arXiv subjects

Dinshaw S. Balsara

Publications and source records attributed to Dinshaw S. Balsara.

At least 37 records · Page 2Linked to original sources

arXiv2019

A simplified Cauchy-Kowalewskaya procedure for the implicit solution of generalized Riemann problems of hyperbolic balance laws

The Cauchy-Kowalewskaya (CK) procedure is a key building block in the design of solvers for the Generalised Rieman Problem (GRP) based on Taylor series expansions in time. The CK procedure allows us to express time derivatives in terms of purely space derivatives. This is a very cumbersome procedure, which often requires the use of software manipulators. In this paper, a simplification of the CK procedure is proposed in the context of implicit Taylor series expansion for GRP, for hyperbolic balance laws in the framework of [Journal of Computational Physics 303 (2015) 146-172]. A recursive formula for the CK procedure, which is straightforwardly implemented in computational codes, is obtained. The proposed GRP solver is used in the context of the ADER approach and several one-dimensional problems are solved to demonstrate the applicability and efficiency of the present scheme. An enhancement in terms of efficiency, is obtained. Furthermore, the expected theoretical orders of accuracy are achieved, conciliating accuracy and stability.

math.NA↗

arXiv2019

Efficient, Divergence-Free, High Order MHD on 3D Spherical Meshes with Optimal Geodesic Meshing

There is a great need in several areas of astrophysics and space-physics to carry out high order of accuracy, divergence-free MHD simulations on spherical meshes. This requires us to pay careful attention to the interplay between mesh quality and numerical algorithms. Methods have been designed that fundamentally integrate high order isoparametric mappings with the other high accuracy algorithms that are needed for divergence-free MHD simulations on geodesic meshes. The goal of this paper is to document such algorithms that are implemented in the geodesic mesh version of the RIEMANN code. The fluid variables are reconstructed using a special kind of WENO-AO algorithm that integrates the mesh geometry into the reconstruction process from the ground-up. A novel divergence-free reconstruction strategy for the magnetic field that performs efficiently at all orders, even on isoparametrically mapped meshes, is then presented. The MHD equations are evolved in space and time using a novel ADER predictor algorithm that is efficiently adapted to the isoparametrically mapped geometry. The application of one-dimensional and multidimensional Riemann solvers at suitable locations on the mesh then provides the corrector step. The corrector step for the magnetic field uses a Yee-type staggering of magnetic fields. This results in a scheme with divergence-free update for the magnetic field. The use of ADER enables a one-step update which only requires one messaging operation per complete timestep. This is very beneficial for parallel processing. Several accuracy tests are presented as are stringent test problems. PetaScale performance is also demonstrated on the largest available supercomputers.

physics.comp-ph↗

arXiv2019

Turbulent dynamo in a weakly ionized medium

The small-scale turbulent dynamo is an important process contributing to the cosmic magnetization. In partially ionized astrophysical plasmas, the dynamo growth of magnetic energy strongly depends on the coupling state between ions and neutrals and the ion-neutral collisional damping effect. A new damping stage of turbulent dynamo in a weakly ionized medium was theoretically predicted by Xu \& Lazarian (2016). By carrying out a 3D two-fluid dynamo simulation, here we for the first time numerically confirmed the physical conditions and the linear-in-time growth of magnetic field strength of the damping stage of dynamo. The dynamo-amplified magnetic field has a characteristic length as the damping scale, which increases with time and can reach the injection scale of turbulence after around eight largest eddy-turnover times given sufficiently low ionization fraction and weak initial magnetic field. Due to the weak coupling between ions and neutrals, most turbulent energy carried by neutrals cannot be converted to the magnetic energy, resulting in a relatively weak magnetic field at the end of dynamo. This result has important implications for the growth of magnetic fields in the partially ionized interstellar medium and shock acceleration of Galactic cosmic rays.

astro-ph.GA↗

arXiv2018

General Relativistic Numerical Simulation of sub-Keplerian Transonic Accretion Flows onto Rotating Black Holes: Kerr Spacetime

