Design of provably physical-constraint-preserving methods for general relativistic hydrodynamics

Wu, Kailiang

doi:10.1103/physrevd.95.103001

Cited by 46 publications

(31 citation statements)

References 47 publications

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“…The convexity of admissible state set is very useful in bound-preserving analysis, because it can help reduce the complexity of analysis if the schemes can be rewritten into certain convex combinations, see e.g., [51,53,40]. For the ideal MHD, the convexity of G * or G can be easily shown by definition.…”

Section: Basic Propertiesmentioning

confidence: 99%

“…where C s = γp/ρ is the sound speed. If true, the LF splitting property would be very useful in analyzing the PP property of the schemes with the LF flux, see its roles in [51,42,40] for the equations of hydrodynamics. Unfortunately, for the ideal MHD, (8) is untrue in general, as evidenced numerically in [11] for ideal gases.…”

Section: Basic Propertiesmentioning

confidence: 99%

“…l 1 -error order l 2 -error order l ∞ -error order 40 2 The first is a 1D vacuum shock tube problem [13] with γ = 5 3 and the initial data given by (ρ, v, p, B)(x, 0) = (10 −12 , 0, 0, 0, 10 −12 , 0, 0, 0), x < 0, (1, 0, 0, 0, 0.5, 0, 1, 0),…”

Section: Meshmentioning

confidence: 99%

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Positivity-Preserving Analysis of Numerical Schemes for Ideal Magnetohydrodynamics

Wu¹

2018

SIAM J. Numer. Anal.

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Numerical schemes provably preserving the positivity of density and pressure are highly desirable for ideal magnetohydrodynamics (MHD), but the rigorous positivity-preserving (PP) analysis remains challenging. The difficulties mainly arise from the intrinsic complexity of the MHD equations as well as the indeterminate relation between the PP property and the divergence-free condition on the magnetic field. This paper presents the first rigorous PP analysis of conservative schemes with the Lax-Friedrichs (LF) flux for one-and multi-dimensional ideal MHD. The significant innovation is the discovery of the theoretical connection between the PP property and a discrete divergence-free (DDF) condition. This connection is established through the generalized LF splitting properties, which are alternatives of the usually-expected LF splitting property that does not hold for ideal MHD. The generalized LF splitting properties involve a number of admissible states strongly coupled by the DDF condition, making their derivation very difficult. We derive these properties via a novel equivalent form of the admissible state set and an important inequality, which is skillfully constructed by technical estimates. Rigorous PP analysis is then presented for finite volume and discontinuous Galerkin schemes with the LF flux on uniform Cartesian meshes. In the 1D case, the PP property is proved for the first-order scheme with proper numerical viscosity, and also for arbitrarily high-order schemes under conditions accessible by a PP limiter. In the 2D case, we show that the DDF condition is necessary and crucial for achieving the PP property. It is observed that even slightly violating the proposed DDF condition may cause failure to preserve the positivity of pressure. We prove that the 2D LF type scheme with proper numerical viscosity preserves both the positivity and the DDF condition. Sufficient conditions are derived for 2D PP high-order schemes, and extension to 3D is discussed. Numerical examples further confirm the theoretical findings.

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Section: Basic Propertiesmentioning

confidence: 99%

Section: Basic Propertiesmentioning

confidence: 99%

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Positivity-Preserving Analysis of Numerical Schemes for Ideal Magnetohydrodynamics

Wu¹

2018

SIAM J. Numer. Anal.

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“…The robustness of that scheme was further demonstrated in [49] by extensive numerical tests and comparisons. In the last few years, significant advances have been made in developing bound-preserving high-order schemes for hyperbolic systems; see the pioneer works by Zhang and Shu [62,63,65], and more recent works, e.g., [31,57,37,15,53,50,59,61]. Balsara [5] proposed a self-adjusting PP limiter to enforce the positivity of the reconstructed solutions in a finite volume method for (1.1).…”

Section: Introductionmentioning

confidence: 99%

Provably positive high-order schemes for ideal magnetohydrodynamics: analysis on general meshes

Shu

2019

Numer. Math.

