arXiv subjects

Praveen Chandrashekar

Publications and source records attributed to Praveen Chandrashekar.

At least 19 recordsLinked to original sources

arXiv2026

Admissible Lax-Wendroff Flux Reconstruction Method with Automatic Differentiation on Adaptive Curved Meshes for Relativistic Hydrodynamics

The relativistic hydrodynamics (RHD) equations can give rise to solutions which have shocks, contact discontinuities, and other sharp structures, which interact and evolve over time. Capturing these sharp waves effectively requires a mesh with high resolution, making the scheme computationally expensive. In this work, adaptive mesh refinement is used with the high-order Lax-Wendroff flux reconstruction (LWFR) method to solve the system of RHD equations, which is closed with general equations of state. To make the scheme Jacobian-free, the idea of automatic differentiation is incorporated for computing the temporal derivatives in the time average flux approximations. The high-order method is blended with an admissible low-order method at the subcell level to control the Gibbs oscillations and maintain the physical admissibility of the solution. Finally, several test cases involving high Lorentz factors, low densities, low pressures, strong shock waves, and other discontinuities are used to demonstrate the robustness, accuracy, and effectiveness of the proposed method. These simulations are performed with AMR using various linear and curved meshes to show the scheme's efficiency and ability to handle complex geometries.

math.NA↗

arXiv2026

A low-dissipation central scheme for ideal MHD

Central schemes for conservation laws are Riemann solver free methods which are simple and easy to implement. In recent work for Euler equations [Kurganov & Xin, J. Sci. Comput., 96:56, 2023] their accuracy has been enhanced in terms of better resolution of contact waves. In this paper, we extend this low dissipation central upwind method to the ideal MHD system in one- and two-dimensions. In the two-dimensional case, we separate the variables into two groups: hydrodynamic and magnetic, which are stored at cell centers and faces, respectively. For the the hydrodynamic variables, we apply the low dissipation central upwind scheme while for the magnetic variables, a constrained transport method is used which maintains the divergence-free property of the magnetic field. The time integration is performed with third order strong stability preserving Runge-Kutta scheme. To validate the proposed scheme, we apply this method to several challenging test cases. The results show that the LDCU correction term plays a useful role at the contact discontinuity and enhances the resolution of waves. We also observe experimental second-order accuracy for smooth solutions and the divergence-free condition is maintained to machine precision.

math.NA↗

arXiv2026

Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domains

The very challenging task of learning solution operators of PDEs on arbitrary domains accurately and efficiently is of vital importance to engineering and industrial simulations. Despite the existence of many operator learning algorithms to approximate such PDEs, we find that accurate models are not necessarily computationally efficient and vice versa. We address this issue by proposing a geometry aware operator transformer (GAOT) for learning PDEs on arbitrary domains. GAOT combines novel multiscale attentional graph neural operator encoders and decoders, together with geometry embeddings and (vision) transformer processors to accurately map information about the domain and the inputs into a robust approximation of the PDE solution. Multiple innovations in the implementation of GAOT also ensure computational efficiency and scalability. We demonstrate this significant gain in both accuracy and efficiency of GAOT over several baselines on a large number of learning tasks from a diverse set of PDEs, including achieving state of the art performance on three large scale three-dimensional industrial CFD datasets.

cs.LG↗

arXiv2025

Constraints Preserving Lax-Wendroff Flux Reconstruction for Relativistic Hydrodynamics with General Equations of State

In the realm of relativistic astrophysics, the ideal equation of state with a constant adiabatic index provides a poor approximation due to its inconsistency with relativistic kinetic theory. However, it is a common practice to use it for relativistic fluid flow equations due to its simplicity. Here we develop a high-order Lax-Wendroff flux reconstruction method on Cartesian grids for solving relativistic hydrodynamics equations with several general equations of state available in the literature. We also study the conversion from conservative to primitive variables, which depends on the equation of state in use, and provide an alternative method of conversion when the existing approach does not succeed. For the admissibility of the solution, we blend the high-order method with a low-order method on sub-cells and prove its physical admissible property in the case of all the equations of state used here. Lastly, we validate the scheme by several test cases having strong discontinuities, large Lorentz factor, and low density or pressure in one and two dimensions.

