Author

# Dinshaw S. Balsara

Other affiliations: National Center for Supercomputing Applications, Joint Institute for Nuclear Astrophysics, University of Illinois at Urbana–Champaign ...read more

Bio: Dinshaw S. Balsara is an academic researcher from University of Notre Dame. The author has contributed to research in topics: Riemann solver & Finite volume method. The author has an hindex of 46, co-authored 171 publications receiving 9200 citations. Previous affiliations of Dinshaw S. Balsara include National Center for Supercomputing Applications & Joint Institute for Nuclear Astrophysics.

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##### Papers

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TL;DR: In this paper, a class of numerical schemes that are higher-order extensions of the weighted essentially non-oscillatory (WENO) schemes of G.-S. Jiang and C.-W. Shu (1996) and X.-D. Liu, S. Osher, and T. T. Chan (1994) are presented.

Abstract: In this paper we design a class of numerical schemes that are higher-order extensions of the weighted essentially non-oscillatory (WENO) schemes of G.-S. Jiang and C.-W. Shu (1996) and X.-D. Liu, S. Osher, and T. Chan (1994). Used by themselves, the schemes may not always be monotonicity preserving but coupled with the monotonicity preserving bounds of A. Suresh and H. T. Huynh (1997) they perform very well. The resulting monotonicity preserving weighted essentially non-oscillatory (MPWENO) schemes have high phase accuracy and high order of accuracy. The higher-order members of this family are almost spectrally accurate for smooth problems. Nevertheless, they, have robust shock capturing ability. The schemes are stable under normal CFL numbers. They are also efficient and do not have a computational complexity that is substantially greater than that of the lower-order members of this same family of schemes. The higher accuracy that these schemes offer coupled with their relatively low computational complexity makes them viable competitors to lower-order schemes, such as the older total variation diminishing schemes, for problems containing both discontinuities and rich smooth region structure. We describe the MPWENO schemes here as well as show their ability to reach their designed accuracies for smooth flow. We also examine the role of steepening algorithms such as the artificial compression method in the design of very high order schemes. Several test problems in one and two dimensions are presented. For multidimensional problems where the flow is not aligned with any of the grid directions it is shown that the present schemes have a substantial advantage over lower-order schemes. It is argued that the methods designed here have great utility for direct numerical simulations and large eddy simulations of compressible turbulence. The methodology developed here is applicable to other hyperbolic systems, which is demonstrated by showing that the MPWENO schemes also work very well on magnetohydrodynamical test problems.

1,173 citations

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TL;DR: This paper presents a staggered mesh strategy which directly uses the properly upwinded fluxes that are provided by a Godunov scheme, and shows that a scheme that involves a collocation of magnetic field variables that is different from the one traditionally favored in the design of higher orderGodunov schemes can nevertheless offer the same robust and accurate performance.

Abstract: The equations of magnetohydrodynamics (MHD) have been formulated as a hyperbolic system of conservation laws. In that form it becomes possible to use higher order Godunov schemes for their solution. This results in a robust and accurate solution strategy. However, the magnetic field also satisfies a constraint that requires its divergence to be zero at all times. This is a property that cannot be guaranteed in the zone centered discretizations that are favored in Godunov schemes without involving a divergence cleaning step. In this paper we present a staggered mesh strategy which directly uses the properly upwinded fluxes that are provided by a Godunov scheme. The process of directly using the upwinded fluxes relies on a duality that exists between the fluxes obtained from a higher order Godunov scheme and the electric fields in a plasma. By exploiting this duality we have been able to construct a higher order Godunov scheme that ensures that the magnetic field remains divergence-free up to the computer's round-off error. We have even presented a variant of the basic algorithm that uses multidimensional features in the flow to design an upwinded strategy that aligns itself with the predominant upwinded direction in the flow. We have devised several stringent test problems to show that the scheme works robustly and accurately in all situations. In doing so we have shown that a scheme that involves a collocation of magnetic field variables that is different from the one traditionally favored in the design of higher order Godunov schemes can nevertheless offer the same robust and accurate performance of higher order Godunov schemes provided the properly upwinded fluxes from the Godunov methodology are used in the scheme's construction.

641 citations

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TL;DR: A conservative least-squares polynomial reconstruction operator is applied to the discontinuous Galerkin method, which yields space–time polynomials for the vector of conserved variables and for the physical fluxes and source terms that can be used in a natural way to construct very efficient fully-discrete and quadrature-free one-step schemes.

