Schemes For Magnetohydrodynamics Research Articles

Lattice kinetic simulation of buoyancy induced MHD flows

A lattice kinetic model is proposed in this article for simulating buoyancy induced classical magnetohydrodynamic (MHD) flow in the low Mach number incompressible limit. The model is derived by coupling the passive scalar approach of He et al. [X. He, S. Chen, G.D. Doolen, A novel thermal model for the lattice Boltzmann method in incompressible limit, J. Comput. Phys. 146 (1998) 282–300] for the flow and thermal fields and Dellar formalism [P.J. Dellar, Lattice kinetic schemes for magnetohydrodynamics, J. Comput. Phys. 179 (2002) 95–126] for the magnetic field. Accordingly, the underlying hydrodynamics is monitored by a conventional single relaxation time lattice Boltzmann (LB) model through a density distribution function (DF), which obeys a scalar kinetic equation (KE) associated with external force fields (Lorentz and buoyancy forces). The magnetic field is represented by a vector DF, which obeys a corresponding vector KE and the thermal field is obtained from a separate temperature DF through another scalar KE incorporating the Joule heating effect. The three distribution functions are coupled through the macroscopic density, momentum, magnetic and thermal fields evaluated at lattice points. This allows a reduced lattice to be used for the magnetic distribution function, with a corresponding saving in the storage. Furthermore, the fluid viscosity, magnetic resistivity and thermal diffusivity may be adjusted independently that renders the model to be applicable for a wide variety of non-isothermal MHD problems. The novelty of the work is the computation of the thermal field in conjunction with the hydro-magnetic fields in the LB framework for the buoyancy driven non-isothermal MHD flows. A 9-bit 2D (d2q9) lattice scheme is used for the numerical computation of the hydrodynamic and thermal fields, whereas the magnetic field is simulated by a reduced 5-bit 2D (d2q5) lattice. Simulation of the magnetoconvective buoyancy induced flow (a) past a vertical flat plate, (b) between two differentially heated vertical walls provide excellent agreement with analytical results. Finally, the model is utilized to solve a classical problem of buoyancy driven MHD flow in a square cavity.

Divergence-free-preserving high-order schemes for magnetohydrodynamics: An artificial magnetic resistivity method

This paper proposes a new strategy that is very simple, divergence-free, high-order accurate, yet has an effective discontinuous-capturing capability for simulating compressible magnetohydrodynamics (MHD). The new strategy is to construct artificial diffusion terms in a physically-consistent manner, such that the artificial terms act as a diffusion term only in the curl of magnetic field to capture numerical discontinuities in the magnetic field while not affecting the divergence field (thus maintaining divergence-free constraint). The physically-consistent artificial diffusion terms are built into the induction equations in a conservation law form at a partial-differential-equation level. The proposed method may be viewed as adding an artificial magnetic resistivity to the induction equations, and is inherently divergence-free both ideal and resistive MHD, with and without shock waves, and also both inviscid and viscous flows. The method is based on finite difference method with co-located variable arrangement, and we show that any linear finite difference scheme in an arbitrary order of accuracy can be used to discretize the modified governing equations to ensures the divergence-free and the global conservation constraints numerically at the discretization level. The artificial magnetic resistivity is designed to localize automatically in regions of discontinuity in the curl of magnetic field and vanish wherever the flow is sufficiently smooth with respect to the grid scale, thereby maintaining the desirable high-order accuracy of the employed discretization scheme in smooth regions. Two-dimensional smooth and non-smooth ideal MHD problems are considered to validate the capability of the proposed method.