math.NA↗

arXiv2025

A generalized Active Flux method of arbitrarily high order in two dimensions

The Active Flux method can be seen as an extended finite volume method. The degrees of freedom of this method are cell averages, as in finite volume methods, and in addition shared point values at the cell interfaces, giving rise to a globally continuous reconstruction. Its classical version was introduced as a one-stage fully discrete, third-order method. Recently, a semi-discrete version of the Active Flux method was presented with various extensions to arbitrarily high order in one space dimension. In this paper we extend the semi-discrete Active Flux method on two-dimensional Cartesian grids to arbitrarily high order, by including moments as additional degrees of freedom (hybrid finite element--finite volume method). The stability of this method is studied for linear advection. For a fully discrete version, using an explicit Runge-Kutta method, a CFL restriction is derived. We end by presenting numerical examples for hyperbolic conservation laws.

math.NA↗

arXiv2025

Active flux for ideal magnetohydrodynamics: A positivity-preserving scheme with the Godunov-Powell source term

The Active Flux (AF) is a compact, high-order finite volume scheme that allows more flexibility by introducing additional point value degrees of freedom at cell interfaces. This paper proposes a positivity-preserving (PP) AF scheme for solving the ideal magnetohydrodynamics, where the Godunov-Powell source term is employed to deal with the divergence-free constraint. For the evolution of the cell average, apart from the standard conservative finite volume method for the flux derivative, the nonconservative source term is built on the quadratic reconstruction in each cell, which maintains the compact stencil in the AF scheme. For the point value update, the local Lax-Friedrichs (LLF) flux vector splitting is adopted for the flux derivative, originally proposed in [Duan, Barsukow, and Klingenberg, SIAM Journal on Scientific Computing, 47(2), A811--A837, 2025], and a central difference is used to discretize the divergence in the source term. A parametrized flux limiter and a scaling limiter are presented to preserve the density and pressure positivity by blending the AF scheme with the first-order PP LLF scheme with the source term. To suppress oscillations, a new shock sensor considering the divergence error is proposed, which is used to compute the blending coefficients for the cell average. Several numerical tests are conducted to verify the third-order accuracy, PP property, and shock-capturing ability of the scheme. The key role of the Godunov-Powell source term and its suitable discretization in controlling divergence error is also validated.

math.NA↗

arXiv2025

Second order divergence constraint preserving schemes for two-fluid relativistic plasma flow equations

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.

math.NA↗

arXiv2025

Bound Preserving Lax-Wendroff Flux Reconstruction Method for Special Relativistic Hydrodynamics

Lax-Wendroff flux reconstruction (LWFR) schemes have high order of accuracy in both space and time despite having a single internal time step. Here, we design a Jacobian-free LWFR type scheme to solve the special relativistic hydrodynamics equations on Cartesian grids. We then blend the scheme with a first-order finite volume scheme to control the oscillations near discontinuities. We also use a scaling limiter to preserve the physical admissibility of the solution after ensuring the scheme is admissible in means. A particular focus is given to designing a discontinuity indicator model to detect the local non-smoothness in the solution of the highly non-linear relativistic hydrodynamics equations. Finally, we present numerical results for a wide range of test cases to show the robustness and efficiency of the proposed scheme.

math.NA↗

arXiv2024

Multi-Derivative Runge-Kutta Flux Reconstruction for hyperbolic conservation laws

We extend the fourth order, two stage Multi-Derivative Runge Kutta (MDRK) scheme to the Flux Reconstruction (FR) framework by writing both stages in terms of a time averaged flux and then using the approximate Lax-Wendroff procedure to compute the time averaged flux. Numerical flux is carefully constructed to enhance Fourier CFL stability and accuracy. A subcell based blending limiter is developed for the MDRK scheme which ensures that the limited scheme is provably admissibility preserving. Along with being admissibility preserving, the blending scheme is constructed to minimize dissipation errors by using Gauss-Legendre solution points and performing MUSCL-Hancock reconstruction on subcells. The accuracy enhancement of the blending scheme is numerically verified on compressible Euler's equations, with test cases involving shocks and small-scale structures.

