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Entropy bounded compact stencil Runge-Kutta discontinuous Galerkin method for hyperbolic conservation laws
Deepak Kumar, Pratima Rai, Arvind Patel
TL;DR
High-order DG methods for Euler equations need to control oscillations while retaining an entropy bound, but the standard SSP-RK argument does not directly cover compact-stencil RKDG's Butcher-form stages. This paper constructs an entropy-bounded cRKDG method using bound-preserving and entropy limiters with convex combinations of forward-Euler updates. It establishes a minimum-entropy principle under stated conditions and reports numerical experiments demonstrating robustness.
Problem
The SSP-RK-based entropy-bound argument does not directly apply to compact-stencil RKDG, which is expressed in Butcher form and need not be a convex combination of forward-Euler steps.
Source: §1 · p. 3Method
The method combines the cRKDG bound-preserving framework with positivity and entropy-bound limiters and proves bounds through convex combinations of DG and local-derivative forward-Euler updates.
Source: §1 · p. 3, §4 · pp. 10–11Results
Under the stated stage and time-step conditions, the cRKDG solution remains in the entropy-admissible set, and numerical experiments are reported to demonstrate robustness.
Source: §4 · p. 11, If the one-dimensional Riemann problem · p. 17, §7 · p. 18Takeaways & Limitations
The approach establishes a minimum-entropy principle for compact-stencil RKDG by treating its Butcher-form stages through the bound-preserving framework.
Source: §1 · p. 3, §4 · p. 11The entropy-bound result assumes each cRKDG stage can be expressed as a convex combination of prior-stage forward-Euler steps and requires limiters after each stage.
Source: §4 · p. 10, If the one-dimensional Riemann problem · p. 16
Abstract
from arXiv · showhide
In this paper we introduce a compact stencil Runge-Kutta discontinuous Galerkin method for the Euler equations of compressible gas dynamics, which preserves a minimum entropy principle similar to that introduced by Tadmor for entropy solution of the Euler equations [4]. Compact stencil RKDG, introduced by Chen et al. in [15], is a variant of standard RKDG method which is known for its smaller stencil size and parallel efficiency. The main challenge is that the cRKDG method has to be expressed in Butcher form and not every Runge-Kutta method of this form is a convex combination of forward-Euler steps. Therefore, in order to establish this property we leverage the bound preserving framework of the cRKDG schemes [8], an entropy bounded limiter [6] and the classical positivity preserving limiter. In the last section numerical experiments are provided to demonstrate the robustness of the method.
1. Introduction
Hyperbolic conservation laws can develop discontinuities that challenge high-order schemes, while standard RKDG often needs limiting or stabilization to control oscillations. The paper develops an entropy-bounded compact-stencil RKDG method by adapting the bound-preserving framework to its Butcher-form stages.
1. Introduction
- High-order schemes can exhibit Gibbs oscillations near discontinuities, and RKDG often uses limiters or stabilization to restrain oscillations and instabilities.The passage names slope limiters, WENO limiters, and artificial viscosity as established approaches.Source: §1 · p. 2
- The method derives a minimum-entropy principle for Lax-Friedrichs, extends it to forward-Euler updates with a local derivative operator, then uses convex combinations of updates to bound cRKDG stages.The proof uses quasi-concavity of specific thermodynamic entropy; numerical experiments are reported to demonstrate accuracy and robustness.Source: §1 · p. 3
- Compact-stencil RKDG introduces a local derivative operator in inner Runge-Kutta stages while retaining the accuracy and local conservation of the original RKDG method.The passage attributes the method to Chen et al..Source: §1 · p. 3
- The cRKDG entropy argument is not directly obtained from the usual SSP-RK approach because cRKDG is expressed in Butcher form, which need not be a convex combination of forward-Euler steps.Prior high-order entropy-bounded DG arguments rely on SSP-RK structure.Source: §1 · p. 3
2. Scheme Formulation
The section motivates entropy admissibility for Euler equations and presents the positivity and entropy-boundedness framework used to constrain numerical solutions. It also explains why the usual high-order DG extension requires care for compact-stencil RKDG.
