Dinshaw S. Balsara

Notes on the eigensystem of magnetohydrodynamics

The eigenstructure of the equations governing one-dimensional ideal magnetohy-drodynamics is examined, motivated by the wish to exploit it for construction of high-resolution computational algorithms. The results are given in simple forms that avoid indeterminacy or degeneracy whenever possible. The unavoidable indeterminacy near the magnetosonic (or triple umbilic) state is analysed and shown to cause no difficulty in evaluating a numerical flux function. The structure of wave paths close to this singularity is obtained, and simple expressions are presented for the structure coefficients that govern wave steepening.

Divergence-free adaptive mesh refinement for Magnetohydrodynamics

Abstract Several physical systems, such as nonrelativistic and relativistic magnetohydrodynamics (MHD), radiation MHD, electromagnetics, and incompressible hydrodynamics, satisfy Stokes 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 Stokes 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 meshs 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 Balsaras 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].

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 Balsaras 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.

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.

Total Variation Diminishing Scheme for Relativistic Magnetohydrodynamics

In this paper we present a total variation diminishing (TVD) scheme for numerical relativistic magnetohydrodynamics (MHD). The eigenstructure of the equations of relativistic MHD has been cataloged here. We also describe a strategy for obtaining the physically relevant eigenvectors. These eigenvectors are then used to build an interpolation strategy that operates on the characteristic fields. The design of a linearized Riemann solver for relativistic MHD is also described. The resulting higher order Godunov scheme has been implemented in the authors RIEMANN code for astrophysical fluid dynamics. The resulting code is second-order accurate both in space and time. A number of design features have been used to make it a high-resolution scheme. It shows efficient and robust performance for several stringent test problems.

Total Variation Diminishing Scheme for Adiabatic and Isothermal Magnetohydrodynamics

In this paper a total variation diminishing (TVD) scheme is constructed for solving the equations of ideal adiabatic and isothermal MHD. It is based on an extremely efficient formulation of the MHD Riemann problems for either case. Piecewise linear interpolation is applied to the characteristic variables along with steepening of linearly degenerate characteristic fields. A predictor-corrector formulation is used to achieve second-order-accurate temporal update. An artificial viscosity and hyperviscosity are formulated using the characteristic variables. The viscosity and hyperviscosity are designed so that they never damage the TVD property. An accurate formulation of the divergence cleaning step is presented. This formulation is more accurate than the one that has been used so far. The scheme designed is second-order accurate in space and time. It has been implemented in the authors RIEMANN code for numerical MHD. A variety of test problems are presented. They test all aspects of numerical MHD including (1) handling of exotic wave structures that occur in MHD, (2) treatment of multiple discontinuities, (3) handling of very strong shocks, and (4) multidimensional problems. The scheme displays robust and accurate behavior in each case. An extremely efficient implementation has been achieved for massively parallel processor (MPP) machines displaying the ability of this scheme to sustain scalable, load-balanced performance in MPP environments.

Multidimensional HLLE Riemann solver: Application to Euler and magnetohydrodynamic flows

Abstract In this work we present a general strategy for constructing multidimensional HLLE Riemann solvers, with particular attention paid to detailing the two-dimensional HLLE Riemann solver. This is accomplished by introducing a constant resolved state between the states being considered, which introduces sufficient dissipation for systems of conservation laws. Closed form expressions for the resolved fluxes are also provided to facilitate numerical implementation. The Riemann solver is proved to be positively conservative for the density variable; the positivity of the pressure variable has been demonstrated for Euler flows when the divergence in the fluid velocities is suitably restricted so as to prevent the formation of cavitation in the flow. We also focus on the construction of multidimensionally upwinded electric fields for divergence-free magnetohydrodynamical (MHD) flows. A robust and efficient second order accurate numerical scheme for two and three-dimensional Euler and MHD flows is presented. The scheme is built on the current multidimensional Riemann solver and has been implemented in the author’s RIEMANN code. The number of zones updated per second by this scheme on a modern processor is shown to be cost-competitive with schemes that are based on a one-dimensional Riemann solver. However, the present scheme permits larger timesteps. Accuracy analysis for multidimensional Euler and MHD problems shows that the scheme meets its design accuracy. Several stringent test problems involving Euler and MHD flows are also presented and the scheme is shown to perform robustly on all of them.

