Browsing by Author "Bhoriya, Deepak"

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An alternative finite difference weno-like scheme with physical constraint preservation for divergence-preserving hyperbolic systems
(2025-06) Bhoriya, Deepak
Alternative finite difference Weighted Essentially Non-Oscillatory (AFD-WENO) schemes allow us to very efficiently update hyperbolic systems even in complex geometries. Recent innovations in AFD-WENO methods allow us to treat hyperbolic system with non-conservative products almost as efficiently as conservation laws. However, some PDE systems,like computational electrodynamics (CED) and magnetohydrodynamics (MHD) and relativistic magnetohydrodynamics (RMHD), have involution constraints that require divergence-free or divergence-preserving evolution of vector fields. In such situations, a Yee-style collocation of variables proves indispensable; and that collocation is retained in this work. In previous works, only higher order finite volume discretization of such involution constrained systems was possible. In this work, we show that substantially more efficient AFD-WENO methods have been extended to encompass divergence-preserving hyperbolic PDEs.
An anisotropic plasma model of the heliospheric interface
(IOP, 2025-05) Bhoriya, Deepak
We present a pioneering model of the interaction between the solar wind and the surrounding interstellar medium that includes the possibility of different pressures in directions parallel and perpendicular to the magnetic field. The outer heliosheath region is characterized by a low rate of turbulent scattering that would permit development of pressure anisotropy. The effect is best seen on the interstellar side of the heliopause, where a narrow region develops with an excessive perpendicular pressure resembling a plasma depletion layer typical of planetary magnetospheres. The magnitude of this effect for typical heliospheric conditions is relatively small owing to proton–proton collisions. We show, however, that if the circumstellar medium is warm and tenuous, a much broader anisotropic boundary layer can exist, with a dominant perpendicular pressure in the southern hemisphere and a dominant parallel pressure in the north.
Chew, goldberger & low equations: eigensystem analysis and applications to one-dimensional test problem
(Elsevier, 2025-06) Bhoriya, Deepak
Chew, Goldberger & Low (CGL) equations describe one of the simplest plasma flow models that allow anisotropic pressure, i.e., pressure is modeled using a symmetric tensor described by two scalar pressure components, one parallel to the magnetic field, another perpendicular to the magnetic field. The system of equations is a non-conservative hyperbolic system. In this work, we analyze the eigensystem of the CGL equations. We present the eigenvalues and the complete set of right eigenvectors. We also prove the linear degeneracy of some of the characteristic fields. Using the eigensystem for CGL equations, we propose HLL and HLLI Riemann solvers for the CGL system. Furthermore, we present the AFD-WENO schemes up to the seventh order in one dimension and demonstrate the performance of the schemes on several one-dimensional test cases.
Efficient alternative finite difference WENO schemes for hyperbolic conservation laws
(Springer, 2024-05) Bhoriya, Deepak
Higher order finite difference Weighted Essentially Non-Oscillatory (FD-WENO) schemes for conservation laws are extremely popular because, for multidimensional problems, they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes. Such schemes come in two formulations. The very popular classical FD-WENO method (Shu and Osher J Comput Phys 83: 32–78, 1989) relies on two reconstruction steps applied to two split fluxes. However, the method cannot accommodate different types of Riemann solvers and cannot preserve free stream boundary conditions on curvilinear meshes. This limits its utility. The alternative FD-WENO (AFD-WENO) method can overcome these deficiencies, however, much less work has been done on this method. The reasons are three-fold. First, it is difficult for the casual reader to understand the intricate logic that requires higher order derivatives of the fluxes to be evaluated at zone boundaries. The analytical methods for deriving the update equation for AFD-WENO schemes are somewhat recondite. To overcome that difficulty, we provide an easily accessible script that is based on a computer algebra system in Appendix A of this paper. Second, the method relies on interpolation rather than reconstruction, and WENO interpolation formulae have not been documented in the literature as thoroughly as WENO reconstruction formulae. In this paper, we explicitly provide all necessary WENO interpolation formulae that are needed for implementing the AFD-WENO up to the ninth order. The third reason is that the AFD-WENO requires higher order derivatives of the fluxes to be available at zone