2. Scheme Formulation
- Euler solutions can become discontinuous, so physically relevant weak solutions are selected using admissibility criteria such as an entropy inequality.The Euler formulation uses density, velocity, total energy, and pressure, with positive density and pressure required for physical entropy to be well defined.Source: §2 · pp. 3–4
- For Euler equations, the entropy solution satisfies a minimum-entropy principle, and high-order DG solutions can be constrained to a convex set with positive density and pressure and entropy no lower than the initial minimum.The specific entropy is quasi-concave, supporting the entropy-bounded framework.Source: §2 · p. 4
- A forward-Euler DG update can bound the element average by the minimum entropy at quadrature points and interfaces, after which a scaling limiter enforces entropy boundedness of the polynomial solution.The described scaling parameter is determined by Newton-Raphson; a later approach in avoids that iterative procedure.Source: §2 · pp. 4–5
- The usual extension to high-order time integration relies on SSP-RK stages as convex combinations of forward-Euler updates, but cRKDG must be expressed in Butcher form to preserve local conservation.Thus the earlier stagewise entropy-limiter methodology does not transfer directly without establishing entropy boundedness at each cRKDG stage.Source: §2 · p. 5
3. Notations and preliminaries
The section sets up the DG and compact-stencil operators, the Runge-Kutta representations, and the entropy-boundedness condition. It recalls how forward-Euler entropy bounds and stagewise limiters support the fully discrete scheme.
3. Notations and preliminaries
- The DG method uses polynomial spaces on mesh elements, with interior and exterior traces denoted by v− and v+ along element edges.The outward edge normal and numerical flux enter the semidiscrete formulation.Source: §3 · p. 5
- The compact-stencil method replaces the DG operator in inner stages with a local derivative operator, while the operators F and G denote the negative DG and local-divergence operators.The method is expressed using Butcher coefficients and stage values.Source: §3 · pp. 5–6
- Entropy boundedness requires the solution at every time level to have specific entropy no lower than the initial minimum; the section illustrates the construction in one dimension on uniform intervals.The setup uses Gauss-Lobatto quadrature and requires positivity of density and pressure at quadrature points for entropy to be defined.Source: §3 · pp. 6–7
- Under its time-step constraint, the forward-Euler DG update satisfies an entropy bound for the element average, which can be enforced for the polynomial solution with an entropy-bound limiter.The limiter is reported not to destroy accuracy or conservation.Source: §3 · p. 7
- For higher-order RKDG, applying positivity and entropy-bound limiters after each stage preserves the entropy bound when the method has the required Shu-Osher convex-combination structure.The usual stagewise construction is distinguished from compact RKDG's Butcher-form treatment.Source: §3 · p. 8
4. Entropy bounded cRKDG method
The section establishes entropy bounds for Lax-Friedrichs and local-derivative forward-Euler updates, then combines these results to show stagewise entropy boundedness for cRKDG under stated conditions. The argument requires convex combinations of prior-stage updates and limiters after each stage.
4. Entropy bounded cRKDG method
- The Lax-Friedrichs average H(Uℓ, Ur; α) belongs to G0 when the Riemann solution is self-similar, remains in G0, and its waves lie within |ξ| ≤ α.The resulting average is used to establish the entropy bound for the Lax-Friedrichs scheme.Source: §4 · pp. 8–9
- The Lax-Friedrichs scheme for one-dimensional Euler equations is positivity preserving and satisfies an entropy principle under its stated CFL constraint.The proof represents the update as a convex combination of admissible states, then invokes the entropy principle.Source: §4 · p. 9
- Forward-Euler updates using the local derivative operator, uh ± ΔtG(uh), satisfy an entropy principle under the stated conditions.The proof uses weak bound preservation, a Riemann-problem construction, the Lax-Friedrichs bound, and quasi-concavity of specific entropy.Source: §4 · pp. 9–10
- A cRKDG stage preserves the entropy bound when it is expressed as a convex combination of forward-Euler steps from earlier stages involving the DG and local derivative operators.The stated stage argument assumes previous-stage values satisfy the entropy bound and uses the corresponding limiter procedure.Source: §4 · pp. 10–11
- Theorem 4.1 establishes that cRKDG remains in the entropy-admissible set when each stage is a convex combination of prior-stage forward-Euler steps using ±G and F, with both bound-preserving and entropy-bound limiters applied after every stage.The theorem imposes a time-step constraint.Source: §4 · p. 11
- The theorem's conclusion places the limited cRKDG solution in the entropy-admissible set on every element.The final-stage argument applies the stage result and concludes membership in G for all K.Source: §4 · p. 12
5. Two dimensional case
The two-dimensional case formulates the Euler equations on a uniform rectangular mesh and develops directional Lax–Friedrichs averages and convex decompositions for the cell-average update.