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

A high-order accurate finite-volume scheme for the compressible ideal magnetohydrodynamics (MHD) equations is proposed. The high-order MHD scheme is based on a central essentially non-oscillatory (CENO) method combined with the generalized Lagrange multiplier divergence cleaning method for MHD. The CENO method uses k-exact multidimensional reconstruction together with a monotonicity procedure that switches from a high-order reconstruction to a limited low-order reconstruction in regions of discontinuous or under-resolved solution content. Both reconstructions are performed on central stencils, and the switching procedure is based on a smoothness indicator. The proposed high-order accurate MHD scheme can be used on general polygonal grids. A highly sophisticated parallel implementation of the scheme is described that is fourth-order accurate on two-dimensional dynamically-adaptive body-fitted structured grids. The hierarchical multi-block body-fitted grid permits grid lines to conform to curved boundaries. High-order accuracy is maintained at curved domain boundaries by employing high-order spline representations and constraints at the Gauss quadrature points for flux integration. Detailed numerical results demonstrate high-order convergence for smooth flows and robustness against oscillations for problems with shocks. A new MHD extension of the well-known Shu–Osher test problem is proposed to test the ability of the high-order MHD scheme to resolve small-scale flow features in the presence of shocks. The dynamic mesh adaptation capabilities of the approach are demonstrated using adaptive time-dependent simulations of the Orszag–Tang vortex problem with high-order accuracy and unprecedented effective resolution.

A solution accurate, efficient and stable unsplit staggered mesh scheme for three dimensional magnetohydrodynamics

In this paper, we extend the unsplit staggered mesh scheme (USM) for 2D magnetohydrodynamics (MHD) [D. Lee, A.E. Deane, An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics, J. Comput. Phys. 228 (2009) 952–975] to a full 3D MHD scheme. The scheme is a finite-volume Godunov method consisting of a constrained transport (CT) method and an efficient and accurate single-step, directionally unsplit multidimensional data reconstruction-evolution algorithm, which extends Colella’s original 2D corner transport upwind (CTU) method [P. Colella, Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys. 87 (1990) 446–466]. We present two types of data reconstruction-evolution algorithms for 3D: (1) a reduced CTU scheme and (2) a full CTU scheme. The reduced 3D CTU scheme is a variant of a simple 3D extension of Collela’s 2D CTU method and is considered as a direct extension from the 2D USM scheme. The full 3D CTU scheme is our primary 3D solver which includes all multidimensional cross-derivative terms for stability. The latter method is logically analogous to the 3D unsplit CTU method by Saltzman [J. Saltzman, An unsplit 3D upwind method for hyperbolic conservation laws, J. Comput. Phys. 115 (1994) 153–168]. The major novelties in our algorithms are twofold. First, we extend the reduced CTU scheme to the full CTU scheme which is able to run with CFL numbers close to unity. Both methods utilize the transverse update technique developed in the 2D USM algorithm to account for transverse fluxes without solving intermediate Riemann problems, which in turn gives cost-effective 3D methods by reducing the total number of Riemann solves. The proposed algorithms are simple and efficient especially when including multidimensional MHD terms that maintain in-plane magnetic field dynamics. Second, we introduce a new CT scheme that makes use of proper upwind information in taking averages of electric fields. Our 3D USM schemes can be easily combined with various reconstruction methods (e.g., first-order Godunov, second-order MUSCL-Hancock, third-order PPM and fifth-order WENO), and a wide choice of 1D based Riemann solvers (e.g., local Lax–Friedrichs, HLLE, HLLC, HLLD, and Roe). The 3D USM-MHD solver is available in the University of Chicago Flash Center’s official FLASH release.

A Comparative Study of Divergence-Cleaning Techniques for Multi-Dimensional MHD Schemes

Several divergence-cleaning techniques for multi-dimensional Godunov-type magnetohydrodynamic schemes are comparatively investigated. We also propose a new divergence-cleaning technique that is improved from an earlier projection method to improve the robustness.