Abstract: In this article, a conservative least-squares polynomial reconstruction operator is applied to the discontinuous Galerkin method. In a first instance, piecewise polynomials of degree N are used as test functions as well as to represent the data in each element at the beginning of a time step. The time evolution of these data and the flux computation, however, are then done with a different set of piecewise polynomials of degree M⩾NM⩾N, which are reconstructed from the underlying polynomials of degree N . This approach yields a general, unified framework that contains as two special cases classical high order finite volume (FV) schemes (N=0)(N=0) as well as the usual discontinuous Galerkin (DG) method (N=M)(N=M). In the first case, the polynomial is reconstructed from cell averages, for the latter, the reconstruction reduces to the identity operator. A completely new class of numerical schemes is generated by choosing N≠0N≠0 and M>NM>N. The reconstruction operator is implemented for arbitrary polynomial degrees N and M on unstructured triangular and tetrahedral meshes in two and three space dimensions.
To provide a high order accurate one-step time integration of the same formal order of accuracy as the spatial discretization operator, the (reconstructed) polynomial data of degree M are evolved in time locally inside each element using a new local continuous space–time Galerkin method. As a result of this approach, we obtain, as a high order accurate predictor, space–time polynomials for the vector of conserved variables and for the physical fluxes and source terms, which then can be used in a natural way to construct very efficient fully-discrete and quadrature-free one-step schemes. This feature is particularly important for DG schemes in three space dimensions, where the cost of numerical quadrature may become prohibitively expensive for very high orders of accuracy.
Numerical convergence studies of all members of the new general class of proposed schemes are shown up to sixth-order of accuracy in space and time on unstructured two- and three-dimensional meshes for two very prominent nonlinear hyperbolic systems, namely for the Euler equations of compressible gas dynamics and the equations of ideal magnetohydrodynamics (MHD). The results indicate that the new class of intermediate schemes (N≠0,M>N)(N≠0,M>N) is computationally more efficient than classical finite volume or DG schemes.
Finally, a large set of interesting test cases is solved on unstructured meshes, where the proposed new time stepping approach is applied to the equations of ideal and relativistic MHD as well as to nonlinear elasticity, using a standard high order WENO finite volume discretization in space to cope with discontinuous solutions.

555 citations

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TL;DR: In this article, a von Neumann stability analysis of the equations of smoothed particle hydrodynamics (SPH) along with a critical discussion of various parts of the algorithm is presented.

Abstract: We present a von Neumann stability analysis of the equations of smoothed particle hydrodynamics (SPH) along with a critical discussion of various parts of the algorithm. The stability analysis is done without any major restrictions and, hence, models the full Euler equations in one dimension. This then allows us to deduce optimal ranges for parameters that need to be used in SPH. Thus we show that for the commonly used M"5 spline the ratio of smoothing length to interparticle distance should range between 1.0 to 1.4. We also show that the linear artificial viscosity coefficient and the coefficient of spatial filtering have to be bounded. The results of this von Neumann stability analysis provide us with several suggestions for future algorithm improvement. Because the SPH method is so unique we provide, wherever possible, comparisons with more familiar and well-used high resolution finite difference methods.

522 citations

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TL;DR: In this paper, Balsara et al. showed that the magnetic field can be updated in divergence-free fashion with a formulation that is better than the one in BalsARA & Spicer.

Abstract: 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.

298 citations

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TL;DR: GADGET-2 as mentioned in this paper is a massively parallel tree-SPH code, capable of following a collisionless fluid with the N-body method, and an ideal gas by means of smoothed particle hydrodynamics.

Abstract: We discuss the cosmological simulation code GADGET-2, a new massively parallel TreeSPH code, capable of following a collisionless fluid with the N-body method, and an ideal gas by means of smoothed particle hydrodynamics (SPH). Our implementation of SPH manifestly conserves energy and entropy in regions free of dissipation, while allowing for fully adaptive smoothing lengths. Gravitational forces are computed with a hierarchical multipole expansion, which can optionally be applied in the form of a TreePM algorithm, where only short-range forces are computed with the ‘tree’ method while long-range forces are determined with Fourier techniques. Time integration is based on a quasi-symplectic scheme where long-range and short-range forces can be integrated with different time-steps. Individual and adaptive short-range time-steps may also be employed. The domain decomposition used in the parallelization algorithm is based on a space-filling curve, resulting in high flexibility and tree force errors that do not depend on the way the domains are cut. The code is efficient in terms of memory consumption and required communication bandwidth. It has been used to compute the first cosmological N-body simulation with more than 10 10 dark matter particles, reaching a homogeneous spatial dynamic range of 10 5 per dimension in a three-dimensional box. It has also been used to carry out very large cosmological SPH simulations that account for radiative cooling and star formation, reaching total particle numbers of more than 250 million. We present the algorithms used by the code and discuss their accuracy and performance using a number of test problems. GADGET-2 is publicly released to the research community. Ke yw ords: methods: numerical ‐ galaxies: interactions ‐ dark matter.

