the stability of high order ESDG schemes.…”

A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations

“…Afterwards, several examples of classical schemes are rephrased as SBP schemes. While there are generalizations of SBP methods [26,27,81,87,88], we concentrate here on nodal collocation schemes where the boundary points are included. Thus, an interval [ min , max ] is discretized using a grid 1…”

“…Consider the Degasperis-Procesi equation [31] ( 26) with periodic boundary conditions, which can also be written as 27) where (I − 2 , ) −1 is the inverse of the elliptic operator I − 2 with periodic boundary conditions. The functionals…”

Entropy Stable Discontinuous Galerkin Methods for Balance Laws in Non-Conservative Form: Applications to the Euler Equations with Gravity

“…Following the pioneering work in Carpenter et al [9], Gassner et al [25] for the compressible Euler equations, the class of entropy-stable DG schemes has been extended to general systems of nonlinear conservation laws by Chen and Shu [16]. Chan [13,14] generalized this to "modal" DG formulations which do not satisfy summation-by-parts. Most entropy stable DG schemes are constructed assuming exact integration in time, and Ranocha et al [45] achieved fully discrete entropy-stability using relaxation Runge-Kutta time integrators.…”