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Limiter-based fully-discrete entropy stable explicit DG schemes for ideal MHD equations
Yuchang Liu, Yan Jiang, Zheng Sun
TL;DR
Entropy-stable explicit DG schemes for MHD are difficult to construct because Godunov’s non-conservative source term complicates entropy analysis. This paper introduces generalized path decomposition and combines it with an ES limiter to prove fully discrete entropy guarantees. The scheme is reported to have low numerical dissipation and strong robustness, and its entropy inequality supports a Lax–Wendroff-type result for solution limits.
Problem
Constructing high-order explicit DG schemes with rigorous fully-discrete entropy guarantees for MHD is challenging because the Godunov source term obstructs the weak entropy inequality.
Source: §1 · pp. 1–2Method
The method interprets the source-term volume integral as a path integral along a polynomial path and combines this analysis with an ES limiter for explicit DG schemes.
Source: §1 · p. 2, §7 · p. 23Results
The proposed scheme enforces the genuine cell entropy inequality, and numerical tests report low numerical dissipation and strong robustness.
Source: §7 · pp. 23–24Takeaways & Limitations
The fully-discrete entropy inequality yields a Lax–Wendroff-type result showing that solution limits satisfy the entropy condition.
Source: §1 · p. 3, §7 · p. 23Extension of the current framework to globally divergence-free methods remains future work.
Source: §7 · p. 24
Abstract
from arXiv · showhide
We propose a class of high-order fully-discrete entropy stable (ES) explicit discontinuous Galerkin (DG) solvers for the compressible ideal magnetohydrodynamics (MHD) equations. Our main theoretical contribution is the introduction of a novel generalized-path-decomposition framework for MHD equations in Godunov's symmetric form. By innovatively interpreting the interior volume integral of the non-conservative source term as a path integral along a generalized path constructed by the solution polynomial, we establish the weak cell entropy inequality for the fully-discrete DG schemes. This overarching framework also accommodates other existing DG solvers based on the symmetric form. Combined with a carefully designed ES limiter, the proposed scheme satisfies the genuine fully-discrete cell entropy inequality. With this property, a Lax--Wendroff-type theorem can be obtained to show that the solution limit satisfies the entropy condition. Finally, the scheme is naturally compatible with the locally divergence-free space. Extensive numerical experiments demonstrate the scheme's low numerical dissipation and strong robustness.
1. Introduction.
Entropy-stable DG methods for MHD must handle the Godunov source term, which makes entropy analysis harder than for standard conservation laws. This work uses generalized path decomposition and an entropy limiter to establish fully discrete entropy guarantees for explicit DG schemes.
1. Introduction.
- The generalized-path-decomposition framework interprets the source-term volume integral as a path integral along a polynomial path, establishing a weak cell entropy inequality for MHD.The DG formulation preserves the prescribed path defining the non-conservative product; generalized paths are an analytical tool for entropy analysis.Source: §1 · p. 2
- The ES limiter yields a genuine fully-discrete cell entropy inequality, and a Lax–Wendroff-type theorem shows that solution limits satisfy the entropy condition.The analysis also establishes fully-discrete entropy inequalities for a class of first-order schemes and a weak cell entropy inequality for forward-Euler DG.Source: §1 · pp. 2–3
- The framework also applies, with minor modifications, to other DG discretizations of Godunov’s symmetric form and is naturally compatible with locally divergence-free magnetic fields.The paper notes that without the locally divergence-free property, schemes may produce nonphysical solutions or become unstable.Source: §1 · pp. 2–3
- Numerical tests report optimal convergence rates, low numerical dissipation, and strong robustness while respecting physical constraints.Source: §1 · p. 2
2. Ideal MHD equations.
The ideal MHD equations describe compressible plasma dynamics and include a magnetic divergence constraint. Their entropy structure motivates a symmetric form with a non-conservative product whose weak interpretation depends on a chosen path.
