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On discretely entropy conservative and entropy stable discontinuous Galerkin methods

Jesse Chan

arXiv:1708.01243v5math.NA

TL;DR

High-order DG methods need discrete entropy control for nonlinear hyperbolic systems, especially beyond diagonal-norm SBP constructions. The paper uses flux differencing with quadrature-based projections, lifting, and SBP-like operators to construct generalized entropy-conservative schemes. The resulting methods are entropy conservative or stable under the selected volume quadrature and show high-order accuracy and improved robustness for compressible Euler tests.

  • Problem

    The paper addresses the need for high-order DG methods with discrete entropy conservation or dissipation beyond diagonal-norm SBP-DG settings.

  • Method

    The method combines flux differencing, quadrature-based projections and lifting, and SBP-like operators for more general approximation spaces and quadrature rules.

  • Results

    The resulting DG schemes satisfy discrete entropy conservation, while compressible Euler experiments indicate high-order accuracy and improved stability and robustness for under-resolved and shock solutions.

  • Takeaways & Limitations

    The framework extends discretely entropy conservative DG methods to general approximation spaces and sufficiently accurate quadrature rules.

  • Takeaways & Limitations

    The required high-order accuracy of the difference between entropy-projected and original conservative variables has not been proven, although experiments indicate r = N + 1 works in one and two dimensions.

Abstract

from arXiv · show

High order methods based on diagonal-norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as nodal discontinuous Galerkin methods with diagonal mass matrices. In this work, we describe how use flux differencing, quadrature-based projections, and SBP-like operators to construct discretely entropy conservative schemes for DG methods under more arbitrary choices of volume and surface quadrature rules. The resulting methods are semi-discretely entropy conservative or entropy stable with respect to the volume quadrature rule used. Numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible Euler equations in one and two dimensions.

1. Introduction

High-order DG methods offer accurate, efficient discretizations for complex hyperbolic problems, but nonlinear stability and suitable entropy-stable constructions remain challenging beyond diagonal-norm SBP settings.

  • High-order methods improve accuracy per degree of freedom while maintaining reasonable computational costs for sufficiently regular solutions.
  • High-order DG methods are well suited to time-dependent hyperbolic problems on unstructured meshes and modern computing architectures.
  • Nonlinear hyperbolic problems can destabilize high-order schemes, whereas low-order schemes gain stabilization from greater numerical dissipation.
  • Diagonal-norm SBP operators remain difficult to construct for some elements, with three-dimensional point sets reported only through N ≤4 and no high-order construction for pyramids.
  • The paper generalizes entropy-conservative DG schemes using over-integrated quadrature, quadrature-based projections, and a decoupled SBP-like operator.
  • Existing entropy-stable methods outside diagonal-norm SBP-DG often rely on continuous-level proofs using exact integration or the chain rule.

2. Entropy stability for systems of hyperbolic PDEs

Entropy stability extends energy-stability ideas to nonlinear conservation laws through convex entropy functions, entropy variables, and discrete entropy conservation or dissipation inequalities.

  • The framework begins from one-dimensional systems of nonlinear conservation laws with n variables and smooth flux functions.
  • For systems with a convex entropy function, entropy variables are defined as v = U′(u), and the conservative-to-entropy-variable mapping is invertible.
  • For smooth solutions, multiplying the conservation law by the entropy variables and applying the chain rule yields entropy conservation.
  • Physically relevant solutions more generally satisfy an entropy inequality rather than equality.
  • The paper targets high-order polynomial DG methods satisfying discrete analogues of entropy conservation and entropy dissipation.

3. Discrete differential operators and quadrature-based matrices

The paper builds quadrature-based projection, lifting, and differentiation operators that reproduce SBP-like identities under general volume and surface quadrature rules.