boundaries. Since those derivatives are usually obtained by finite differencing the zone-centered fluxes, they become susceptible to a Gibbs phenomenon when the solution is non-smooth. The inclusion of those fluxes is also crucially important for preserving the order property when the solution is smooth. This has limited the utility of the AFD-WENO in the past even though the method per se has many desirable features. Some efforts to mitigate the effect of finite differencing of the fluxes have been tried, but so far they have been done on a case by case basis for the PDE being considered. In this paper we find a general-purpose strategy that is based on a different type of the WENO interpolation. This new WENO interpolation takes the first derivatives of the fluxes at zone centers as its inputs and returns the requisite non-linearly hybridized higher order derivatives of flux-like terms at the zone boundaries as its output. With these three advances, we find that the AFD-WENO becomes a robust and general-purpose solution strategy for large classes of conservation laws. It allows any Riemann solver to be used. The AFD-WENO has a computational complexity that is entirely comparable to the classical FD-WENO, because it relies on two interpolation steps which cost the same as the two reconstruction steps in the classical FD-WENO. We apply the method to several stringent test problems drawn from Euler flow, relativistic hydrodynamics (RHD), and ten-moment equations. The method meets its design accuracy for smooth flow and can handle stringent problems in one and multiple dimensions.
Efficient alternative finite difference WENO schemes for hyperbolic systems with non-conservative products
(Springer, 2024-05) Bhoriya, Deepak
Higher order finite difference Weighted Essentially Non-oscillatory (WENO) schemes for conservation laws represent a technology that has been reasonably consolidated. They are extremely popular because, when applied to multidimensional problems, they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes. They come in two flavors. There is the classical finite difference WENO (FD-WENO) method (Shu and Osher in J. Comput. Phys. 83: 32–78, 1989). However, in recent years there is also an alternative finite difference WENO (AFD-WENO) method which has recently been formalized into a very useful general-purpose algorithm for conservation laws (Balsara et al. in Efficient alternative finite difference WENO schemes for hyperbolic conservation laws, submitted to CAMC, 2023). However, the FD-WENO algorithm has only very recently been formulated for hyperbolic systems with non-conservative products (Balsara et al. in Efficient finite difference WENO scheme for hyperbolic systems with non-conservative products, to appear CAMC, 2023). In this paper, we show that there are substantial advantages in obtaining an AFD-WENO algorithm for hyperbolic systems with non-conservative products. Such an algorithm is documented in this paper. We present an AFD-WENO formulation in a fluctuation form that is carefully engineered to retrieve the flux form when that is warranted and nevertheless extends to non-conservative products. The method is flexible because it allows any Riemann solver to be used. The formulation we arrive at is such that when non-conservative products are absent it reverts exactly to the formulation in the second citation above which is in the exact flux conservation form. The ability to transition to a precise conservation form when non-conservative products are absent ensures, via the Lax-Wendroff theorem, that shock locations will be exactly captured by the method. We present two formulations of AFD-WENO that can be used with hyperbolic systems with non-conservative products and stiff source terms with slightly differing computational complexities. The speeds of our new AFD-WENO schemes are compared to the speed of the classical FD-WENO algorithm from the first of the above-cited papers. At all orders, AFD-WENO outperforms FD-WENO. We also show a very desirable result that higher order variants of AFD-WENO schemes do not cost that much more than their lower order variants. This is because the larger number of floating point operations associated with larger stencils is almost very efficiently amortized by the CPU when the AFD-WENO code is designed to be cache friendly. This should have great, and very beneficial, implications for the role of our AFD-WENO schemes in the Peta- and Exascale computing. We apply the method to several stringent test problems drawn from the Baer-Nunziato system, two-layer shallow water equations, and the multicomponent debris flow. The method meets its design accuracy for the smooth flow and can handle stringent problems in one and multiple dimensions. Because of the pointwise nature of its update, AFD-WENO for hyperbolic systems with non-conservative products is also shown to be a very efficient performer on problems with stiff source terms.