The Distribution of Pressures in a Supernova-driven Interstellar Medium. I. Magnetized Medium

Observations have suggested substantial departures from pressure equilibrium in the interstellar medium (ISM) in the plane of the Galaxy, even on scales under 50 pc. Nevertheless, multi-phase models of the ISM assume at least locally isobaric gas. The pressure then determines the density reached by gas cooling to stable thermal equilibrium. We use two different sets of numerical models of the ISM to examine the consequences of supernova driving for interstellar pressures. The first set of models is hydrodynamical, and uses adaptive mesh refinement to allow computation of a 1 x 1 x 20 kpc section of a stratified galactic disk. The second set of models is magnetohydrodynamical, using an independent code framework, and examines a 200 pc cubed periodic domain threaded by magnetic fields. Both of these models show broad pressure distributions with roughly log-normal functional forms produced by both shocks and rarefaction waves, rather than the power-law distributions predicted by previous work, with rather sharp thermal pressure gradients. The width of the distribution of the logs of pressure in gas with log T < 3.9 is proportional to the rms Mach number in that gas, while the distribution in hotter gas is broader, but not so broad as would be predicted by the Mach numbers in that gas. Individual parcels of gas reach widely varying points on the thermal equilibrium curve: no unique set of phases is found, but rather a dynamically-determined continuum of densities and temperatures. Furthermore, a substantial fraction of the gas remains entirely out of thermal equilibrium. Our results appear consistent with observations of interstellar pressures, and suggest that the pressures observed in molecular clouds may be due to ram pressure rather than gravitational confinement.

Fast and accurate discrete ordinates methods for multidimensional radiative transfer. Part I, basic methods

Abstract The ability to solve the radiative transfer equation in a fast and accurate fashion is central to several important applications in combustion physics, controlled thermonuclear fusion and astrophysics. Most practitioners see the value of using discrete ordinates methods for such applications. However, previous efforts at designing discrete ordinates methods that are both fast and accurate have met with limited success. This is especially so when parts of the application satisfy the free streaming limit in which case most solution strategies become unacceptably diffusive or when parts of the application have high absorption or scattering opacities in which case most solution strategies converge poorly. Designing a single solution strategy that retains second-order accuracy and converges with optimal efficiency in the free streaming limit as well as the optically thick limit is a challenge. Recent results also indicate that schemes that are less than second-order accurate will not retrieve the radiation diffusion limit. In this paper we analyze several of the challenges involved in doing multidimensional numerical radiative transfer. It is realized that genuinely multidimensional discretizations of the radiative transfer equation that are second-order accurate exist. Because such discretizations are more faithful to the physics of the problem they help minimize the diffusion in the free streaming limit. Because they have a more compact stencil, they have superior convergence properties. The ability of the absorption and scattering terms to couple strongly to the advection terms is examined. Based on that we find that operator splitting of the scattering and advection terms damages the convergence in several situations. Newton–Krylov methods are shown to provide a natural way to incorporate the effects of nonlinearity as well as strong coupling in a way that avoids operator splitting. Used by themselves, Newton–Krylov methods converge slowly. However, when the Newton–Krylov methods are used as smoothers within a full approximation scheme multigrid method, the convergence is vastly improved. The combination of a genuinely multidimensional, nonlinearly positive scheme that uses Full Approximation Scheme multigrid in conjunction with the Newton–Krylov method is shown to result in a discrete ordinates method for radiative transfer that is highly accurate and converges very rapidly in all circumstances. Several convergence studies are carried out which show that the resultant method has excellent convergence properties. Moreover, this excellent convergence is retained in the free streaming limit as well as in the limit of high optical depth. The presence of strong scattering terms does not slow down the convergence rate for our method. In fact it is shown that without operator splitting, the presence of a strong scattering opacity enhances the convergence rate in quite the same way that the convergence is enhanced when a high absorption opacity is present! We show that the use of differentiable limiters results in substantial improvement in the convergence rate of the method. By carrying out an accuracy analysis on meshes with increasing resolution it is further shown that the accuracy that one obtains seems rather close to the designed second-order accuracy and does not depend on the specific choice of limiter. The methods for multidimensional radiative transfer that are presented here should improve the accuracy of several radiative transfer calculations while at the same time improving their convergence properties. Because the methods presented here are similar to those used for simulating neutron transport problems and problems involving rarefied gases, those fields should also see improvements in their numerical capabilities by assimilation of the methods presented here.

ADER-WENO finite volume schemes with space–time adaptive mesh refinement

Abstract We present the first high order one-step ADER-WENO finite volume scheme with adaptive mesh refinement (AMR) in multiple space dimensions. High order spatial accuracy is obtained through a WENO reconstruction, while a high order one-step time discretization is achieved using a local space–time discontinuous Galerkin predictor method. Due to the one-step nature of the underlying scheme, the resulting algorithm is particularly well suited for an AMR strategy on space–time adaptive meshes, i.e. with time-accurate local time stepping. The AMR property has been implemented ‘cell-by-cell’, with a standard tree-type algorithm, while the scheme has been parallelized via the message passing interface (MPI) paradigm. The new scheme has been tested over a wide range of examples for nonlinear systems of hyperbolic conservation laws, including the classical Euler equations of compressible gas dynamics and the equations of magnetohydrodynamics (MHD). High order in space and time have been confirmed via a numerical convergence study and a detailed analysis of the computational speed-up with respect to highly refined uniform meshes is also presented. We also show test problems where the presented high order AMR scheme behaves clearly better than traditional second order AMR methods. The proposed scheme that combines for the first time high order ADER methods with space–time adaptive grids in two and three space dimensions is likely to become a useful tool in several fields of computational physics, applied mathematics and mechanics.