  • The approximation uses degree N polynomials on reference elements, with element-dependent spaces in higher dimensions.
  • Volume and surface quadrature rules are assumed sufficiently accurate to ensure integration by parts for degree-N polynomial pairs.
  • The mass matrix is symmetric and positive definite under the quadrature assumptions, without requiring it to be diagonal.
  • Quadrature-based projection and lifting matrices map function evaluations at volume or surface points to polynomial coefficients.
  • The quadrature-based differentiation operator D_i^q = V_qD_iP_q approximates derivatives and satisfies an SBP property with respect to diagonal quadrature weights.
  • The projection matrix transforms a modal SBP property with potentially dense M into a quadrature-based SBP property involving diagonal W and projected boundary evaluations.
  • The SBP-like operator cannot be used directly as a differentiation operator, motivating a separate treatment of discrete differentiation.

4. Entropy conservative DG methods on a single element

The method interprets flux differencing through quadrature-based projections and constructs a single-element DG formulation whose entropy balance follows from symmetric two-point fluxes and SBP-like operators.

  • 4. Entropy conservative DG methods on a single element: The construction uses a symmetric, consistent two-point flux and Tadmor’s entropy-conservative condition to obtain discrete entropy conservation.
  • 4. Entropy conservative DG methods on a single element: Flux differencing is reformulated through quadrature-based matrices, Hadamard products, and diagonal entries of matrix products.
  • 4. Entropy conservative DG methods on a single element: The DG formulation combines projection and lifting matrices with the decoupled SBP-like operator D_N for polynomial approximations.
  • 4. Entropy conservative DG methods on a single element: Entropy-projected conservative variables are evaluated at volume and surface quadrature points and used to form the symmetric flux matrix F_S.
  • 4. Entropy conservative DG methods on a single element: Theorem 2 establishes semi-discrete entropy conservation for the formulation when the two-point flux is entropy conservative and time is continuous.
  • 4. Entropy conservative DG methods on a single element: The resulting entropy balance approximates continuous entropy conservation using numerical quadrature and a numerical boundary flux.

5. Entropy stable DG methods on multiple elements and in higher dimensions

The paper extends entropy-conservative and entropy-stable DG constructions from single-element settings to multiple elements and higher dimensions using quadrature-based, SBP-like formulations. Interface fluxes and added dissipation yield local conservation and discrete entropy inequalities under stated boundary and mapping conditions.

  • 5.1. Multiple elements: Multiple-element schemes combine elementwise entropy conservation with interface-term cancellation to obtain local conservation and entropy conservation.For periodic domains, shared interface contributions cancel after summing the one-element identities over all elements.
  • 5.1. Multiple elements: The formulation supports entropy stability by adding dissipative terms, including matrix dissipation or Lax-Friedrichs penalization in entropy variables.The local Lax-Friedrichs flux is entropy dissipative when its wave-speed estimate is chosen appropriately.
  • 5.1. Multiple elements: Entropy-stable boundary fluxes imply a discrete global entropy inequality for the semi-discrete scheme, with an analogous result for periodic boundaries.The result follows from the multiple-element entropy statement when boundary conditions make f ∗ entropy stable.
  • 5.2. Higher dimensions: Higher-dimensional schemes use affine element and face mappings, geometric factors, Jacobian-weighted quadrature matrices, and higher-dimensional entropy-conservative fluxes.Theorem 4 states that the resulting scheme is locally conservative and satisfies a quadrature-based higher-dimensional entropy-conservation relation.
  • 5.2. Higher dimensions: The multidimensional construction preserves the same structural outcome: local conservation and entropy conservation relative to numerical quadrature and boundary numerical fluxes.The result is expressed through SBP-like operators and entropy-projected conservative variables at volume and surface quadrature points.

6. Numerical experiments: the compressible Euler equations

The numerical experiments study the entropy conservation and accuracy of the proposed schemes for the one-dimensional compressible Euler equations. Implementations use fourth-order five-stage low-storage Runge–Kutta time integration and show that the discretization is determined by quadrature choices rather than basis choice.

