Hyperbolic Divergence Cleaning for the MHD Equations

Dedner, Andreas; Kemm, Friedemann; Kröner, Dietmar; Munz, Claus‐Dieter; Schnitzer, T.; Wesenberg, Matthias

doi:10.1006/jcph.2001.6961

Cited by 1,235 publications

(1,285 citation statements)

References 41 publications

(74 reference statements)

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“…In this work, the hyperbolic system of ideal MHD equations is solved using a GLM-MHD formulation [12], which couples the divergence constraint, ∇· B = 0, with the induction equation through the introduction of a new potential variable, ψ. Thus, the system of conservation laws for which the solution is sought here may be expressed in conservation (or divergence) form as…”

Section: Glm Formulation Of Ideal Mhd Governing Equationsmentioning

confidence: 99%

“…(1)- (4) can have a so-called elliptic, parabolic or hyperbolic nature. Moreover, it can be shown [12] that the coefficient c h determines how fast the divergence of the magnetic field is advected out of the domain, whereas c p controls its dissipation. Note that Eqs.…”

Section: Glm Formulation Of Ideal Mhd Governing Equationsmentioning

confidence: 99%

“…Consequently, the practise adopted here is to use one quadrature point (N g = 1) for second-order schemes (K = 1) and four quadrature points (N g = 4) for thirdand fourth-order schemes (K = 2 and K = 3). To calculate the MHD numerical flux, F num · n, at each of the N g Gauss quadrature points the procedure outlined in [12,13] is used. Thus, a local Riemann problem with the left and right reconstructed solution states, U l and U r , as initial data is solved at each point by decoupling the equations for B x and ψ from the rest of the system and applying a Lax-Friedrichs numerical flux function to the other seven MHD variables.…”

Section: High-order Residual Evaluation For General Hexahedral Cells mentioning

confidence: 99%

“…General hexahedral cells are handled by adopting a trilinear description of the nonplanar cell faces, which allows us to compute fluxes, areas and volumes with high-order accuracy by transforming to a reference cubic cell. The resulting high-order 3D finite-volume CENO scheme is applied to MHD flows on cubed-sphere grids by combining it with the generalized Lagrange multiplier (GLM) divergence cleaning method for MHD that was proposed by Dedner et al [12], following our recent work on a 2D CENO method for MHD [13]. The proposed 3D CENO scheme is implemented with fourth-order accuracy in the highly sophisticated parallel and adaptive 3D cubed-sphere grid simulation framework that we presented in [14,15], resulting in a 3D cubed-sphere grid code for conservation laws that is novel in that it provides solution accuracy with order higher than two uniformly in all three directions.…”

Section: Introductionmentioning

confidence: 99%

“…It can be extended with secondorder accuracy to logically Cartesian grids and to triangular or tetrahedral unstructured grids [36,37], but extensions beyond second-order [38] and to general polygonal grids are far from trivial. A fourth method is the generalized Lagrange multiplier (GLM) divergence cleaning method proposed by Dedner et al [12].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Glm Formulation Of Ideal Mhd Governing Equationsmentioning

confidence: 99%

Section: Glm Formulation Of Ideal Mhd Governing Equationsmentioning

confidence: 99%

Section: High-order Residual Evaluation For General Hexahedral Cells mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Susanto

Ivan

Sterck

et al. 2013

Journal of Computational Physics

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Multiphysics Modeling with Computational Electromagnetics

Jin,

Yan

2024

Encyclopedia of RF and Microwave Engineering

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As computational methods for solving Maxwell's equations become rather mature, the time has come to tackle much more challenging multiphysics problems, which have a great range of applications in science and technology. In this chapter, we use four examples to illustrate the challenges and resolutions of multiphysics modeling. The first example is related to electromagnetic hyperthermia for cancer treatment, which requires solving electromagnetic and bioheat transfer equation for the planning and optimization of the treatment process. The second example concerns the heat problem in integrated circuits due to electromagnetic dissipated power and its effects and a potential cooling solution, which requires an electrical–thermal–fluid–mechanical co‐simulation. The third example considers modeling of monolithic microwave integrated circuits, which consist of both distributive and lumped circuit components. The fourth example simulates the air and dielectric breakdown in high‐power microwave devices by coupling electromagnetic modeling with various plasma models. With these examples, we discuss in detail some of the technical challenges typically encountered in multiphysics modeling and their potential solutions.

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Inner heliosphere MHD modeling system applicable to space weather forecasting for the other planets

et al. 2014

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We developed a magnetohydrodynamic (MHD) solar wind model which can be used for practical use in real-time space weather forecasting at Earth's orbit and those of other planets. The MHD simulation covering 3 years (2007)(2008)(2009)) was performed to test the accuracy, and the numerical results show reasonable agreement with in situ measurements of the solar wind at Earth's orbit and with measurements at Venus and Mars by Venus Express and Mars Express, respectively. The comparison also shows that the numerical results can be used to detect stream interfaces, which is useful for space weather forecast of killer electrons in the outer Van Allen belt.

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Hyperbolic Divergence Cleaning for the MHD Equations

Cited by 1,235 publications

References 41 publications

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

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

Multiphysics Modeling with Computational Electromagnetics

Inner heliosphere MHD modeling system applicable to space weather forecasting for the other planets

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