A Provably Positive Discontinuous Galerkin Method for Multidimensional Ideal Magnetohydrodynamics

Wu, Kailiang; Shu, Chi‐Wang

doi:10.1137/18m1168042

Cited by 50 publications

(93 citation statements)

References 57 publications

(196 reference statements)

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“…which is consistent with the one introduced in [51,52] on the Cartesian meshes. The PP property of the scheme (4.1) is shown as follows.…”

Section: First-order Schemessupporting

confidence: 90%

“…This is motivated from our theoretical analysis, and is very important for achieving the provably PP property, as we will see the proof of Theorem 4.2 and Remark 4.3. If the LF flux is employed, i.e., σ − = −σ + , then η K (x) ≡ − 1 2 , and the penalty term reduces to the one used in [52].…”

Section: Locally Divergence-free Schemesmentioning

confidence: 99%

“…The last term in (4.16) is relatively small compared to the maximum signal speed, and thus does not cause strict restriction on the time step-size; see the detailed justification and numerical evidence in [52].…”

Section: Positivity-preserving Schemesmentioning

confidence: 99%

“…For the conservative system (1.1), Janhunen [32] noticed that the exact solutions to 1D Riemann problems can have negative pressure if the initial data has a jump in the normal component of the magnetic field (i.e., a nonzero divergence). Recently in [52], we first observed that the exact smooth solution of (1.1) may also fail to be PP if the divergence-free constraint (1.2) is (slightly) violated.…”

Section: Introductionmentioning

confidence: 99%

“…We prove that the strong solution to the initial-value problem of the modified MHD system (1.4) preserves the positivity of density and pressure even if the divergence-free condition (1.2) is not satisfied. This feature, not enjoyed by the conservative system (1.1) (see [52]), can serve as a justification for designing provably PP multidimensional schemes based on the modified system (1.4).…”

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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“…which is consistent with the one introduced in [51,52] on the Cartesian meshes. The PP property of the scheme (4.1) is shown as follows.…”

Section: First-order Schemessupporting

confidence: 90%

Section: Locally Divergence-free Schemesmentioning

confidence: 99%

Section: Positivity-preserving Schemesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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On physical-constraints-preserving schemes for special relativistic magnetohydrodynamics with a general equation of state

Tang

2018

Z. Angew. Math. Phys.

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The paper studies the physical-constraints-preserving (PCP) schemes for multidimensional special relativistic magnetohydrodynamics with a general equation of state (EOS) on more general meshes. It is an extension of the work (Ref. [45]) which focuses on the ideal EOS and uniform Cartesian meshes. The general EOS without a special expression poses some additional difficulties in discussing the mathematical properties of admissible state set with the physical constraints on the fluid velocity, density and pressure. Rigorous analyses are provided for the PCP property of finite volume or discontinuous Galerkin schemes with the Lax-Friedrichs (LxF) type flux on a general mesh with non-self-intersecting polytopes. Those are built on a more general form of generalized LxF splitting property and a different convex decomposition technique. It is shown in theory that the PCP property is closely connected with a discrete divergence-free condition, which is proposed on the general mesh and milder than that in Ref. [45].where the conservative vector U = D, m, B, E ⊤ and the flux in the x i -direction F i (U) is defined byCorresponding author. Tel: +86-10-62757018; Fax: +86-10-62751801.where the adiabatic index Γ ∈ (1, 2]. The system (1) takes into account the relativistic description for the dynamics of electrically-conducting fluid (plasma) at nearly speed of light in vacuum in the presence of magnetic fields. The relativistic magneto-fluid flow appears in investigating numerous astrophysical phenomena from stellar to galactic scales, e.g., core collapse super-novae, coalescing neutron stars, Xray binaries, active galactic nuclei, formation of black holes, super-luminal jets and gamma-ray bursts etc. However, due to the relativistic effect, especially the appearance of Lorentz factor, the system (1) involves strong nonlinearity, making its analytic treatment extremely difficult. Numerical simulation is a primary and powerful approach to improve our understanding of the physical mechanisms in the RMHDs. In comparison with the non-relativistic MHD case, the numerical difficulties mainly come from highly nonlinear coupling between the RMHD equations in (1), which leads to no explicit expression of the primitive variables (ρ, v, p) and the flux F i in terms of U.Since nearly 2000s, numerical study of the RMHDs has attracted considerable attention, and various modern shock-capturing methods have been developed for the RMHD equations. They include but are not limited to: the Godunov-type scheme based on the linear Riemann solver [22], the total variation diminishing scheme [2], the third-order accurate central-type scheme based on two-speed approximate Riemann solver [12], the high-order kinetic flux-splitting method [33], the exact Riemann solver [16], the HLLC (Harten-Lax-van Leer-contact) type schemes [20,21,29], the adaptive methods with mesh refinement [1,40], the adaptive moving mesh method [18], the locally divergence-free Runge-Kutta discontinuous Galerkin (RKDG) method and exactly divergence-free central RKDG method with ...

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Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations

Shu

2021

Numer. Math.

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A Provably Positive Discontinuous Galerkin Method for Multidimensional Ideal Magnetohydrodynamics

Cited by 50 publications

References 57 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

On physical-constraints-preserving schemes for special relativistic magnetohydrodynamics with a general equation of state

Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations

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