Realistic magnetohydrodynamical simulation of solar local supergranulation

Three-dimensional numerical simulations of solar surface magnetoconvection using realistic model physics are conducted. The thermal structure of convective motions into the upper radiative layers of the photosphere, the main scales of convective cells and the penetration depths of convection are investigated. We take part of the solar photosphere with a size of 60×60 Mm2 in the horizontal direction and of depth 20 Mm from the level of the visible solar surface. We use a realistic initial model of the sun and apply the equation of state and opacities of stellar matter. The equations of fully compressible radiation magnetohydrodynamics (MHD) with dynamical viscosity and gravity are solved. We apply (i) the conservative total variation diminishing (TVD) difference scheme for MHD, (ii) the diffusion approximation for radiative transfer and (iii) dynamical viscosity from subgrid-scale modeling. In simulation, we take a uniform two-dimensional grid in the horizontal plane and a nonuniform grid in the vertical direction with the number of cells being 600×600×204. We use 512 processors with distributed memory multiprocessors on the supercomputer MVS-100k at the Joint Computational Centre of the Russian Academy of Sciences.

A semi-discrete central scheme for magnetohydrodynamics on orthogonal–curvilinear grids

This work describes a novel scheme for the equations of magnetohydrodynamics on orthogonal–curvilinear grids within a finite-volume framework. The scheme is based on a combination of central-upwind techniques for hyperbolic conservation laws and projection–evolution methods originally developed for Hamilton–Jacobi equations. The scheme is derived in semi-discrete form, and a full-fledged version is obtained by applying any stable and accurate solver for integration in time. The divergence-free condition of the magnetic field is a built-in property of the scheme by virtue of a constrained-transport ansatz for the induction equation. From the general formulation second-order accurate schemes for cylindrical grids and spherical grids are introduced in some more detail pointing out their potential importance in many applications. Special emphasis in this context is put to a treatment of the geometric axis implying severe complications because of the presence of coordinate singularities and associated grid degeneracy. An attempt to tackle these problems is presented. Numerical experiments illustrate the overall robustness and performance of the scheme for a small suite of tests.

Emergence of high peaks in the axial velocity for an ideal magnetohydrodynamic disk configuration

We study the profile of a thin disk configuration as described by an axisymmetric ideal magnetohydrodynamics steady equilibrium. We consider the disk like a differentially rotating system dominated by the Keplerian term, but allowing for a nonzero radial and vertical matter flux. As a result, the steady state allows for the existence of local peaks for the vertical velocity of the plasma particles, though it prevents the radial matter accretion rate. This ideal magnetohydrodynamics scheme is therefore unable to solve the angular momentum-transport problem, but we suggest that it provides a mechanism for the generation of matter-jet seeds.

Stable discretization of magnetohydrodynamics in bounded domains

We study a semi-implicit time-difference scheme for magnetohydrodynamics of a viscous and resistive incompressible fluid in a bounded smooth domain with a perfectly conducting boundary. In the scheme, the velocity and magnetic fields are updated by solving simple Helmholtz equations. Pressure is treated explicitly in time, by solving Poisson equations corresponding to a recently de- veloped formula for the Navier-Stokes pressure involving the commutator of Laplacian and Leray projection operators. We prove stability of the time-difference scheme, and deduce a local-time well- posedness theorem for MHD dynamics extended to ignore the divergence-free constraint on velocity and magnetic fields. These fields are divergence-free for all later time if they are initially so.

Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics

In a pair of earlier papers the author showed the importance of divergence-free reconstruction in adaptive mesh refinement problems for magnetohydrodynamics (MHD) and the importance of the same for designing robust second order schemes for MHD. Second order accurate divergence-free schemes for MHD have shown themselves to be very useful in several areas of science and engineering. However, certain computational MHD problems would be much benefited if the schemes had third and higher orders of accuracy. In this paper we show that the reconstruction of divergence-free vector fields can be carried out with better than second order accuracy. As a result, we design divergence-free weighted essentially non-oscillatory (WENO) schemes for MHD that have order of accuracy better than second. A multi-stage Runge–Kutta time integration is used to ensure that the temporal accuracy matches the spatial accuracy. While this is achieved quite simply up to third order in time, going beyond third order is most simply achieved by using the ADER-WENO schemes that are detailed in a companion paper. (ADER stands for Arbitrary Derivative Riemann Problem.) Accuracy analysis is carried out and it is shown that the schemes meet their design accuracy for smooth problems. Stringent tests are also presented showing that the schemes perform well on those tests.

