Bound-Preserving High-Order Schemes

Xu, Zhao Di; Zhang, Xiangxiong

doi:10.1016/bs.hna.2016.08.002

Cited by 28 publications

(16 citation statements)

References 77 publications

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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.…”

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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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confidence: 99%

A Provably Positive Discontinuous Galerkin Method for Multidimensional Ideal Magnetohydrodynamics

Wu¹,

Shu²

2018

SIAM J. Sci. Comput.

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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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“…Most of high order schemes fail to preserve positivity because of interpolation errors when near vacuum states occur or in the presence of strong shocks. Recently, some positivity-preserving techniques for high order schemes were developed by Zhang and Shu as in [73,74,76,45,69] for homogeneous compressible Euler equations, and, in [75,62,77] for compressible Euler equations with several types of source terms, like the energetic reactive products in gaseous detonations for example.…”

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confidence: 99%

Solution property preserving reconstruction BVD+MOOD scheme for compressible euler equations with source terms and detonations

et al. 2020

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In "Solution Property Reconstruction for Finite Volume scheme: a BVD+MOOD framework", Int. J. Numer. Methods Fluids, 2020, we have designed a novel solution property preserving reconstruction, so-called multi-stage BVD-MOOD scheme. The scheme is able to maintain a high accuracy in smooth profile, eliminate the oscillations in the vicinity of discontinuity, capture sharply discontinuity and preserve some physical properties like the positivity of density and pressure for the Euler equations of compressible gas dynamics. In this paper, we present an extension of this approach for the compressible Euler equations supplemented with source terms (e.g., gravity, chemical reaction). One of the main challenges when simulating these models is the occurrence of negative density or pressure during the time evolution, which leads to a blow-up of the computation. General compressible Euler equations with different type of source terms are considered as models for physical situations such as detonation waves. Then, we illustrate the performance of the proposed scheme via a numerical test suite including genuinely demanding numerical tests. We observe that the present scheme is able to preserve the physical properties of the numerical solution still ensuring robustness and accuracy when and where appropriate.

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Bound-Preserving High-Order Schemes

Cited by 28 publications

References 77 publications

A Provably Positive Discontinuous Galerkin Method for Multidimensional Ideal Magnetohydrodynamics

A Provably Positive Discontinuous Galerkin Method for Multidimensional Ideal Magnetohydrodynamics

Positivity-Preserving Analysis of Numerical Schemes for Ideal Magnetohydrodynamics

Solution property preserving reconstruction BVD+MOOD scheme for compressible euler equations with source terms and detonations

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