Globally Divergence-Free Discontinuous Galerkin Methods for Ideal Magnetohydrodynamic Equations

Fu, Pei; Li, Fengyan; Xu, Yan

doi:10.1007/s10915-018-0750-6

Cited by 21 publications

(17 citation statements)

References 33 publications

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“…[11,24,8,46,34]. In the past several decades, many numerical techniques were proposed to control the divergence error or enforce the divergence-free condition in the discrete sense, including but not limited to: the projection method [11], the locally divergence-free methods [34,60], the hyperbolic divergence cleaning method [20], the constrained transport method [24] and its variants (e.g., [43,8,3,27,45,1,35,17,25]), and the eight-wave methods (e.g., [40,41,12,38]). The eight-wave method was first proposed by Powell [40,41], based on appropriate discretization of the modified MHD equations of Godunov [28]: U t + ∇ · F(U) = −(∇ · B) S(U), (1.4) where S(U) = (0, B, v, v · B) .…”

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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Section: Introductionmentioning

confidence: 99%

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

Shu

2019

Numer. Math.

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“…We refer to the discussion in [18] for a comparison of various version of CT methods. See also the recent work on global divergence-free DG methods for compressible MHD [8,13], which can be interpreted as high-order CT-type methods. The key idea to achieve a divergence-free magnetic field for the CT methods is to first advance in time the normal component of the magnetic field on the mesh interfaces, then apply a (locally defined) divergencefree reconstruction procedure.…”

Section: Introductionmentioning

confidence: 99%

An Explicit Divergence-Free DG Method for Incompressible Magnetohydrodynamics

2019

J Sci Comput

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We extend the recently introduced explicit divergence-free DG scheme for incompressible hydrodynamics [7] to the incompressible magnetohydrodynamics (MHD). A globally divergence-free finite element space is used for both the velocity and the magnetic field. Highlights of the scheme includes global and local conservation properties, high-order accuracy, energy-stability, pressure-robustness. When forward Euler time stepping is used, we need two symmetric positive definite (SPD) hybrid-mixed Poisson solvers (one for velocity and one for magnetic field) to advance the solution to the next time level.Since we treat both viscosity in the momentum equation and resistivity in the magnetic induction equation explicitly, the method shall be best suited for inviscid or high-Reynolds number, low resistivity flows so that the CFL constraint is not too restrictive.

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“…The key step there is to design a special local divergence-free test function space for the magnetic field which can make the divergence-free condition automatically hold in each cell. This strategy has been further explored in the past years and many local and global divergence-free DG methods have been proposed, see [27,28,8,45,36,19]. However, all the above mentioned works are on Cartesian meshes and for 2D MHD computation.…”

Section: Introductionmentioning

confidence: 99%

Locally Divergence-Free Spectral-DG Methods for Ideal Magnetohydrodynamic Equations on Cylindrical Coordinates

Liu¹

2019

CiCP

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In this paper, we propose a class of high order locally divergence-free spectraldiscontinuous Galerkin (DG) methods for three dimensional (3D) ideal magnetohydrodynamic (MHD) equations on cylindrical geometry. Under the conventional cylindrical coordinates (r, ϕ, z), we adopt the Fourier spectral method in the ϕ-direction and discontinuous Galerkin (DG) approximation in the (r, z) plane, motivated by the structure of the particular physical flows of magnetically confined plasma. By a careful design of the DG approximation function space, our spectral-DG methods are divergence-free inside each element for the magnetic field. Numerical examples with third order strong-stability-preserving Runge-Kutta methods are provided to demonstrate the efficiency and performance of our proposed methods.

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Globally Divergence-Free Discontinuous Galerkin Methods for Ideal Magnetohydrodynamic Equations

Cited by 21 publications

References 33 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

An Explicit Divergence-Free DG Method for Incompressible Magnetohydrodynamics

Locally Divergence-Free Spectral-DG Methods for Ideal Magnetohydrodynamic Equations on Cylindrical Coordinates

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