We study time evolution of sub-Keplerian transonic accretion flows onto black holes using a general relativistic numerical simulation code. We perform simulations around the black holes having non-zero rotation. We first compare one-dimensional simulation results with theoretical results and validate the performance of our code. Next, we present results of axisymmetric, two-dimensional simulation of advective flows. In the literature, there is no solution which describes steady shock solutions in two dimensions. However, our simulations produce these centrifugal force supported steady shock waves even in presence of strong dragging of inertial frames. Since the post-shock region could be hot and upscatter photons through Comptonization, these shock would put imprints on the spectra. Thus, our solutions, which represent truly new results, could be useful to measure spins through radiation spectrum of accreting Kerr black holes.

astro-ph.HE↗

arXiv2018

Globally constraint-preserving FR/DG scheme for Maxwell's equations at all orders

Computational electrodynamics (CED), the numerical solution of Maxwell's equations, plays an incredibly important role in several problems in science and engineering. High accuracy solutions are desired, and the discontinuous Galerkin (DG) method is one of the better ways of delivering high accuracy in CED. Maxwell's equations have a pair of involution constraints for which mimetic schemes that globally satisfy the constraints at a discrete level are highly desirable. Balsara and Kappeli presented a von Neumann stability analysis of globally constraint-preserving DG schemes for CED up to 4'th order which was focused on developing the theory and documenting the superior dissipation and dispersion of DGTD schemes in media with constant permittivity and permeability. In this paper we present DGTD schemes for CED that go up to 5'th order of accuracy and analyze their performance when permittivity and permeability vary strongly in space. Our DGTD schemes achieve constraint preservation by collocating the electric displacement and magnetic induction as well as their higher order modes in the faces of the mesh. Our first finding is that at 4'th and higher orders, one has to evolve some zone-centered modes in addition to the face-centered modes. It is well-known that the limiting step in DG schemes causes a reduction of the optimal accuracy of the scheme. In this paper we document simulations where permittivity and permeability vary by almost an order of magnitude without requiring any limiting of the DG scheme. This very favorable finding ensures that DGTD schemes retain optimal accuracy even in the presence of large spatial variations in permittivity/permeability. Our third finding shows that the electromagnetic energy is conserved very well even when permittivity and permeability vary strongly in space; as long as the conductivity is zero.

math.NA↗

arXiv2018

von Neumann Stability Analysis of Globally Constraint-Preserving DGTD and PNPM Schemes for the Maxwell Equations using Multidimensional Riemann Solvers

The time-dependent equations of computational electrodynamics (CED) are evolved consistent with the divergence constraints. As a result, there has been a recent effort to design finite volume time domain (FVTD) and discontinuous Galerkin time domain (DGTD) schemes that satisfy the same constraints and, nevertheless, draw on recent advances in higher order Godunov methods. This paper catalogues the first step in the design of globally constraint-preserving DGTD schemes. The algorithms presented here are based on a novel DG-like method that is applied to a Yee-type staggering of the electromagnetic field variables in the faces of the mesh. The other two novel building blocks of the method include constraint-preserving reconstruction of the electromagnetic fields and multidimensional Riemann solvers; both of which have been developed in recent years by the first author. We carry out a von Neumann stability analysis of the entire suite of DGTD schemes for CED at orders of accuracy ranging from second to fourth. A von Neumann stability analysis gives us the maximal CFL numbers that can be sustained by the DGTD schemes presented here at all orders. It also enables us to understand the wave propagation characteristics of the schemes in various directions on a Cartesian mesh. We find that the CFL of DGTD schemes decreases with increasing order. To counteract that, we also present constraint-preserving PNPM schemes for CED. We find that the third and fourth order constraint-preserving DGTD and P1PM schemes have some extremely attractive properties when it comes to low-dispersion, low-dissipation propagation of electromagnetic waves in multidimensions. Numerical accuracy tests are also provided to support the von Neumann stability analysis.

physics.comp-ph↗

arXiv2018

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

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 multi-dimensional 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 all terms related to the pressure in the momentum equation and the total energy equation implicitly, making again use of a properly staggered grid for pressure and velocity. 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. Our new method is 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 due to the increasingly 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.

math.NA↗

arXiv2018

An Efficient, Second Order Accurate, Universal Generalized Riemann Problem Solver Based on the HLLI Riemann Solver