2.1. Governing equation.
- The governing equations describe d-dimensional compressible ideal MHD, with state quantities including density, momentum density, total energy density, pressure, and magnetic field.Source: §2.1 · p. 3
2.2. Divergence-free property.
- The magnetic field remains divergence-free over time if it is initially divergence-free, and preserving this constraint discretely is critical to avoiding nonphysical artifacts and numerical instabilities.Source: §2.2 · p. 3
2.3. Entropy structure and symmetrizable form.
- The entropy relation for the conservative MHD form does not satisfy the stated entropy-pair definition, so Godunov’s symmetric form introduces a non-conservative product and the scheme targets that form while locally enforcing divergence freedom.At discontinuities the non-conservative product must be defined carefully; the symmetric form is equivalent to the conservative form for smooth solutions.Source: §2.3 · pp. 4–5, §2.4 · p. 5
2.4. Non-conservative product and paths.
- Weak solutions of the non-conservative system depend on the chosen path, which should reflect physical considerations; the numerical tests use a linear path although the scheme is derived for any given path.Source: §2.4 · p. 5
3. First-order building block in one dimension.
First-order three-point schemes provide building blocks for entropy-stable DG methods, but the MHD source term makes their entropy analysis more challenging. The section introduces path-conservative and generalized-path formulations to support cell entropy inequalities.
3. First-order building block in one dimension.
- First-order three-point entropy-stable schemes serve as building blocks for high-order methods, with MHD analysis complicated by its entropy pair and non-conservative source term.Source: §3 · p. 6
3.1. Path-conservative scheme.
- The path-conservative scheme uses D± operators to approximate Godunov’s source term and satisfy path-conservation requirements; the analysis also applies to the LF flux.Source: §3.1 · pp. 6–7
3.2. A generalized scheme.
- Generalized paths may vary across interfaces, allowing the high-order analysis to represent connections induced by the solution polynomial; this formulation is analytical and is not implemented.Source: §3.2 · p. 7
3.3. A cell entropy inequality.
- The three-point scheme is entropy stable under a sufficient CFL condition, which can be checked after computation by rejecting a violating step and recomputing with a smaller time step.Source: §3.3 · pp. 8–9
4. High-order ES scheme in one dimension.
The one-dimensional DG analysis interprets the interior source-term integral through a generalized path induced by the solution polynomial, proving a weak cell entropy inequality. A scaling limiter then yields the genuine inequality, and the result extends to high-order time stepping and entropy-stable solution limits under stated assumptions.
4.1. A weak cell entropy inequality for cell averages.
- The forward-Euler DG update satisfies a weak cell entropy inequality under the cases and CFL conditions of Theorem 4.1.The proof decomposes the update into three-point schemes; convexity and telescoping intermediate entropy fluxes yield the inequality.Source: §4.1 · pp. 11–13
- The interior source-term integral is treated analytically as a generalized path between internal states, enabling a three-point decomposition without changing the standard path-conservative DG scheme.The generalized path is induced by the solution polynomial and is an analytical tool for the entropy proof.Source: §4.1 · p. 13
- The analysis preserves the weak entropy inequality for any prescribed interface path defining the Godunov-form non-conservative product, including the physically relevant path.The polynomial generalized path is used for the interior source-term analysis.Source: §4.1 · p. 13
4.2. Scaling limiter and genuine cell entropy inequality.
- Scaling the predicted solution toward its cell average produces a limited solution satisfying the genuine discrete cell entropy inequality.The scaling does not increase cell entropy.Source: §4.2 · p. 14
4.3. High-order time discretization.
- An SSP multistep method extends the construction to high-order time discretization: under the corresponding forward-Euler CFL conditions, it preserves the weak inequality, and with the limiter it satisfies the genuine inequality.For smooth solutions, the limiter preserves spatial accuracy under appropriate assumptions.Source: §4.3 · p. 15
4.4. Entropy stability of the solution limit.
- Under the stated convergence and TVB-like assumptions, limits of the forward-Euler or multistep schemes with the entropy-stable limiter satisfy the entropy inequality in distributions.For the multistep scheme, the starting values must also be consistent.Source: §4.4 · p. 15
5. Extension to multi-dimensions.
The one-dimensional framework extends direction by direction to multiple dimensions, where a fully-discrete DG scheme with the entropy-stable limiter satisfies a cell entropy inequality. The locally divergence-free space is compatible with the construction and limiter.
