The Effect of Nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations

Brackbill, J. U.; Barnes, D. C.

doi:10.1016/0021-9991(80)90079-0

Cited by 667 publications

(608 citation statements)

References 3 publications

(3 reference statements)

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“…(6) which is obtained numerically, so that any error in matching these values by ψ(x, y, z) reduces to a finite jump close to the boundaries. Finally, we notice that this method is often used to remove the divergence of a vector field (Brackbill & Barnes 1980, sometimes referred to as "projection method"), and it has the property of conserving the current, i.e., ∇ × B J = ∇ × B J,s .…”

Section: Helmholtz Decomposition Of the Potential Part Of The Fieldmentioning

confidence: 99%

Accuracy of magnetic energy computations

Valori

Démoulin

Pariat

et al. 2013

A&A

118

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Context. For magnetically driven events, the magnetic energy of the system is the prime energy reservoir that fuels the dynamical evolution. In the solar context, the free energy (i.e., the energy in excess of the potential field energy) is one of the main indicators used in space weather forecasts to predict the eruptivity of active regions. A trustworthy estimation of the magnetic energy is therefore needed in three-dimensional (3D) models of the solar atmosphere, e.g., in coronal fields reconstructions or numerical simulations. Aims. The expression of the energy of a system as the sum of its potential energy and its free energy (Thomson's theorem) is strictly valid when the magnetic field is exactly solenoidal. For numerical realizations on a discrete grid, this property may be only approximately fulfilled. We show that the imperfect solenoidality induces terms in the energy that can lead to misinterpreting the amount of free energy present in a magnetic configuration. Methods. We consider a decomposition of the energy in solenoidal and nonsolenoidal parts which allows the unambiguous estimation of the nonsolenoidal contribution to the energy. We apply this decomposition to six typical cases broadly used in solar physics. We quantify to what extent the Thomson theorem is not satisfied when approximately solenoidal fields are used. Results. The quantified errors on energy vary from negligible to significant errors, depending on the extent of the nonsolenoidal component of the field. We identify the main source of errors and analyze the implications of adding a variable amount of divergence to various solenoidal fields. Finally, we present pathological unphysical situations where the estimated free energy would appear to be negative, as found in some previous works, and we identify the source of this error to be the presence of a finite divergence. Conclusions. We provide a method of quantifying the effect of a finite divergence in numerical fields, together with detailed diagnostics of its sources. We also compare the efficiency of two divergence-cleaning techniques. These results are applicable to a broad range of numerical realizations of magnetic fields.

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Section: Helmholtz Decomposition Of the Potential Part Of The Fieldmentioning

confidence: 99%

Accuracy of magnetic energy computations

Valori

Démoulin

Pariat

et al. 2013

A&A

118

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“…However, this system may generate some uncertainty [42]. The projection method proposed in [43] has been widely used. The projection involves the solution of a Poisson equation and also restricts the choice of boundary conditions.…”

Section: E-cusp Scheme For Mhd Equationsmentioning

confidence: 99%

E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

Shen

Zha

Huerta

2012

Journal of Computational Physics

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a b s t r a c tAn E-CUSP (energy-convective upwind and split pressure) scheme is developed to solve the equations of magnetohydrodynamics. Fifth order WENO reconstructions are employed to calculate the fluxes in order to achieve high order spacial accuracy. A characteristic speed of sound by averaging the fast wave speed and the acoustic speed of sound is suggested to evaluate the Mach number, which will yield robust and accurate solutions. The numerical experiments have demonstrated the accuracy and the capability of the new scheme to capture complex interactions of multiple shocks and vortices.

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“…High-order numerical discretizations for MHD space-physics flows must properly handle the solenoidal constraint for the magnetic field (i.e., ∇ · B = 0) [16] so as to provide stability to the discrete system of differential equations and to avoid unphysical plasma transport effects [17]. They must efficiently provide both solution accuracy and monotonicity even in the presence of large solution gradients and/or discontinuous solutions (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…A variety of approaches have been proposed to handle the ∇ · B constraint. One option is to employ an elliptic correction scheme, called the "Hodge Projection", which essentially projects a vector field onto its solenoidal part [17,32]. While the elliptic correction scheme maintains solenoidality up to machine accuracy (in the chosen discretization), it requires a Poisson equation to be solved at each hyperbolic step.…”

Section: Introductionmentioning

confidence: 99%

High-order central ENO finite-volume scheme for ideal MHD

Susanto

Ivan

Sterck

et al. 2013

Journal of Computational Physics

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A high-order central essentially non-oscillatory (CENO) finite-volume scheme is developed for the compressible ideal magnetohydrodynamics (MHD) equations solved on threedimensional (3D) cubed-sphere grids. The proposed formulation is an extension to 3D geometries of a recent high-order MHD CENO scheme developed on two-dimensional (2D) grids. The main technical challenge in extending the 2D method to 3D cubed-sphere grids is to properly handle the nonplanar cell faces that arise in cubed-sphere grids. This difficulty is solved by considering general hexahedral cells with trilinear faces, which allow us to compute fluxes, areas and volumes with high-order accuracy by transforming to a reference cubic cell. The 3D CENO scheme is implemented within a flexible multi-block cubed-sphere grid framework to fourth-order accuracy, resulting in a high-order solution method for cubed-sphere grids with unique capabilities in terms of adaptive refinement and parallel scalability. The high-order method is applicable to the solution of general hyperbolic conservation laws with, in principle, arbitrary order. The CENO scheme is based on a hybrid solution reconstruction procedure that provides high-order accuracy in smooth regions, even for smooth extrema, and non-oscillatory transitions at discontinuities. The scheme is applied herein to MHD in combination with a GLM divergence correction technique to control the solenoidal condition for the magnetic field while preserving the high-order accuracy of the numerical procedure. The cubed-sphere simulation framework features a flexible design based on a genuine multi-block implementation, leading to high-order accuracy, flux calculation, adaptivity and parallelism that are fully transparent to the boundaries between the six sectors of the cubedsphere grid. The proposed 3D MHD CENO scheme is shown to achieve uniform fourth-order accuracy on cubed-sphere grids. Parallel domain partitioning and grid adaptivity are achieved on the 3D cubed-sphere grids using a hierarchical block-based division strategy with blocks of equal size. Numerical results to demonstrate the high-order accuracy, robustness and capability of the proposed high-order framework are presented and discussed for several test problems.

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The Effect of Nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations

Cited by 667 publications

References 3 publications

Accuracy of magnetic energy computations

Accuracy of magnetic energy computations

E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

High-order central ENO finite-volume scheme for ideal MHD

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