The Riemann problem, and the associated generalized Riemann problem, are increasingly seen as the important building blocks for modern higher order Godunov-type schemes. In the past, building a generalized Riemann problem solver was seen as an intricately mathematical task for complicated physical or engineering problems because the associated Riemann problem is different for each hyperbolic system of interest. This paper changes that situation. The HLLI Riemann solver is a recently-proposed Riemann solver that is universal in that it is applicable to any hyperbolic system, whether in conservation form or with non-conservative products. The HLLI Riemann solver is also complete in the sense that if it is given a complete set of eigenvectors, it represents all waves with minimal dissipation. It is, therefore, very attractive to build a generalized Riemann problem solver version of the HLLI Riemann solver. This is the task that is accomplished in the present paper. We show that at second order, the generalized Riemann problem version of the HLLI Riemann solver is easy to design. Our GRP solver is also complete and universal because it inherits those good properties from original HLLI Riemann solver. We also show how our GRP solver can be adapted to the solution of hyperbolic systems with stiff source terms. Our generalized HLLI Riemann solver is easy to implement and performs robustly and well over a range of test problems. All implementation-related details are presented. Results from several stringent test problems are shown. These test problems are drawn from many different hyperbolic systems, and include hyperbolic systems in conservation form; with non-conservative products; and with stiff source terms. The present generalized Riemann problem solver performs well on all of them.

math.NA↗

arXiv2017

General Relativistic Numerical Simulation of sub-Keplerian Transonic Accretion Flows onto Black Holes: Schwarzschild Spacetime

We study time evolution of sub-Keplerian transonic accretion flows onto black holes using a general relativistic numerical simulation code. We perform simulations in Schwarzschild spacetime. We first compare one-dimensional simulation results with theoretical results and validate the performance of our code. Next, we present results of axisymmetric, two-dimensional simulation of advective flows. We find that even in this case, for which no complete theoretical analysis is present in the literature, steady state shock formation is possible.

astro-ph.HE↗

arXiv2017

Higher Order Accurate Space-Time Schemes for Computational Astrophysics -- Part I -- Finite Volume Methods

As computational astrophysics comes under pressure to become a precision science, there is an increasing need to move to high accuracy schemes for computational astrophysics. Hence the need for a specialized review on higher order schemes for computational astrophysics. The focus here is on weighted essentially non-oscillatory (WENO) schemes, discontinuous Galerkin (DG) schemes and PNPM schemes. WENO schemes are higher order extensions of traditional second order finite volume schemes which are already familiar to most computational astrophysicists. DG schemes, on the other hand, evolve all the moments of the solution, with the result that they are more accurate than WENO schemes. PNPM schemes occupy a compromise position between WENO and PNPM schemes. They evolve an Nth order spatial polynomial, while reconstructing higher order terms up to Mth order. As a result, the timestep can be larger. Time-dependent astrophysical codes need to be accurate in space and time. This is realized with the help of SSP-RK (strong stability preserving Runge-Kutta) schemes and ADER (Arbitrary DERivative in space and time) schemes. The most popular approaches to SSP-RK and ADER schemes are also described. The style of this review is to assume that readers have a basic understanding of hyperbolic systems and one-dimensional Riemann solvers. Such an understanding can be acquired from a sequence of prepackaged lectures available from http://www.nd.edu/~dbalsara/Numerical-PDE-Course. We then build on this understanding to give the reader a practical introduction to the schemes described here. The emphasis is on computer-implementable ideas, not necessarily on the underlying theory, because it was felt that this would be most interesting to most computational astrophysicists.

astro-ph.IM↗

arXiv2017

On the Linear Stability of Magnetized Jets Without Current Sheets - Relativistic Case

In our prior papers, we considered the non-relativistic linear stability analysis of magnetized jets that do not have current sheet at the boundary. In this paper, we extend our analysis to relativistic jets. In order to find the unstable modes of current sheet-free, magnetized relativistic jets, we linearize full relativistic magnetohydrodynamics equations and solve them numerically. We find the dispersion relation of the pinch and kink mode instabilities. By comparing the dispersion relations of mildly relativistic jet (Lorentz factor 2) with moderately relativistic jet (Lorentz factor 10), we find that the jet with higher Lorentz factor is significantly more stable in both pinch and kink modes. We show that inclusion of the current sheet-free magnetic field in the jet further enhances the stability. Both pinch and kink mode instabilities become progressively more stable with increasing magnetization. We also show a scaling relation between the maximum temporal growth rate of the unstable mode and the Lorentz factor of the jet. The maximum temporal growth rates of the unstable modes are inversely proportion to the Lorentz factors for most of the modes that we study. However, for the fundamental pinch mode it is inversely proportional to the square of the Lorentz factor. This very beneficial scaling relation holds regardless of the presence of a magnetic field.