5. Extension to multi-dimensions.
- The multidimensional construction extends the one-dimensional framework direction by direction, with the 2D example using a tensor-product mesh and directional operators.The forward-Euler fully-discrete DG scheme is given by the one-dimensional update.Source: §5.1 · p. 16
5.2. Fully-discrete ES scheme.
- The two-dimensional scheme with forward-Euler time discretization and the entropy-stable limiter satisfies the discrete cell entropy inequality and is globally entropy stable.The extension to high-order temporal accuracy with an SSP multistep method is analogous to the one-dimensional case.Source: §5.2 · p. 18
5.3. Locally divergence-free methods.
- In the locally divergence-free space, the magnetic field has zero divergence within each cell, so the corresponding interior source integral vanishes.The limiter preserves the magnetic field's divergence-free property while enforcing the genuine cell entropy inequality.Source: §5.3 · p. 18
6. Numerical tests.
The numerical tests show that the entropy-stable schemes preserve cell entropy and remain robust across smooth, shock-driven, turbulent, and extreme MHD flows. The LDF variant also reduces numerical dissipation and divergence errors in several tests.
Example 6.1: Smooth MHD vortex
- For the smooth vortex, all schemes achieve optimal third-order accuracy for k = 2, while ES and ES-LDF keep entropy violation at machine precision.LDF variants yield smaller divergence errors, and all schemes except Base show second-order divergence convergence.Source: §6 · p. 19, Figure 1 · p. 20, Figure 2 · p. 20
Example 6.2: Orszag–Tang vortex
- In the Orszag–Tang test, ES and ES-LDF remain stable after shocks appear near T ≈ 1.08, while non-ES variants blow up; their later density results capture small structures.The semi-discrete ES scheme also blows up without a shock limiter.Source: §6 · pp. 19–20, Figure 4 · p. 21
Example 6.3: Kelvin–Helmholtz instability
- In the Kelvin–Helmholtz test, ES and ES-LDF remain stable to T = 20 while non-ES variants blow up; ES-LDF captures finer structures than ES.ES-LDF also avoids the unnatural magnetic-energy flattening seen in ES near t ≈ 5.Source: §6 · pp. 20–21, Figure 5 · p. 22
Example 6.4: Cloudshock interaction
- With the positivity-preserving limiter, all six cloudshock schemes remain stable and produce similar results; ES-LDF resolves complex structures with low numerical dissipation.Its solution agrees with results in the literature [25].Source: §6 · p. 22, Figure 7 · p. 23
Example 6.5: Extreme MHD jet
- In the extreme MHD jet, ES and ES-LDF with positivity preservation remain stable and track the jet and bow shock, whereas non-ES schemes blow up.The non-ES variants blow up even with the positivity-preserving limiter.Source: Figure 8 · p. 23, §6 · p. 23
7. Concluding remarks.
The paper introduces a generalized-path-decomposition framework for MHD and combines it with an entropy-stable limiter to establish fully-discrete entropy guarantees. The method is compatible with locally divergence-free methods, while globally divergence-free and strictly conservative extensions remain future work.
7. Concluding remarks.
- The generalized-path-decomposition framework interprets the source-term volume integral as an exact integral along a polynomial path constructed by the DG solution.This establishes the weak entropy inequality for cell-average update schemes.Source: §7 · p. 23
- With the entropy-stable limiter, the explicit scheme enforces the genuine cell entropy inequality and global entropy stability, supporting a Lax–Wendroff-type theorem for entropy-consistent solution limits.Source: §7 · p. 23
- The framework also applies to several classical solvers viewed as using low-order approximations of path integrals, and the proposed scheme combines with locally divergence-free methods.The locally divergence-free methods enforce zero divergence within each cell.Source: §7 · p. 23
- Numerical tests report low numerical dissipation for resolving complex structures and strong robustness.Source: §7 · pp. 23–24
- Future work targets extending the framework to globally divergence-free methods and constructing fully-discrete entropy-stable, globally divergence-free, strictly conservative DG schemes.Source: §7 · p. 24