A characteristic-based nonconvex entropy-fix upwind scheme for the ideal magnetohydrodynamic equations

In this paper we perform an analysis of the wave structure of the ideal magnetohydrodynamic (MHD) equations. We present an analytical expression of the nonlinearity term associated to each characteristic field derived from a scaled version of the complete system of eigenvectors proposed by Brio and Wu [M. Brio, C.C. Wu, An upwind differencing scheme for the equations of ideal magnetohydrodynamics, J. Comput. Phys. 75 (2) (1988) 400–422] and adopting the eight wave approach by Powell et al. [K.G. Powell, P.L. Roe, R.S. Myong, T. Gombosi, D. deZeeuw, An upwind scheme for magnetohydrodynamics, AIAA 12th Computational Fluid Dynamics Conference, San Diego, CA, 1995, pp. 661–674]. A criterion for the detection of local regions containing points for which a nonlinear characteristic field becomes nonconvex is formulated for the two-dimensional case. We then design a characteristic-based upwind scheme for the ideal MHD equations that resolves the wave dynamics by local characteristic wavefields. The new scheme is able to detect local regions containing nonconvex singularities and to handle an entropy correction through a prescription of a local viscosity ensuring convergence to the entropy solution. A third order accurate version of the scheme performs satisfactorily in resolving one and two-dimensional MHD problems. Numerical results indicate that the proposed scheme behaves low dissipative, stable and accurate under high CFL numbers.

Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics

The present paper introduces a class of finite volume schemes of increasing order of accuracy in space and time for hyperbolic systems that are in conservation form. The methods are specially suited for efficient implementation on structured meshes. The hyperbolic system is required to be non-stiff. This paper specifically focuses on Euler system that is used for modeling the flow of neutral fluids and the divergence-free, ideal magnetohydrodynamics (MHD) system that is used for large scale modeling of ionized plasmas. Efficient techniques for weighted essentially non-oscillatory (WENO) interpolation have been developed for finite volume reconstruction on structured meshes. We have shown that the most elegant and compact formulation of WENO reconstruction obtains when the interpolating functions are expressed in modal space. Explicit formulae have been provided for schemes having up to fourth order of spatial accuracy. Divergence-free evolution of magnetic fields requires the magnetic field components and their moments to be defined in the zone faces. We draw on a reconstruction strategy developed recently by the first author to show that a high order specification of the magnetic field components in zone-faces naturally furnishes an appropriately high order representation of the magnetic field within the zone. We also present a new formulation of the ADER (for Arbitrary Derivative Riemann Problem) schemes that relies on a local continuous space–time Galerkin formulation instead of the usual Cauchy–Kovalewski procedure. We call such schemes ADER-CG and show that a very elegant and compact formulation results when the scheme is formulated in modal space. Explicit formulae have been provided on structured meshes for ADER-CG schemes in three dimensions for all orders of accuracy that extend up to fourth order. Such ADER schemes have been used to temporally evolve the WENO-based spatial reconstruction. The resulting ADER-WENO schemes provide temporal accuracy that matches the spatial accuracy of the underlying WENO reconstruction. In this paper we have also provided a point-wise description of ADER-WENO schemes for divergence-free MHD in a fashion that facilitates computer implementation. The schemes reported here have all been implemented in the RIEMANN framework for computational astrophysics. All the methods presented have a one-step update, making them low-storage alternatives to the usual Runge–Kutta time-discretization. Their one-step update also makes them suitable building blocks for adaptive mesh refinement (AMR) calculations. We demonstrate that the ADER-WENO meet their design accuracies. Several stringent test problems of Euler flows and MHD flows are presented in one, two and three dimensions. Many of our test problems involve near infinite shocks in multiple dimensions and the higher order schemes are shown to perform very robustly and accurately under all conditions. It is shown that the increasing computational complexity with increasing order is handily offset by the increased accuracy of the scheme. The resulting ADER-WENO schemes are, therefore, very worthy alternatives to the standard second order schemes for compressible Euler and MHD flow.