Efficient finite difference weno scheme for hyperbolic systems with non-conservative products
(Springer, 2023-07) Bhoriya, Deepak
Higher order finite difference weighted essentially non-oscillatory (WENO) schemes have been constructed for conservation laws. For multidimensional problems, they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy. This makes them quite attractive for several science and engineering applications. But, to the best of our knowledge, such schemes have not been extended to non-linear hyperbolic systems with non-conservative products. In this paper, we perform such an extension which improves the domain of the applicability of such schemes. The extension is carried out by writing the scheme in fluctuation form. We use the HLLI Riemann solver of Dumbser and Balsara (J. Comput. Phys. 304: 275–319, 2016) as a building block for carrying out this extension. Because of the use of an HLL building block, the resulting scheme has a proper supersonic limit. The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme, thus expanding its domain of the applicability. Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions, making it very easy for users to transition over to the present formulation. For conservation laws, the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO, with two major advantages: (i) It can capture jumps in stationary linearly degenerate wave families exactly. (ii) It only requires the reconstruction to be applied once. Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows. Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow, multiphase debris flow and two-layer shallow water equations are also shown to document the robustness of the method. For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results. Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.
Entropy stable discontinuous Galerkin schemes for the special relativistic hydrodynamics equations
(Elsevier, 2022-04) Bhoriya, Deepak
This article presents entropy stable discontinuous Galerkin numerical schemes for equations of special relativistic hydrodynamics with the ideal equation of state. The numerical schemes use the summation by parts (SBP) property of the Gauss-Lobatto quadrature rules. To achieve entropy stability for the scheme, we use two-point entropy conservative numerical flux inside the cells and a suitable entropy stable numerical flux at the cell interfaces. The resulting semi-discrete scheme is then shown to be entropy stable. Time discretization is performed using SSP Runge-Kutta methods. Several numerical test cases are presented to validate the accuracy and stability of the proposed schemes.
Entropy stable discontinuous Galerkin schemes for two-fluid relativistic plasma flow equations
(Springer, 2023-11) Bhoriya, Deepak
This article proposes entropy stable discontinuous Galerkin schemes (DG) for two-fluid relativistic plasma flow equations. These equations couple the flow of relativistic fluids via electromagnetic quantities evolved using Maxwell’s equations. The proposed schemes are based on the Gauss–Lobatto quadrature rule, which has the summation by parts property. We exploit the structure of the equations having the flux with three independent parts coupled via nonlinear source terms. We design entropy stable DG schemes for each flux part, coupled with the fact that the source terms do not affect entropy, resulting in an entropy stable scheme for the complete system. The proposed schemes are then tested on various test problems in one and two dimensions to demonstrate their accuracy and stability.
Entropy stable finite difference schemes for chew, goldberger and low anisotropic plasma flow equations
(Springer, 2025-01) Bhoriya, Deepak
In this article, we consider the Chew, Goldberger and Low (CGL) plasma flow equations, which is a set of nonlinear, non-conservative hyperbolic PDEs modeling anisotropic plasma flows. These equations incorporate the double adiabatic approximation for the evolution of the pressure, making them very valuable for plasma physics, space physics, and astrophysical applications. We first present the entropy analysis for the weak solutions. We then propose entropy-stable finite-difference schemes for the CGL equations. The key idea is to reformulate the CGL equations by rewriting some of the conservative terms in the non-conservation form. The conservative part of the reformulated equations is very similar to the magnetohydrodynamics (MHD) equations which is then symmetrized using Godunov’s symmetrization process for the MHD equations. The resulting equations are in the form where the conservative part combined with non-conservative Godunov’s terms is compatible with the entropy equation and the rest of the non-conservative terms do not contribute to the entropy equations. The final set of reformulated equations is then discretized by designing entropy conservative numerical flux and entropy diffusion operator based on the entropy scaled eigenvectors of the conservative part. We then prove the semi-discrete entropy stability of the schemes for the reformulated CGL equations. The schemes are then tested using several test problems derived from the corresponding MHD test cases.