5. Two dimensional case
- The two-dimensional Euler equations are considered on a uniform rectangular mesh, with each cell’s solution represented by a polynomial approximation.The cell average is defined using quadrature on the reference interval, and Gauss–Lobatto points are used for the element quadrature.Source: §5 · pp. 12–13
- The scheme defines admissible-state conditions on cell quadrature points and traces, then writes the DG update using one-dimensional numerical fluxes.The fluxes used are Lax–Friedrichs numerical fluxes.Source: §5 · p. 13
- The cell-average update is rewritten through Jiang–Liu convex decomposition and one-dimensional slice averages.The text notes that general or optimal convex decompositions are alternatives.Source: §5 · pp. 13–14
- Lemma 5.1 introduces a result for each unit vector, providing a preceding lemma for the two-dimensional entropy analysis.Source: §5 · p. 14
If the one-dimensional Riemann problem
This section establishes entropy bounds for forward-Euler updates and extends them to cRKDG stages under convex-combination and time-step conditions. The resulting theorem requires applying the bound-preserving and entropy-bound limiters after each stage.
If the one-dimensional Riemann problem
- The section introduces directional Lax–Friedrichs averages and a preceding lemma that supplies entropy principles for those averages under CFL constraints.The stated constraints bound λ1α0 and λ2α0 by 1.Source: If the one-dimensional Riemann problem · pp. 14–15
- The forward-Euler update with ±G satisfies the entropy principle when the solution is admissible on Sij and the CFL constraint holds.The proof uses the weak bound-preserving property for the operators ±G and quasi-concavity of entropy.Source: If the one-dimensional Riemann problem · pp. 15–16
- The update with F satisfies the entropy principle when the solution is admissible on the enlarged set ˜Sij.Its cell average is expressed as a convex combination of finite-volume updates and admissible states.Source: If the one-dimensional Riemann problem · p. 16
- For admissible left and right states, the intermediate state H(Uℓ, Ur; α) belongs to G0 under the stated assumptions.Source: If the one-dimensional Riemann problem · pp. 14, 16
- Under the stated time-step constraint, each cRKDG stage written as a convex combination of prior forward-Euler updates preserves the entropy principle when both limiters are applied after every stage.The theorem includes the operators ±G and F and requires the bound-preserving limiter together with the entropy-bound limiter.Source: If the one-dimensional Riemann problem · p. 17
6. Numerical Experiments
The numerical experiments apply EBcRKDG to one- and two-dimensional test cases to assess accuracy and its ability to restrain spurious oscillations. The one-dimensional subsection presents a density plot for the Sod shock tube.
6. Numerical Experiments
- The experiments apply the positivity-preserving limiter and entropy-bound limiter consecutively after every Runge–Kutta stage update.Two-dimensional problems use rectangular meshes.Source: §6 · p. 17
6.1 One-dimensional test cases
- The Sod shock tube is computed on 100 elements to final time T = 0.2, with the entropy bound set to s0 = ln(1).The bound is the minimum entropy of the initial condition; the displayed scheme uses k = 2 and a third-order RK method in Butcher form.Source: §6.1 · p. 17
- Figure 1 plots density for the entropy-bounded cRKDG scheme with k = 2.Source: Figure 1 · p. 17, §6.1 · p. 17
7. Summary and Conclusion
The conclusion presents EBcRKDG for hyperbolic conservation laws and explains how its stage updates and limiters yield a fully discrete entropy principle. The construction extends entropy bounding to cRKDG schemes expressed in Butcher form.
7. Summary and Conclusion
- The paper develops an entropy-bounded compact Runge–Kutta discontinuous Galerkin method for hyperbolic conservation laws.Source: §7 · p. 18
- The authors report that cRKDG element averages with a local derivative operator satisfy an entropy principle similar to the DG operator’s.Source: §7 · p. 18
- The bound-preserving framework expresses each Runge–Kutta stage as a convex combination of prior-stage forward-Euler updates involving local derivative and DG operators.Source: §7 · p. 18
- Applying the entropy-bound and positivity-preserving limiters yields the entropy principle for the fully discrete schemes.Source: §7 · p. 18
- The conclusion identifies cRKDG’s Butcher-form stages as a case not covered by earlier techniques based on method-of-lines DG and SSP Runge–Kutta methods.Source: §7 · p. 18