A Two‐dimensional Magnetohydrodynamics Scheme for General Unstructured Grids

We report a new finite-difference scheme for two-dimensional magnetohydrodynamics (MHD) simulations, with and without rotation, in unstructured grids with quadrilateral cells. The new scheme is implemented within the code VULCAN/2D, which already includes radiation-hydrodynamics in various approximations and can be used with arbitrarily moving meshes (ALE). The MHD scheme, which consists of cell-centered magnetic field variables, preserves the nodal finite difference representation of $div(\bB)$ by construction, and therefore any initially divergence-free field remains divergence-free through the simulation. In this paper, we describe the new scheme in detail and present comparisons of VULCAN/2D results with those of the code ZEUS/2D for several one-dimensional and two-dimensional test problems. The code now enables two-dimensional simulations of the collapse and explosion of the rotating, magnetic cores of massive stars. Moreover, it can be used to simulate the very wide variety of astrophysical problems for which multi-D radiation-magnetohydrodynamics (RMHD) is relevant.

A numerical scheme for ionizing shock waves

A two-dimensional (2D) visual computer code to solve the steady state (SS) or transient shock problems including partially ionizing plasma is presented. Since the flows considered are hypersonic and the resulting temperatures are high, the plasma is partially ionized. Hence the plasma constituents are electrons, ions and neutral atoms. It is assumed that all the above species are in thermal equilibrium, namely, that they all have the same temperature. The ionization degree is calculated from Saha equation as a function of electron density and pressure by means of a nonlinear Newton type root finding algorithms. The code utilizes a wave model and numerical fluctuation distribution (FD) scheme that runs on structured or unstructured triangular meshes. This scheme is based on evaluating the mesh averaged fluctuations arising from a number of waves and distributing them to the nodes of these meshes in an upwind manner. The physical properties (directions, strengths, etc.) of these wave patterns are obtained by a new wave model: ION-A developed from the eigen-system of the flux Jacobian matrices. Since the equation of state (EOS) which is used to close up the conservation laws includes electronic effects, it is a nonlinear function and it must be inverted by iterations to determine the ionization degree as a function of density and temperature. For the time advancement, the scheme utilizes a multi-stage Runge–Kutta (RK) algorithm with time steps carefully evaluated from the maximum possible propagation speed in the solution domain. The code runs interactively with the user and allows to create different meshes to use different initial and boundary conditions and to see changes of desired physical quantities in the form of color and vector graphics. The details of the visual properties of the code has been published before (see [N. Aslan, A visual fluctuation splitting scheme for magneto-hydrodynamics with a new sonic fix and Euler limit, J. Comput. Phys. 197 (2004) 1–27]). The two-dimensional nature of ION-A was presented by a planar shock wave propagating over a circular obstacle. It was demonstrated that including the effects of ionization in calculating complex flows is important, even when they appear initially negligible. This code can be used to accurately simulate the nonlinear time dependent evolution of neutral or ionized plasma flows from supersonic to hypersonic regimes.

A solution-adaptive central-constraint transport scheme for magnetohydrodynamics

The central-constraint transport scheme for magnetohydrodynamics in Ziegler [J. Comput. Phys. 196 (2004) 393] is made adaptive employing a block-structured mesh refinement method. Based on the guidelines in Berger and Collela [J. Comput. Phys. 82 (1989) 64] a mesh refinement variant has been developed which combines a flexible grid adaptation by using small blocks as refinement elements with integration speed by reducing the inherent overhead via block clustering techniques. The algorithms are discussed in detail and the efficiency of the implementation is benchmarked in terms of an efficiency parameter which takes into account both the obtained speedup factor and an error estimate. The three-dimensional benchmark problems are a spherical implosion problem and a shock-cloud collision problem in a magnetic medium. Further examples of astrophysical interest are presented which demonstrate the robustness and versatility of the new adaptive grid code.