  • 6. Numerical experiments: the compressible Euler equations: The one-dimensional compressible Euler experiments use a fourth-order five-stage low-storage Runge–Kutta method.The timestep is defined using a trace-inequality constant and a user-defined CFL parameter.
  • 6. Numerical experiments: the compressible Euler equations: The implementation is basis-independent because the discretization is specified completely by the volume and surface quadrature choices.Gauss–Lobatto quadrature recovers entropy-conservative or entropy-stable DG-SEM discretizations, while Gauss quadrature with N + 1 points recovers generalized SBP-DG methods.

6.1. One-dimensional experiments

The one-dimensional experiments assess entropy-conservative and Lax–Friedrichs fluxes across smooth, discontinuous, and shock-containing Euler problems using GLL and over-integrated Gauss quadratures. Results show high-order accuracy and entropy behavior, but also expose oscillation, positivity, timestep, and projection-sensitivity limitations.

  • Setup: The Euler experiments assume an ideal gas with γ = 1.4 and require discrete density and pressure to remain bounded away from zero.Positivity-preserving limiters could enforce this condition, but they were not implemented in the simulations.
  • Discontinuous profile on a periodic domain: For the periodic discontinuous profile, entropy-conservative entropy changes decrease as CFL and timestep decrease, approaching the semi-discrete zero-entropy-change behavior.Lax–Friedrichs retains a non-zero entropy change that is comparatively insensitive to CFL.
  • Sod shock tube: In the Sod shock tube, entropy-conservative simulations diverge as oscillations drive density and temperature negative, while Lax–Friedrichs avoids observed blowup.Cell averages for both quadratures nevertheless agree relatively well with the exact solution, with spurious oscillations remaining.
  • Sine-shock interaction: In the sine-shock interaction, GLL and GQ-(N + 2) cell averages remain close to the WENO reference, but Gauss quadrature requires CFL .05 to prevent blowup.GQ-(N + 2) produces smoother, smaller oscillations, while GLL is stable at the larger CFL .125.

6.2. Two-dimensional experiments

The two-dimensional experiments assess entropy conservation, convergence, and shock-capturing behavior for the proposed high-order DG schemes for compressible Euler equations. They show time-step-dependent entropy conservation, near-optimal vortex convergence, and qualitative agreement for a two-dimensional Riemann problem.

  • Experimental setup: The experiments use degree-2N-exact triangular volume quadrature with (N + 1)-point one-dimensional Gauss quadrature on faces.L2 errors are evaluated with a degree-2N + 2 volume quadrature rule.
  • 6.2.1. Discontinuous profile on a periodic domain: The entropy change decreases with the time-step and converges to zero as O(∆t4) at T = 2 for the entropy conservative scheme.A jump near t = 1.86 does not affect the final-time convergence order, matching the fourth-order time integrator.
  • 6.2.2. Isentropic vortex problem: For N = 4, the vortex experiment uses a periodic rectangular domain [0, 20]×[−5, 5] and evolves the solution to T = 5.The mesh is formed by subdividing uniform quadrilateral elements into triangles.
  • 6.2.2. Isentropic vortex problem: The isentropic vortex achieves optimal O(hN+1) L2 convergence for N = 1, . . . , 3.For N = 4, the observed rate lies between O(hN+1) and O(hN+1/2), and halving the time-step changes the rate only from 4.785 to 4.8.
  • Two-dimensional Riemann problem: The entropy-stable Riemann-problem computation uses Lax-Friedrichs penalization without additional stabilization, artificial dissipation, or limiting.On an 8192-element triangular mesh with N = 3 and CFL .125, the physical-domain result qualitatively agrees with results in the literature.

6.3. On accuracy and computational cost

The proposed schemes achieve high-order accuracy in numerical experiments, while over-integrated quadrature broadens admissible approximation-space and quadrature pairings at increased computational cost. Accuracy depends on conditions involving entropy projection, especially when entropy and conservative variables differ.

