High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics

Wu, Kailiang; Tang, Huazhong

doi:10.1016/j.jcp.2015.06.012

Cited by 74 publications

(103 citation statements)

References 69 publications

(131 reference statements)

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“…Successfully simulating such jet flows is indeed a challenge, cf. [63,5,53,55]. We consider the Mach 800 MHD jets proposed in [51,52] and extended from the gas dynamical jet of Balsara [5] by adding a magnetic field.…”

Section: Astrophysical Jetsmentioning

confidence: 99%

“…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%

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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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Section: Astrophysical Jetsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Shu

2019

Numer. Math.

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“…The extension of the cut-off limiter to the GRHD case is similar to the special RHD case [35]. In the following, we mainly focus on developing parametrized PCP flux limiter, because the parametrized limiter works well in maintaining the high-order accuracy [41].…”

Section: Pcp Flux Limitermentioning

confidence: 99%

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

2017

Phys. Rev. D

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The paper develops high-order physical-constraint-preserving (PCP) methods for general relativistic hydrodynamic (GRHD) equations, equipped with a general equation of state. Here the physical constraints, describing the admissible states of GRHD, are referred to the subluminal constraint on the fluid velocity and the positivity of the density, pressure and specific internal energy. Preserving these constraints is very important for robust computations, otherwise violating one of them will lead to the ill-posed problem and numerical instability. To overcome the difficulties arising from the inherent strong nonlinearity contained in the constraints, we derive an equivalent definition of the admissible states. Using this definition, we prove the convexity, scaling invariance and LaxFriedrichs (LxF) splitting property of the admissible state set G, and discover the dependence of G on the spacetime metric. Unfortunately, such dependence yields the non-equivalence of G at different points in curved spacetime, and invalidates the convexity of G in analyzing PCP schemes. This obstacle is effectively overcame by introducing a new formulation of the GRHD equations. Based on this formulation and the above theories, a first-order LxF scheme is designed on general unstructured mesh and rigorously proved to be PCP under a CFL condition. With two types of PCP limiting procedures, we design high-order, provably (not probably) PCP methods under discretization on the proposed new formulation. These high-order methods include the PCP finite difference, finite volume and discontinuous Galerkin methods.

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“…From a relaxation system, Bouchut et al [7,8] derived a multiwave approximate Riemann solver for 1D ideal MHD, and deduced sufficient conditions for the solver to satisfy discrete entropy inequalities and the PP property. Recent years have witnessed some significant advances in developing bound-preserving high-order schemes for hyperbolic systems (e.g., [50,51,52,21,47,28,42,31,44,48]). Highorder limiting techniques were well developed in [4,11] for the finite volume or DG methods of MHD, to enforce the admissibility 1 of the reconstructed or DG polynomial solutions at certain nodal points.…”

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confidence: 99%

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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High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics

Cited by 74 publications

References 69 publications

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

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

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

Positivity-Preserving Analysis of Numerical Schemes for Ideal Magnetohydrodynamics

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