A non-staggered, conservative, [formula omitted], finite-volume scheme for 3D implicit extended magnetohydrodynamics in curvilinear geometries

In the development of spatial difference schemes for magnetohydrodynamics (MHD), the preservation of continuum properties such as conservation of mass, momentum, and energy, as well as required electromagnetic constraints ( ∇ ⋅ B → = ∇ ⋅ j → = 0 , where j → = ∇ × B → is the electrical current), is desirable to preserve numerical accuracy. Moreover, simplicity of the scheme is also a desirable feature, particularly when an implicit implementation is considered (the focus of this paper). We propose here a finite-volume, cell-centered (non-staggered) scheme for the extended MHD formulation that: (1) is suitable for implicit implementations in arbitrary curvilinear geometries, (2) is conservative, (3) preserves both the magnetic field and the electrical current solenoidal to machine precision, and (4) is linearly and nonlinearly stable in the absence of numerical and physical dissipation. Crucial to the viability of the scheme is the use of a clever interpolation scheme (ZIP [Hirt, J. Comput. Phys. 2 (1968) 339–355]), the proper treatment of boundary conditions in curvilinear geometry, and a novel treatment of geometric source terms in the momentum equation that ensures their exact cancellation in the absence of pressure forces.

A visual fluctuation splitting scheme for magnetohydrodynamics with a new sonic fix and Euler limit

This paper presents a two dimensional visual computer code developed to solve magnetohydrodynamic (MHD) equations. This code runs on structured and unstructured triangles and operates by a fluctuation splitting (FS) scheme. The FS scheme originally introduced by Roe [in: K.W. Morton, M.J. Baines (Eds.), Numerical Methods for Fluid Dynamics II, Academic Press, New York, 1982] to solve Euler equations was extended by Aslan [J. Comput. Phys. 153 (1999) 437] for solving ideal MHD equations. Aslan's method included a wave model, called MHD-A, consisting of slow and fast magneto-acoustic waves as well as an entropy and artificial magnetic monopole wave. In this work, Aslan's method was extended to include external sources, a new sonic fix, and a careful normalization in the Euler limit. It is shown by numerical experiments that VIS-MHD-A is able to work accurately for a wide range of problems including discontinuities, shock structures, and problems including smooth solutions (e.g., Rayleigh–Taylor and Kelvin–Helmholtz instability).

Second‐Order–accurate Schemes for Magnetohydrodynamics with Divergence‐free Reconstruction

While working on an adaptive mesh refinement (AMR) scheme for divergence-free magnetohydrodynamics (MHD), Balsara discovered a unique strategy for the reconstruction of divergence-free vector fields. Balsara also showed that for one-dimensional variations in flow and field quantities the reconstruction reduces exactly to the total variation diminishing (TVD) reconstruction. In a previous paper by Balsara the innovations were put to use in studying AMR-MHD. While the other consequences of the invention especially as they pertain to numerical scheme design were mentioned, they were not explored in any detail. In this paper we begin such an exploration. We study the problem of divergence-free numerical MHD and show that the work done so far still has four key unresolved issues. We resolve those issues in this paper. It is shown that the magnetic field can be updated in divergence-free fashion with a formulation that is better than the one in Balsara & Spicer. The problem of reconstructing MHD flow variables with spatially second-order accuracy is also studied. Some ideas from weighted essentially non-oscillatory (WENO) reconstruction, as they apply to numerical MHD, are developed. Genuinely multidimensional reconstruction strategies for numerical MHD are also explored. The other goal of this paper is to show that the same well-designed second-order-accurate schemes can be formulated for more complex geometries such as cylindrical and spherical geometry. Being able to do divergence-free reconstruction in those geometries also resolves the problem of doing AMR in those geometries; the appendices contain detailed formulae for the same. The resulting MHD scheme has been implemented in Balsara's RIEMANN framework for parallel, self-adaptive computational astrophysics. The present work also shows that divergence-free reconstruction and the divergence-free time update can be done for numerical MHD on unstructured meshes. As a result, we establish important analogies between MHD on structured meshes and MHD on unstructured meshes because such analogies can guide the design of MHD schemes and AMR-MHD techniques on unstructured meshes. The present paper also lays out the roadmap for designing MHD schemes for structured and unstructured meshes that have better than second-order accuracy in space and time. All the schemes designed here are shown to be second-order-accurate. We also show that the accuracy does not depend on the quality of the Riemann solver. We have compared the numerical dissipation of the unsplit MHD schemes presented here with the dimensionally split MHD schemes that have been used in the past and found the former to be superior. The dissipation does depend on the Riemann solver, but the dependence becomes weaker as the quality of the interpolation is improved. Several stringent test problems are presented to show that the methods work, including problems involving high-velocity flows in low-plasma-? magnetospheric environments. Similar advances can be made in other fields, such as electromagnetics, radiation MHD, and incompressible flow, that rely on a Stokes-law type of update strategy.