math.NA↗

arXiv2024

Lax-Wendroff Flux Reconstruction on adaptive curvilinear meshes with error based time stepping for hyperbolic conservation laws

Lax-Wendroff Flux Reconstruction (LWFR) is a single-stage, high order, quadrature free method for solving hyperbolic conservation laws. This work extends the LWFR scheme to solve conservation laws on curvilinear meshes with adaptive mesh refinement (AMR). The scheme uses a subcell based blending limiter to perform shock capturing and exploits the same subcell structure to obtain admissibility preservation on curvilinear meshes. It is proven that the proposed extension of LWFR scheme to curvilinear grids preserves constant solution (free stream preservation) under the standard metric identities. For curvilinear meshes, linear Fourier stability analysis cannot be used to obtain an optimal CFL number. Thus, an embedded-error based time step computation method is proposed for LWFR method which reduces fine-tuning process required to select a stable CFL number using the wave speed based time step computation. The developments are tested on compressible Euler's equations, validating the blending limiter, admissibility preservation, AMR algorithm, curvilinear meshes and error based time stepping.

math.NA↗

arXiv2024

Generalized framework for admissibility preserving Lax-Wendroff Flux Reconstruction for hyperbolic conservation laws with source terms

Lax-Wendroff Flux Reconstruction (LWFR) is a single-stage, high order, quadrature free method for solving hyperbolic conservation laws. We perform a cell average decomposition of the LWFR scheme that is similar to the one used in the admissibility preserving framework of Zhang and Shu (2010). By performing a flux limiting of the time averaged numerical flux, the decomposition is used to obtain an admissibility preserving LWFR scheme. The admissibility preservation framework is further extended to a newly proposed extension of LWFR scheme for conservation laws with source terms. This is the first extension of the high order LW scheme that can handle source terms. The admissibility and accuracy are verified by numerical experiments on the Ten Moment equations of Livermore et al.

math.NA↗

arXiv2024

Equivalence of ADER and Lax-Wendroff in DG / FR framework for linear problems

ADER (Arbitrary high order by DERivatives) and Lax-Wendroff (LW) schemes are two high order single stage methods for solving time dependent partial differential equations. ADER is based on solving a locally implicit equation to obtain a space-time predictor solution while LW is based on an explicit Taylor's expansion in time. We cast the corrector step of ADER Discontinuous Galerkin (DG) scheme into an equivalent quadrature free Flux Reconstruction (FR) framework and then show that the obtained ADER-FR scheme is equivalent to the LWFR scheme with D2 dissipation numerical flux for linear problems. This also implies that the two schemes have the same Fourier stability limit for time step size. The equivalence is verified by numerical experiments.

math.NA↗

arXiv2024

Lax-Wendroff Flux Reconstruction for advection-diffusion equations with error-based time stepping

This work introduces an extension of the high order, single stage Lax-Wendroff Flux Reconstruction (LWFR) of Babbar et al., JCP (2022) to solve second order time-dependent partial differential equations in conservative form on curvilinear meshes. The method uses BR1 scheme to reduce the system to first order so that the earlier LWFR scheme can be applied. The work makes use of the embedded error-based time stepping introduced in Babbar, Chandrashekar (2024) which becomes particularly relevant in the absence of CFL stability limit for parabolic equations. The scheme is verified to show optimal order convergence and validated with transonic flow over airfoil and unsteady flow over cylinder.