  • Accuracy: For v ≠ u, high-order accuracy requires the difference between entropy-projected and conservative variables to be high-order accurate in smooth regions.The authors report numerical evidence suggesting r = N + 1 in one and two dimensions, but do not provide a proof.
  • Accuracy: The accuracy experiment uses smooth projected conservation variables on uniform one- and two-dimensional meshes with quadrature exact for degree 2N polynomials in two dimensions.The error is evaluated with a quadrature rule two degrees higher.
  • Accuracy: O(h^N+1) convergence rates are observed for N = 1, . . . , 5 in the L2 error between conservative and entropy-projected conservative variables.This indicates high-order approximation of the conservative variables and flux in the reported one- and two-dimensional experiments.
  • Accuracy: The entropy-projection error depends on the nonlinearity of the mapping between conservative and entropy variables and is expected to increase near nonphysical thermodynamic-variable limits.The mapping is described as non-invertible when ρ ≤ 0 or E − 1/(2ρ)|u|^2 ≤ 0.
  • Computational cost: Over-integrated quadrature permits fewer approximation-space degrees of freedom than quadrature-space dimension but requires L2 projections and projection/lifting operations for each right-hand-side evaluation.Consequently, the proposed methods cost more than SBP methods using under-integrated quadrature, although over-integration can improve accuracy in some cases.

7. Conclusions

The paper generalizes discretely entropy conservative DG methods to more general approximation spaces and sufficiently accurate quadrature rules. Its compressible-Euler experiments indicate high-order accuracy, improved stability and robustness, and identify limitations that entropy stability alone does not resolve.

  • Conclusions: Quadrature-based projection and lifting matrices define SBP-like operators that generalize discretely entropy conservative DG schemes beyond diagonal-norm SBP-DG methods.The resulting schemes satisfy a discrete conservation of entropy for more general approximation-space and quadrature pairings.
  • Conclusions: Compressible-Euler experiments indicate high-order accuracy for smooth solutions and improved stability and robustness for under-resolved and shock solutions compared with non-entropy conservative and non-entropy stable schemes.The paper also reports differing sensitivities between Gauss-Lobatto- and Gauss-quadrature-based methods.
  • Limitations: Entropy conservative and entropy stable schemes do not by themselves address spurious oscillations in high-order shock approximations or positivity preservation for density and pressure.Filtering, artificial viscosity, or limiting can address these issues but may introduce reliance on regularization techniques.
  • Limitations: Three-dimensional generalizations and analytical error estimates between conservative and entropy-conservative variables remain future work.Such estimates are identified as necessary for error estimates of the proposed entropy-stable methods.

Appendix A. Recovery of known schemes for Burgers’ equation

The appendix shows how flux differencing recovers known split-form schemes for Burgers’ equation. The recovery uses entropy-conservative two-point fluxes, projection identities, and differential operators satisfying D1 = 0.

  • Split-form recovery: For smooth solutions, the split form of Burgers’ equation conserves the square entropy U(u) = u^2/2.The split formulation can also be recovered through flux differencing with an appropriate two-point flux.
  • Operator condition: Flux differencing recovers the split form when combined with a differential operator D satisfying D1 = 0.This property removes constant-state contributions in the Hadamard-product formulation.
  • Known schemes: The decoupled SBP operator recovers existing entropy-stable Burgers’ schemes under periodic boundary conditions and an energy-conserving flux.The appendix defines the flux matrix entries from squared and cross products of solution values at quadrature points.
  • Multi-element formulation: Applying the one-dimensional multi-element formulation produces a volume Hadamard-product term together with a surface correction involving the entropy-stable flux and projected face flux.The appendix then simplifies these terms using the SBP-like operator structure and projection relations.
  • Collocated Gauss case: When Gauss quadrature has N + 1 points with a co-located Lagrange basis, the projection and interpolation operators become identity matrices.The formulation then reduces to the weak split form with the corresponding Burgers’ correction terms.
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