Entropy stable schemes for the shear shallow water model equations
(Springer, 2023-11) Bhoriya, Deepak
The shear shallow water model is an extension of the classical shallow water model to include the effects of vertical shear. It is a system of six non-linear hyperbolic PDE with non-conservative products. We develop a high-order entropy stable finite difference scheme for this model in one dimension and extend it to two dimensions on rectangular grids. The key idea is to rewrite the system so that non-conservative terms do not contribute to the entropy evolution. Then, we first develop an entropy conservative scheme for the conservative part, which is then extended to the complete system using the fact that the non-conservative terms do not contribute to the entropy production. The entropy dissipative scheme, which leads to an entropy inequality, is then obtained by carefully adding dissipative flux terms. The proposed schemes are then tested on several one and two-dimensional problems to demonstrate their stability and accuracy.
Entropy-stable schemes for relativistic hydrodynamics equations
(Springer, 2020-01) Bhoriya, Deepak
In this article, we propose high-order finite difference schemes for the equations of relativistic hydrodynamics, which are entropy stable. The crucial components of these schemes are a computationally efficient entropy conservative flux and suitable high-order entropy dissipative operators. We first design a higher-order entropy conservative flux. For the construction of appropriate entropy dissipative operators, we derive entropy scaled right eigenvectors. This is then used with ENO-based sign-preserving reconstruction of scaled entropy variables, which results in higher-order entropy-stable schemes. Several numerical results are presented up to fourth order to demonstrate entropy stability and performance of these schemes.
Going beyond the MHD approximation: physics-based numerical solution of the CGL equations
(IOP, 2024-07) Bhoriya, Deepak
We present a new numerical model for solving the Chew–Goldberger–Low system of equations describing a bi-Maxwellian plasma in a magnetic field. Heliospheric and geospace environments are often observed to be in an anisotropic state with distinctly different parallel and perpendicular pressure components. The Chew–Goldberger–Low (CGL) system represents the simplest leading order correction to the common isotropic MHD model that still allows the incorporation of the latter’s most desirable features. However, the CGL system presents several numerical challenges: the system is not in conservation form, the source terms are stiff, and unlike MHD, it is prone to a loss of hyperbolicity if the parallel and perpendicular pressures become too different. The usual cure is to bring the parallel and perpendicular pressures closer to one another, but that has usually been done in an ad hoc manner. We present a physics-informed method of pressure relaxation based on the idea of pitch-angle scattering that keeps the numerical system hyperbolic and naturally leads to zero anisotropy in the limit of very large plasma beta. Numerical codes based on the CGL equations can, therefore, be made to function robustly for any magnetic field strength, including the limit where the magnetic field approaches zero. The capabilities of our new algorithm are demonstrated using several stringent test problems that provide a comparison of the CGL equations in the weakly and strongly collisional limits. This includes a test problem that mimics the interaction of a shock with a magnetospheric environment in 2D.
High-order finite-difference entropy stable schemes for two-fluid relativistic plasma flow equations
(Elsevier, 2023-09) Bhoriya, Deepak
In this article, we propose high-order finite-difference entropy stable schemes for the two-fluid relativistic plasma flow equations. This is achieved by exploiting the structure of the equations, which consists of three independent flux components. The first two components describe the ion and electron flows, which are modeled using the relativistic hydrodynamics equation and the third component is Maxwell's equations. The coupling of the ion and electron flows and electromagnetic fields is via source terms only, but the source terms do not affect the entropy evolution. To design semi-discrete entropy stable schemes, we extend the entropy stable schemes for relativistic hydrodynamics in [1] to three dimensions. This is then coupled with entropy stable discretization of the Maxwell's equations. Finally, we use SSP-RK schemes to discretize in time. We also propose ARK-IMEX schemes to treat the stiff source terms; the resulting nonlinear set of algebraic equations is local (at each discretization point) and hence can be solved cheaply using the Newton's Method. The proposed schemes are then tested using various test problems to demonstrate their stability, accuracy and efficiency.