6,196 citations

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TL;DR: In this article, the theory and application of Smoothed particle hydrodynamics (SPH) since its inception in 1977 are discussed, focusing on the strengths and weaknesses, the analogy with particle dynamics and the numerous areas where SPH has been successfully applied.

Abstract: In this review the theory and application of Smoothed particle hydrodynamics (SPH) since its inception in 1977 are discussed. Emphasis is placed on the strengths and weaknesses, the analogy with particle dynamics and the numerous areas where SPH has been successfully applied.

4,070 citations

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TL;DR: The aim of this paper is to present the reader with a perspective on how JFNK may be applicable to applications of interest and to provide sources of further practical information.

Abstract: Jacobian-free Newton-Krylov (JFNK) methods are synergistic combinations of Newton-type methods for superlinearly convergent solution of nonlinear equations and Krylov subspace methods for solving the Newton correction equations. The link between the two methods is the Jacobian-vector product, which may be probed approximately without forming and storing the elements of the true Jacobian, through a variety of means. Various approximations to the Jacobian matrix may still be required for preconditioning the resulting Krylov iteration. As with Krylov methods for linear problems, successful application of the JFNK method to any given problem is dependent on adequate preconditioning. JFNK has potential for application throughout problems governed by nonlinear partial differential equations and integro-differential equations. In this survey paper, we place JFNK in context with other nonlinear solution algorithms for both boundary value problems (BVPs) and initial value problems (IVPs). We provide an overview of the mechanics of JFNK and attempt to illustrate the wide variety of preconditioning options available. It is emphasized that JFNK can be wrapped (as an accelerator) around another nonlinear fixed point method (interpreted as a preconditioning process, potentially with significant code reuse). The aim of this paper is not to trace fully the evolution of JFNK, nor to provide proofs of accuracy or optimal convergence for all of the constituent methods, but rather to present the reader with a perspective on how JFNK may be applicable to applications of interest and to provide sources of further practical information.

1,803 citations

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TL;DR: In this article, a moving unstructured mesh defined by the Voronoi tessellation of a set of discrete points is used to solve the hyperbolic conservation laws of ideal hydrodynamics with a finite volume approach, based on a second-order unsplit Godunov scheme with an exact Riemann solver.

Abstract: Hydrodynamic cosmological simulations at present usually employ either the Lagrangian smoothed particle hydrodynamics (SPH) technique or Eulerian hydrodynamics on a Cartesian mesh with (optional) adaptive mesh refinement (AMR). Both of these methods have disadvantages that negatively impact their accuracy in certain situations, for example the suppression of fluid instabilities in the case of SPH, and the lack of Galilean invariance and the presence of overmixing in the case of AMR. We here propose a novel scheme which largely eliminates these weaknesses. It is based on a moving unstructured mesh defined by the Voronoi tessellation of a set of discrete points. The mesh is used to solve the hyperbolic conservation laws of ideal hydrodynamics with a finite-volume approach, based on a second-order unsplit Godunov scheme with an exact Riemann solver. The mesh-generating points can in principle be moved arbitrarily. If they are chosen to be stationary, the scheme is equivalent to an ordinary Eulerian method with second-order accuracy. If they instead move with the velocity of the local flow, one obtains a Lagrangian formulation of continuum hydrodynamics that does not suffer from the mesh distortion limitations inherent in other mesh-based Lagrangian schemes. In this mode, our new method is fully Galilean invariant, unlike ordinary Eulerian codes, a property that is of significant importance for cosmological simulations where highly supersonic bulk flows are common. In addition, the new scheme can adjust its spatial resolution automatically and continuously, and hence inherits the principal advantage of SPH for simulations of cosmological structure growth. The high accuracy of Eulerian methods in the treatment of shocks is also retained, while the treatment of contact discontinuities improves. We discuss how this approach is implemented in our new code arepo, both in 2D and in 3D, and is parallelized for distributed memory computers. We also discuss techniques for adaptive refinement or de-refinement of the unstructured mesh. We introduce an individual time-step approach for finite-volume hydrodynamics, and present a high-accuracy treatment of self-gravity for the gas that allows the new method to be seamlessly combined with a high-resolution treatment of collisionless dark matter. We use a suite of test problems to examine the performance of the new code and argue that the hydrodynamic moving-mesh scheme proposed here provides an attractive and competitive alternative to current SPH and Eulerian techniques.

1,778 citations