astro-ph.HE↗

arXiv2016

A High-Order Relativistic Two-Fluid Electrodynamic Scheme with Consistent Reconstruction of Electromagnetic Fields and a Multidimensional Riemann Solver for Electromagnetism

In various astrophysics settings it is common to have a two-fluid relativistic plasma that interacts with the electromagnetic field. While it is common to ignore the displacement current in the ideal, classical magnetohydrodynamic limit, when the flows become relativistic this approximation is less than absolutely well-justified. In such a situation, it is more natural to consider a positively charged fluid made up of positrons or protons interacting with a negatively charged fluid made up of electrons. The two fluids interact collectively with the full set of Maxwell's equations. As a result, a solution strategy for that coupled system of equations is sought and found here. Our strategy extends to higher orders, providing increasing accuracy. Three important innovations are reported here. In our first innovation, the magnetic field within each zone is reconstructed in a divergence-free fashion while the electric field within each zone is reconstructed in a form that is consistent with Gauss' law. In our second innovation, a multidimensionally upwinded strategy is presented which ensures that the magnetic field can be updated via a discrete interpretation of Faraday's law and the electric field can be updated via a discrete interpretation of the generalized Ampere's law. Our third innovation consists of an efficient design for several popular IMEX schemes so that they provide strong coupling between the finite-volume-based fluid solver and the electromagnetic fields at high order. Several accuracy analyses are presented showing that our method meets its design accuracy in the MHD limit as well as in the limit of electromagnetic wave propagation. Several stringent test problems are also presented. We also present a relativistic version of the GEM problem, which shows that our algorithm can successfully adapt to challenging problems in high energy astrophysics.

physics.comp-ph↗

arXiv2016

On the Linear Stability of Sheared and Magnetized Jets Without Current Sheets - Non-Relativistic Case

In a prior paper (Kim et al. 2015) we considered the linear stability of magnetized jets that carry no net electric current and do not have current sheets. In this paper, in addition to physically well-motivated magnetic field structures, we also include the effects of jet shear. The jets we study have finite thermal pressure in addition to having realistic magnetic field structures and velocity shear. We find that shear has a strongly stabilizing effect on various modes of jet instability. Increasing shear stabilizes the fundamental pinch modes at long wavelengths and short wavelengths. Increasing shear also stabilizes the first reflection pinch modes at short wavelengths. Increasing shear has only a very modest stabilizing effect on the fundamental kink modes at long wavelengths; however, increasing shear does have a strong stabilizing effect on the fundamental kink modes at short wavelengths. The first reflection kink modes are strongly stabilized by increasing shear at shorter wavelengths. Overall, we find that the combined effect of magnetic field and shear stabilizes jets more than shear alone. In addition to the results from a formal linear stability analysis, we present a novel way of visualizing and understanding jet stability. This gives us a deeper understanding of the enhanced stability of sheared, magnetized jets. We also emphasize the value of our numerical approach in understanding the linear stability of jets with realistic structure.

astro-ph.HE↗

arXiv2016

A Subluminal Relativistic Magnetohydrodynamics Scheme with ADER-WENO Predictor and Multidimensional Riemann Solver-Based Corrector

The relativistic magnetohydrodynamics (RMHD) set of equations has recently seen increased use in astrophysical computations. Even so, RMHD codes remain fragile. The reconstruction can sometimes yield superluminal velocities in certain parts of the mesh. In this paper we present a reconstruction strategy that overcomes this problem by making a single conservative to primitive transformation per cell followed by higher order WENO reconstruction on a carefully chosen set of primitives that guarantee subluminal reconstruction of the flow variables. For temporal evolution via a predictor step we also present second, third and fourth order accurate ADER methods that keep the velocity subluminal during the predictor step. The RMHD system also requires the magnetic field to be evolved in a divergence-free fashion. In the treatment of classical numerical MHD the analogous issue has seen much recent progress with the advent of multidimensional Riemann solvers. By developing multidimensional Riemann solvers for RMHD, we show that similar advances extend to RMHD. As a result, the face-centered magnetic fields can be evolved much more accurately using the edge-centered electric fields in the corrector step. Those edge-centered electric fields come from a multidimensional Riemann solver for RMHD which we present in this paper. In this paper we also develop several new test problems for RMHD. We show that RMHD vortices can be designed that propagate on the computational mesh as self-preserving structures. These RMHD vortex test problems provide a means to do truly multidimensional accuracy testing for RMHD codes. Several other stringent test problems are presented. We show the importance of resolution in certain test problems. Our tests include a demonstration that RMHD vortices are stable when they interact with shocks.

