A new semi-implicit method for MHD computations

Lerbinger, K.; Luciani, J. F.

doi:10.1016/0021-9991(91)90008-9

Cited by 57 publications

(66 citation statements)

References 13 publications

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“…Finite elements and flux-aligned grids have recently been shown to be valuable for resolving the anisotropic operators ͑various forms of B · ٌ͒ of nonideal MHD. 28 Specialized implicit methods have been developed to deal with temporal stiffness, [21][22][23][24][25][26][27][28][40][41][42][43][44] and have enabled accurate computations to proceed with CFL numbers in excess of 10. 4,5 While all of these approaches require solution of large algebraic systems at every time step, the Hermitian character of the relevant operators can be used to facilitate computation.…”

Section: A General Considerationsmentioning

confidence: 99%

“…The analytic small-island evolution from Ref. 43 with an estimated ⌬Ј is also plotted in ͑b͒. The Poincaré surface of section for the final magnetic configuration is shown in ͑c͒.…”

Section: B Spatial Discretizationmentioning

confidence: 99%

“…41 Another method implicitly advances all dynamics that are linear with respect to an initial equilibrium and explicitly advances all nonlinear dynamics. 25 The 'semiimplicit' method [21][22][23]28,40,43,55 is based on the leapfrog method for wave propagation and in time-discrete form applies a positive-definite self-adjoint differential operator to changes in the flow. The semi-implicit operator can be used to stabilize linear and nonlinear wave propagation, and its spectrum governs temporal convergence properties in multiscale computations.…”

Section: Temporal Discretizationmentioning

confidence: 99%

“…22,24 A benefit that arises with finite elements is that the anisotropic operator that distinguishes the fast, shear, and slow modes of the MHD spectrum 43 is straightforward to implement and, by construction, results in a Hermitian matrix for the velocity advance. 28 For the subdominant plasma flow in resistive MHD, predictor-corrector steps for advective terms can be retained.…”

Section: Temporal Discretizationmentioning

confidence: 99%

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Computational modeling of fully ionized magnetized plasmas using the fluid approximation

et al. 2006

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Strongly magnetized plasmas are rich in spatial and temporal scales, making a computational approach useful for studying these systems. The most accurate model of a magnetized plasma is based on a kinetic equation that describes the evolution of the distribution function for each species in six-dimensional phase space. High dimensionality renders this approach impractical for computations for long time scales. Fluid models are an approximation to the kinetic model. The reduced dimensionality allows a wider range of spatial and/or temporal scales to be explored. Computational modeling requires understanding the ordering and closure approximations, the fundamental waves supported by the equations, and the numerical properties of the discretization scheme. Several ordering and closure schemes are reviewed and discussed, as are their normal modes, and algorithms that can be applied to obtain a numerical solution.

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Section: A General Considerationsmentioning

confidence: 99%

“…The analytic small-island evolution from Ref. 43 with an estimated ⌬Ј is also plotted in ͑b͒. The Poincaré surface of section for the final magnetic configuration is shown in ͑c͒.…”

Section: B Spatial Discretizationmentioning

confidence: 99%

Section: Temporal Discretizationmentioning

confidence: 99%

Section: Temporal Discretizationmentioning

confidence: 99%

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Computational modeling of fully ionized magnetized plasmas using the fluid approximation

et al. 2006

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“…The proposed improvements involve the use of new predictor steps that include a fraction of the wave terms. An algorithm that did not suffer from this instability was presented by Lerbinger and Luciani [6].…”

Section: Introductionmentioning

confidence: 99%

Stability of Algorithms for Waves with Large Flows

Lionello

Mikić

Linker

1999

Journal of Computational Physics

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A divergence‐free semi‐implicit finite volume scheme for ideal, viscous, and resistive magnetohydrodynamics

Dumbser

Balsara

Tavelli

et al. 2018

Numerical Methods in Fluids

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Summary In this paper, we present a novel pressure‐based semi‐implicit finite volume solver for the equations of compressible ideal, viscous, and resistive magnetohydrodynamics (MHD). The new method is conservative for mass, momentum, and total energy, and in multiple space dimensions, it is constructed in such a way as to respect the divergence‐free condition of the magnetic field exactly, also in the presence of resistive effects. This is possible via the use of multidimensional Riemann solvers on an appropriately staggered grid for the time evolution of the magnetic field and a double curl formulation of the resistive terms. The new semi‐implicit method for the MHD equations proposed here discretizes the nonlinear convective terms as well as the time evolution of the magnetic field explicitly, whereas all terms related to the pressure in the momentum equation and the total energy equation are discretized implicitly, making again use of a properly staggered grid for pressure and velocity. Inserting the discrete momentum equation into the discrete energy equation then yields a mildly nonlinear symmetric and positive definite algebraic system for the pressure as the only unknown, which can be efficiently solved with the (nested) Newton method of Casulli et al. The pressure system becomes linear when the specific internal energy is a linear function of the pressure. The time step of the scheme is restricted by a CFL condition based only on the fluid velocity and the Alfvén wave speed and is not based on the speed of the magnetosonic waves. Being a semi‐implicit pressure‐based scheme, our new method is therefore particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, for which it is well known that explicit density‐based Godunov‐type finite volume solvers become increasingly inefficient and inaccurate because of the more and more stringent CFL condition and the wrong scaling of the numerical viscosity in the incompressible limit. We show a relevant MHD test problem in the low Mach number regime where the new semi‐implicit algorithm is a factor of 50 faster than a traditional explicit finite volume method, which is a very significant gain in terms of computational efficiency. However, our numerical results confirm that our new method performs well also for classical MHD test cases with strong shocks. In this sense, our new scheme is a true all Mach number flow solver.

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A new semi-implicit method for MHD computations

Cited by 57 publications

References 13 publications

Computational modeling of fully ionized magnetized plasmas using the fluid approximation

Computational modeling of fully ionized magnetized plasmas using the fluid approximation

Stability of Algorithms for Waves with Large Flows

A divergence‐free semi‐implicit finite volume scheme for ideal, viscous, and resistive magnetohydrodynamics

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