Implicit symmetrized streamfunction formulations of magnetohydrodynamics

Kang, Kyungsu; Keyes, David E.

doi:10.1002/fld.1755

Cited by 5 publications

(4 citation statements)

References 26 publications

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“…Numerical methods for compressible case were developed in [8,9,10,11,12,13,14,15,16,17,18], etc. One can also find similar results for the incompressible case in [1,19,20,21,22,23]. The Lagrange invariance of a generalized ion magnetic helicity was established for Hall MHD in [24].…”

Section: Introductionsupporting

confidence: 58%

Asymmetric and moving-frame approaches to MHD equations

Cao

2011

Acta. Math. Sin.-English Ser.

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The magnetohydrodynamic (MHD) equations of incompressible viscous fluids with finite electrical conductivity describe the motion of viscous electrically conducting fluids in a magnetic field. In this paper, we find twelve families of solutions of these equations by Xu's asymmetric and moving frame methods. A family of singular solutions may reflect basic characteristics of vortices. The other solutions are globally analytic with respect to the spacial variables. In particular, Bernoulli equation and Wronskian determinants play important roles in our approaches. Our solutions may also help engineers to develop more effective algorithms to find physical numeric solutions to practical models.

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Section: Introductionsupporting

confidence: 58%

Asymmetric and moving-frame approaches to MHD equations

Cao

2011

Acta. Math. Sin.-English Ser.

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“…Various numerical algorithms have been used in MHD simulations; examples include finite difference methods, finite volume methods, finite element methods, and Fourier-based spectral and pseudo-spectral methods [27]. In [12,13,14,15,21], two-dimensional, incompressible MHD problems are studied in terms of finite element approximations of the stream function-vorticity advection formulation. Since MHD flow often develop sharp interfaces, adaptive h-refinement techniques have been applied in MHD simulations [16,25,33].…”

Section: Introductionmentioning

confidence: 99%

A nonconforming finite element method for fourth order curl equations in ℝ³

Zheng

2011

Math. Comp.

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Abstract. In this paper we present a nonconforming finite element method for solving fourth order curl equations in three dimensions arising from magnetohydrodynamics models. We show that the method has an optimal error estimate for a model problem involving both (∇×) 2 and (∇×) 4 operators. The element has a very small number of degrees of freedom, and it imposes the inter-element continuity along the tangential direction which is appropriate for the approximation of magnetic fields. We also provide explicit formulae of basis functions for this element.

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“…They gave an efficient solution algorithm, valid for both high and low magnetic Reynolds numbers, with various types of formulations such as the Helmholtz, the vector potential and the conservative formulation. Kang and Keyes [5] gave an FEM solution for the 2D incompressible MHD flow using a hybrid stream function approach and preferred an implicit time difference scheme with Newton's method in order to solve the resulting nonlinear equations. Navarro et al [6] dealt with a stream function-vorticity formulation and presented an extension of the generalized Peaceman and Rachford alternating-direction implicit scheme (ADI) in comparison with ADI scheme for the solution of the MHD flow equations at low magnetic Reynolds numbers.…”

Section: Introductionmentioning

confidence: 99%

The DRBEM solution of incompressible MHD flow equations

Bozkaya

Tezer‐Sezgin

2010

Numerical Methods in Fluids

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SUMMARYThis paper presents a dual reciprocity boundary element method (DRBEM) formulation coupled with an implicit backward difference time integration scheme for the solution of the incompressible magnetohydrodynamic (MHD) flow equations. The governing equations are the coupled system of Navier-Stokes equations and Maxwell's equations of electromagnetics through Ohm's law. We are concerned with a stream function-vorticity-magnetic induction-current density formulation of the full MHD equations in 2D. The stream function and magnetic induction equations which are poisson-type, are solved by using DRBEM with the fundamental solution of Laplace equation. In the DRBEM solution of the time-dependent vorticity and current density equations all the terms apart from the Laplace term are treated as nonhomogeneities. The time derivatives are approximated by an implicit backward difference whereas the convective terms are approximated by radial basis functions. The applications are given for the MHD flow, in a square cavity and in a backward-facing step. The numerical results for the square cavity problem in the presence of a magnetic field are visualized for several values of Reynolds, Hartmann and magnetic Reynolds numbers. The effect of each parameter is analyzed with the graphs presented in terms of stream function, vorticity, current density and magnetic induction contours. Then, we provide the solution of the step flow problem in terms of velocity field, vorticity, current density and magnetic field for increasing values of Hartmann number.

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Implicit symmetrized streamfunction formulations of magnetohydrodynamics

Cited by 5 publications

References 26 publications

Asymmetric and moving-frame approaches to MHD equations

Asymmetric and moving-frame approaches to MHD equations

A nonconforming finite element method for fourth order curl equations in ℝ³

The DRBEM solution of incompressible MHD flow equations

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