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This paper proposes and analyzes arbitrarily high-order discontinuous Galerkin (DG) and finite volume methods which provably preserve the positivity of density and pressure for the ideal magnetohydrodynamics (MHD) on general meshes. Unified auxiliary theories are built for rigorously analyzing the positivity-preserving (PP) property of numerical MHD schemes with a Harten-Lax-van Leer (HLL) type flux on polytopal meshes in any space dimension. The main challenges overcome here include establishing certain relation between the PP property and a discrete divergence of magnetic field on general meshes, and estimating proper wave speeds in the HLL flux to ensure the PP property. In the 1D case, we prove that the standard DG and finite volume methods with the proposed HLL flux are PP, under a condition accessible by a PP limiter. For the multidimensional conservative MHD system, the standard DG methods with a PP limiter are not PP in general, due to the effect of unavoidable divergence error in the magnetic field. We construct provably PP high-order DG and finite volume schemes by proper discretization of the symmetrizable MHD system, with two divergence-controlling techniques: the locally divergence-free elements and an important penalty term. The former technique leads to zero divergence within each cell, while the latter controls the divergence error across cell interfaces. Our analysis reveals in theory that a coupling of these two techniques is very important for positivity preservation, as they exactly contribute the discrete divergence terms which are absent in standard multidimensional DG schemes but crucial for ensuring the PP property. Several numerical tests further confirm the PP property and the effectiveness of the proposed PP schemes. Unlike the conservative MHD system, the exact smooth solutions of the symmetrizable MHD system are proved to retain the positivity even if the divergence-free condition is not satisfied. Our analysis and findings further the understanding, at both discrete and continuous levels, of the relation between the PP property and the divergence-free constraint.

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“…The robustness of that scheme was further demonstrated in [42] by extensive benchmark tests and comparisons. Recent years have witnessed significant progresses in developing high-order bound-preserving methods for hyperbolic systems (see, e.g., [54,55,48,56,25,50,31,51,44,53]) including the ideal MHD system [3,13,15,14] and the relativistic MHD system [46]. Two PP limiting techniques were developed in [3,13] for the finite volume or discontinuous Galerkin (DG) methods for (1) to enforce the admissibility 1 of the reconstructed or DG solutions at certain nodal points.…”

mentioning

confidence: 99%

A Provably Positive Discontinuous Galerkin Method for Multidimensional Ideal Magnetohydrodynamics

Wu¹,

Shu²

2018

SIAM J. Sci. Comput.

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The density and pressure are positive physical quantities in magnetohydrodynamics (MHD). Design of provably positivity-preserving (PP) numerical schemes for ideal compressible MHD is highly desirable, but remains a challenge especially in the multidimensional cases. In this paper, we first develop uniformly high-order discontinuous Galerkin (DG) schemes which provably preserve the positivity of density and pressure for multidimensional ideal MHD. The schemes are constructed by using the locally divergence-free DG schemes for the symmetrizable ideal MHD equations as the base schemes, a PP limiter to enforce the positivity of the DG solutions, and the strong stability preserving methods for time discretization. The significant innovation is that we discover and rigorously prove the PP property of the proposed DG schemes by using a novel equivalent form of the admissible state set and very technical estimates. Several two-dimensional numerical examples further confirm the PP property, and demonstrate the accuracy, effectiveness and robustness of the proposed PP methods.

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Design of provably physical-constraint-preserving methods for general relativistic hydrodynamics

Cited by 46 publications

References 47 publications

Positivity-Preserving Analysis of Numerical Schemes for Ideal Magnetohydrodynamics

Positivity-Preserving Analysis of Numerical Schemes for Ideal Magnetohydrodynamics

Provably positive high-order schemes for ideal magnetohydrodynamics: analysis on general meshes

A Provably Positive Discontinuous Galerkin Method for Multidimensional Ideal Magnetohydrodynamics

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