astro-ph.HE↗

arXiv2015

ADER-WENO Finite Volume Schemes with Space-Time Adaptive Mesh Refinement

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.

math.NA↗

arXiv2015

On the Linear Stability of Magnetized Jets Without Current Sheets: Non-Relativistic Case

In this paper we consider stability of magnetized jets that carry no net electric current and do not have current sheets (Gourgouliatos et al. 2012). The non-relativistic MHD equations are linearized around the background velocity and the magnetic field structure of the jet. The resulting linear equations are solved numerically inside the jet. We find that introduction of current-sheet-free magnetic field significantly improves jet stability relative to unmagnetized jets or magnetized jets with current sheets at their surface. This particularly applies to the fundamental pinch and kink modes - they become completely suppressed in a wide range of long wavelengths that are known to become most pernicious to jet stability when the evolution enters the non-linear regime. The reflection modes, both for the pinch and kink instability, also become progressively more stable with increased magnetization.

astro-ph.HE↗

arXiv2013

Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers

In this paper we use the genuinely multidimensional HLL Riemann solvers recently developed by Balsara et al. to construct a new class of computationally efficient high order Lagrangian ADER-WENO one-step ALE finite volume schemes on unstructured triangular meshes. A nonlinear WENO reconstruction operator allows the algorithm to achieve high order of accuracy in space, while high order of accuracy in time is obtained by the use of an ADER time-stepping technique based on a local space-time Galerkin predictor. The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the grid, considering the entire Voronoi neighborhood of each node and allows for larger time steps than conventional one-dimensional Riemann solvers. The results produced by the multidimensional Riemann solver are then used twice in our one-step ALE algorithm: first, as a node solver that assigns a unique velocity vector to each vertex, in order to preserve the continuity of the computational mesh; second, as a building block for genuinely multidimensional numerical flux evaluation that allows the scheme to run with larger time steps compared to conventional finite volume schemes that use classical one-dimensional Riemann solvers in normal direction. A rezoning step may be necessary in order to overcome element overlapping or crossing-over. We apply the method presented in this article to two systems of hyperbolic conservation laws, namely the Euler equations of compressible gas dynamics and the equations of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to fourth order of accuracy in space and time have been carried out. Several numerical test problems have been solved to validate the new approach.

math.NA↗

arXiv2013

Observational Diagnostics for Two-Fluid Turbulence in Molecular Clouds As Suggested by Simulations

We present high resolution simulations of two-fluid (ion-neutral) MHD turbulence with resolutions as large as 512^3. The simulations are supersonic and mildly sub-Alfvenic, in keeping with the conditions present in molecular clouds. Such turbulence is thought to influence star formation processes in molecular clouds because typical cores form on length scales that are comparable to the dissipation scales of this turbulence in the ions. The simulations are motivated by the fact that recent studies of isophotologue lines in molecular clouds have found significant differences in the linewidth-size relationship for neutral and ion species. The goals of this paper are to explain those observations using simulations and analytic theory, present a new set of density-based diagnostics by drawing on similar diagnostics that have been obtained by studying single-fluid turbulence, and show that our two-fluid simulations play a vital role in reconciling alternative models of star formation. The velocity-dependent diagnostics display a complementarity with the density-dependent diagnostics. We find that the linewidth-size relationships show a prominent difference between ions and neutrals when the line of sight is orthogonal to the mean field. This is because the MHD waves in the ions differ from hydrodynamic waves in the neutrals. We also find that the density probability distribution functions (PDFs) show prominent differences between the ions and neutrals when the line of sight is parallel to the mean field. This is because the velocity fluctuations in the ions tend to produce column density fluctuations along field lines. When the magnetic field makes an angle to the line of sight, both observable differences should be visible. These diagnostics should be easy for observers to test. This analysis assumes optically thin lines and a mean magnetic field that is uniform in direction within a cloud.

astro-ph.GA↗