Divergence-Free Adaptive Mesh Refinement for Magnetohydrodynamics

Several physical systems, such as nonrelativistic and relativistic magnetohydrodynamics (MHD), radiation MHD, electromagnetics, and incompressible hydrodynamics, satisfy Stoke's law type equations for the divergence-free evolution of vector fields. In this paper we present a full-fledged scheme for the second-order accurate, divergence-free evolution of vector fields on an adaptive mesh refinement (AMR) hierarchy. We focus here on adaptive mesh MHD. However, the scheme has applicability to the other systems of equations mentioned above. The scheme is based on making a significant advance in the divergence-free reconstruction of vector fields. In that sense, it complements the earlier work of D. S. Balsara and D. S. Spicer (1999, J. Comput. Phys.7, 270) where we discussed the divergence-free time-update of vector fields which satisfy Stoke's law type evolution equations. Our advance in divergence-free reconstruction of vector fields is such that it reduces to the total variation diminishing (TVD) property for one-dimensional evolution and yet goes beyond it in multiple dimensions. For that reason, it is extremely suitable for the construction of higher order Godunov schemes for MHD. Both the two-dimensional and three-dimensional reconstruction strategies are developed. A slight extension of the divergence-free reconstruction procedure yields a divergence-free prolongation strategy for prolonging magnetic fields on AMR hierarchies. Divergence-free restriction is also discussed. Because our work is based on an integral formulation, divergence-free restriction and prolongation can be carried out on AMR meshes with any integral refinement ratio, though we specialize the expressions for the most popular situation where the refinement ratio is two. Furthermore, we pay attention to the fact that in order to efficiently evolve the MHD equations on AMR hierarchies, the refined meshes must evolve in time with time steps that are a fraction of their parent mesh's time step. An electric field correction strategy is presented for use on AMR meshes. The electric field correction strategy helps preserve the divergence-free evolution of the magnetic field even when the time steps are subcycled on refined meshes. The above-mentioned innovations have been implemented in Balsara's RIEMANN framework for parallel, self-adaptive computational astrophysics, which supports both nonrelativistic and relativistic MHD. Several rigorous, three-dimensional AMR-MHD test problems with strong discontinuities have been run with the RIEMANN framework showing that the strategy works very well. In our AMR-MHD scheme, the adaptive mesh hierarchy can change in response to discontinuities that move rapidly with respect to the mesh. Time-step subcycling permits efficient processing of the AMR hierarchy. Our AMR-MHD scheme parallelizes very well as shown by Balsara and Norton [8].

A lattice Boltzmann analysis of the effects of the internal angular momentum on rheology of magnetic fluids

We study the rheological properties of the magnetic fluids with the internal angular momentum, using a new lattice Boltzmann model. By the appropriate extension of the lattice Boltzmann scheme for magnetohydrodynamics, it is possible to deal with the internal angular momentum caused by the magnetic dipole moment efficiently. This model is suitable for a wide use for simulating various behaviors of magnetic fluids with the internal angular momentum. We present here a two-dimensional lattice Boltzmann analysis of the effects of internal angular momentum in magnetic fluids in order to understand the rheological properties of the magnetic colloid driven by the external magnetic field.