math.NA↗

arXiv2024

Admissibility preserving subcell limiter for Lax-Wendroff flux reconstruction

Lax-Wendroff Flux Reconstruction (LWFR) is a single-stage, high order, quadrature free method for solving hyperbolic conservation laws. We develop a subcell based limiter by blending LWFR with a lower order scheme, either first order finite volume or MUSCL-Hancock scheme. While the blending with a lower order scheme helps to control oscillations, it may not guarantee admissibility of discrete solution, e.g., positivity property of quantities like density and pressure. By exploiting the subcell structure and admissibility of lower order schemes, we devise a strategy to ensure that the blended scheme is admissibility preserving for the mean values and then use a scaling limiter to obtain admissibility of the polynomial solution. For MUSCL-Hancock scheme on non-cell-centered subcells, we develop a slope limiter, time step restrictions and suitable blending of higher order fluxes, that ensures admissibility of lower order updates and hence that of the cell averages. By using the MUSCL-Hancock scheme on subcells and Gauss-Legendre points in flux reconstruction, we improve small-scale resolution compared to the subcell-based RKDG blending scheme with first order finite volume method and Gauss-Legendre-Lobatto points. We demonstrate the performance of our scheme on compressible Euler's equations, showcasing its ability to handle shocks and preserve small-scale structures.

math.NA↗

arXiv2023

Entropy stable schemes for the shear shallow water model Equations

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.

math.NA↗

arXiv2023

High-order finite-difference entropy stable schemes for two-fluid relativistic plasma flow equations

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. The third component is Maxwell's equations, which are linear systems. The coupling of the ion and electron flows, and electromagnetic fields is via source terms only. Furthermore, we also show that the source terms do not affect the entropy evolution. To design semi-discrete entropy stable schemes, we extend the RHD entropy stable schemes in Bhoriya et al. 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). These equations are solved using the Newton's Method, which results in an efficient method. The proposed schemes are then tested using various test problems to demonstrate their stability, accuracy and efficiency.

math.NA↗

arXiv2023

Multidimensional Generalized Riemann Problem Solver for Maxwell's Equations

Approximate multidimensional Riemann solvers are essential building blocks in designing globally constraint-preserving finite volume time domain (FVTD) and discontinuous Galerkin time domain (DGTD) schemes for computational electrodynamics (CED). In those schemes, we can achieve high-order temporal accuracy with the help of Runge-Kutta or ADER time-stepping. This paper presents the design of a multidimensional approximate Generalized Riemann Problem (GRP) solver for the first time. The multidimensional Riemann solver accepts as its inputs the four states surrounding an edge on a structured mesh, and its output consists of a resolved state and its associated fluxes. In contrast, the multidimensional GRP solver accepts as its inputs the four states and their gradients in all directions; its output consists of the resolved state and its corresponding fluxes and the gradients of the resolved state. The gradients can then be used to extend the solution in time. As a result, we achieve second-order temporal accuracy in a single step. In this work, the formulation is optimized for linear hyperbolic systems with stiff, linear source terms because such a formulation will find maximal use in CED. Our formulation produces an overall constraint-preserving time-stepping strategy based on the GRP that is provably L-stable in the presence of stiff source terms. We present several stringent test problems, showing that the multidimensional GRP solver for CED meets its design accuracy and performs stably with optimal time steps. The test problems include cases with high conductivity, showing that the beneficial L-stability is indeed realized in practical applications.

math.NA↗

arXiv2022

Lax-Wendroff flux reconstruction method for hyperbolic conservation laws

The Lax-Wendroff method is a single step method for evolving time dependent solutions governed by partial differential equations, in contrast to Runge- Kutta methods that need multiple stages per time step. We develop a flux reconstruction version of the method in combination with a Jacobian-free Lax- Wendroff procedure that is applicable to general hyperbolic conservation laws. The method is of collocation type, is quadrature free and can be cast in terms of matrix and vector operations. Special attention is paid to the construction of numerical flux, including for non-linear problems, resulting in higher CFL numbers than existing methods, which is shown through Fourier analysis and yielding uniform performance at all orders. Numerical results up to fifth order of accuracy for linear and non-linear problems are given to demonstrate the performance and accuracy of the method.

math.NA↗