Multidimensional HLLI generalized riemann problem solver for conservation laws – the two-dimensional case for structured meshes
(Elsevier, 2025-10) Bhoriya, Deepak
The Riemann problem, and the associated generalized Riemann problem (GRP), are increasingly seen as important building blocks for modern higher order Godunov-type schemes. While most solutions of the GRP are specific to a particular hyperbolic law, a general-purpose GRP that can be applied to any hyperbolic conservation law has emerged in the form of the one-dimensional HLLI-GRP. Approximate multidimensional Riemann solvers have also been designed by the first author and his colleagues. However, a multidimensional GRP that is applicable to any hyperbolic conservation law has never been designed until now to the best of our knowledge. It this paper, we accomplish such a task. The study of the multidimensional Riemann problem entails the study of the strongly-interacting state. Starting with the multidimensional HLL-based Riemann solver, we present all the steps for endowing spatial gradients to the strongly-interacting state. This is accomplished through application of Rankine-Hugoniot shock jump conditions to the higher order terms in a Taylor series expansion of the strongly-interacting state. A linearized formulation is also used to obtain the spatial gradients to the strongly-interacting state. With the spatial gradients in hand, it is possible to specify the multidimensional HLL flux as well its time-derivative. This results in a multidimensional HLL-GRP solver. We then utilize intermediate waves to reduce the dissipation of the multidimensional HLL-GRP. This gives us an HLLI-GRP solver which significantly reduces dissipation and is complete in multiple dimensions.
A new first-order formulation of the Einstein equations: comparison among different high order numerical schemes
(IOP, 2025) Bhoriya, Deepak
The solution of the Einstein-Euler equations by the majority of numerical codes is still based on traditional finite difference schemes for the Einstein sector, while it relies on conservative schemes for the matter part. This is due to the fact that the celebrated BSSNOK formulation of the Einstein equations has second order in space derivatives. We present a first-order (in space derivatives) formulation of the BSSNOK Einstein equations that is strongly hyperbolic and it allows for the implementation of a monolithic numerical scheme for its solution. The new formulation is compatible with quite different numerical schemes, such as Central WENO finite differences or Discontinuous Galerkin schemes, that we have analyzed in terms of accuracy and computational performances.
Physical constraint preserving higher order finite volume schemes for divergence-free astrophysical MHD and RMHD
(2025-06) Bhoriya, Deepak
Higher order finite volume schemes for magnetohydrodynamics (MHD) and relativistic magnetohydrodynamics (RMHD) are very valuable because they allow us to carry out astrophysical simulations with very high accuracy. However, astrophysical problems sometimes have unusually large Mach numbers, exceptionally high Lorentz factors and very strong magnetic fields. All these effects cause higher order codes to become brittle and prone to code crashes. In this paper we document physical constraint preserving (PCP) methods for treating numerical MHD and RMHD. While unnecessary for standard problems, for stringent astrophysical problems these methods show their value. We describe higher order methods that allow divergence-free evolution of the magnetic field. We present a novel two-dimensional Riemann solver. This two-dimensional Riemann solver plays a key role in the design of PCP schemes for MHD and RMHD. We present a very simple PCP formulation and show how it is amalgamated with the evolution of face-centered magnetic fields. The methods presented here are time-explicit and do not add much to the computational cost. We show that the methods meet their design accuracies and work well on problems that would otherwise be considered too extreme for typical higher order Godunov methods of the type used in computational astrophysics.
Pressure anisotropies in the very local interstellar medium
(The SAO Astrophysics Data System, 2024-07) Bhoriya, Deepak
Voyager 2 observations indicate that the plasma of the very local interstellar medium might be hotter than predicted by MHD models. However, based on experience with planetary magnetospheres, a region could exist on the interstellar side of the heliopause, where due to compression of the magnetic field the plasma density is depleted and the temperature is different in the directions parallel and normal to the magnetic field. The pressure anisotropy could also have a measurable effect on magnetic field draping around the heliopause and the resulting asymmetries of the structure of the global heliosphere. In this presentation we compare results from a conventional MHD model with a newly developed model based on the Chew-Goldberger-Low (CGL) approximation. The new model was constructed in such a way as to steer the system evolution away from kinetically unstable regions using information about the turbulence responsible for scattering and isotropization. It represents an important step toward developing a general purpose CGL framework for space and astrophysical applications. CGL results indicate that a plasma depletion layer develops in the outer heliosheath where the perpendicular temperature of the plasma is higher than the parallel temperature, while in the inner heliosheath certain regions can have an excess of parallel temperature.
Second order divergence constraint preserving entropy stable finite difference schemes for ideal two-fluid plasma flow equations
(Springer, 2024-10) Bhoriya, Deepak
Two-fluid plasma flow equations describe the flow of ions and electrons with different densities, velocities, and pressures. We consider the ideal plasma flow i.e. we ignore viscous, resistive and collision effects. The resulting system of equations has flux consisting of three independent components, one for ions, one for electrons, and a linear Maxwell’s equation flux for the electromagnetic fields. The coupling of these components is via source terms. In this article, we present conservative second-order finite difference schemes that ensure the consistent evolution of the divergence constraints on the electric and magnetic fields. The key idea is to design a numerical solver for Maxwell’s equations using the multidimensional Riemann solver at the vertices, ensuring discrete divergence constraints; for the fluid parts, we use an entropy-stable discretization. The proposed schemes are co-located, second-order accurate, entropy stable, and ensure divergence-free evolution of the magnetic field. We use explicit and IMplicit–EXplicit (IMEX) schemes for time discretizations. To demonstrate the accuracy, stability, and divergence constraint-preserving ability of the proposed schemes, we present several test cases in one and two dimensions. We also compare the numerical results with those obtained from schemes with no divergence cleaning and those employing perfectly hyperbolic Maxwell (PHM) equations-based divergence cleaning methods for Maxwell’s equations.
Second order divergence constraint preserving schemes for two-fluid relativistic plasma flow equations
(2025-03) Bhoriya, Deepak
Two-fluid relativistic plasma flow equations combine the equations of relativistic hydrodynamics with Maxwell's equations for electromagnetic fields, which involve divergence constraints for the magnetic and electric fields. When developing numerical schemes for the model, the divergence constraints are ignored, or Maxwell's equations are reformulated as Perfectly Hyperbolic Maxwell's (PHM) equations by introducing additional equations for correction potentials. In the latter case, the divergence constraints are preserved only as the limiting case. In this article, we present second-order numerical schemes that preserve the divergence constraint for electric and magnetic fields at the discrete level. The schemes are based on using a multidimensional Riemann solver at the vertices of the cells to define the numerical fluxes on the edges. The second-order accuracy is obtained by reconstructing the electromagnetic fields at the corners using a MinMod limiter. The discretization of Maxwell's equations can be combined with any consistent and stable discretization of the fluid parts. In particular, we consider entropy-stable schemes for the fluid part. The resulting schemes are second-order accurate, entropy stable, and preserve the divergence constraints of the electromagnetic fields. We use explicit and IMEX-based time discretizations. We then test these schemes using several one- and two-dimensional test cases. We also compare the divergence constraint errors of the proposed schemes with schemes having no divergence constraints treatment and schemes based on the PHM-based divergence cleaning.
Techniques, tricks, and algorithms for efficient GPU-based processing of higher order hyperbolic PDEs
(Springer, 2023-03) Bhoriya, Deepak
GPU computing is expected to play an integral part in all modern Exascale supercomputers. It is also expected that higher order Godunov schemes will make up about a significant fraction of the application mix on such supercomputers. It is, therefore, very important to prepare the community of users of higher order schemes for hyperbolic PDEs for this emerging opportunity. Not every algorithm that is used in the space-time update of the solution of hyperbolic PDEs will take well to GPUs. However, we identify a small core of algorithms that take exceptionally well to GPU computing. Based on an analysis of available options, we have been able to identify weighted essentially non-oscillatory (WENO) algorithms for spatial reconstruction along with arbitrary derivative (ADER) algorithms for time extension followed by a corrector step as the winning three-part algorithmic combination. Even when a winning subset of algorithms has been identified, it is not clear that they will port seamlessly to GPUs. The low data throughput between CPU and GPU, as well as the very small cache sizes on modern GPUs, implies that we have to think through all aspects of the task of porting an application to GPUs. For that reason, this paper identifies the techniques and tricks needed for making a successful port of this very useful class of higher order algorithms to GPUs.