arXiv subjects

Arpit Babbar

Publications and source records attributed to Arpit Babbar.

15 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.

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arXiv2026

Compact Runge-Kutta flux reconstruction methods with entropy and/or kinetic energy preserving fluxes

Compact Runge-Kutta (cRK) methods are a class of high order methods for solving hyperbolic conservation laws characterized by their compact stencil including only immediate neighboring finite elements. A Compact Runge-Kutta flux reconstruction (cRKFR) method for solver hyperbolic conservation laws was introduced in [Babbar, A., Chen, Q., Journal of Scientific Computing, 2025] which uses a time average flux formulation to perform evolution using a single numerical flux computation at each step, making it a single stage method. Entropy or kinetic energy preserving numerical fluxes are often used for construction of high order entropy stable or kinetic energy preserving methods for hyperbolic conservation laws, and are known to enhance the robustness of numerical methods for under-resolved simulations. In this work, we show how these fluxes can be incorporated into the cRKFR framework for general hyperbolic equations that consist of fluxes and non-conservative products. We test the effectiveness of this new class of methods through numerical experiments for the compressible Euler equations, magnetohydronamics (MHD) equations and multi-ion MHD equations. It is observed that the application of entropy or kinetic energy preserving fluxes enhances the robustness of the cRKFR methods.

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arXiv2026

Jin-Xin relaxation as a shock-capturing method for high-order DG/FR schemes

Jin-Xin relaxation is a method for approximating non-linear hyperbolic conservation laws by a linear system of hyperbolic equations with an $\varepsilon$ dependent stiff source term. The system formally relaxes to the original conservation law as $\varepsilon \to 0$. An asymptotic analysis of the Jin-Xin relaxation system shows that it can be seen as a convection-diffusion equation with a diffusion coefficient that depends on the relaxation parameter $\varepsilon$. This work makes use of this property to use the Jin-Xin relaxation system as a shock-capturing method for high-order discontinuous Galerkin (DG) or flux reconstruction (FR) schemes. The idea is to use a smoothness indicator to choose the $\varepsilon$ value in each cell, so that we can use larger $\varepsilon$ values in non-smooth regions to add extra numerical dissipation. We show how this can be done by using a single stage method by using the compact Runge-Kutta FR method that handles the stiff source term by using IMplicit-EXplicit Runge-Kutta (IMEX-RK) schemes. Numerical results involving Burgers' equation and the compressible Euler equations are shown to demonstrate the effectiveness of the proposed method.

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arXiv2025

Compact Runge-Kutta flux reconstruction methods for non-conservative hyperbolic equations

Compact Runge-Kutta (cRK) Flux Reconstruction (FR) methods are a variant of RKFR methods for hyperbolic conservation laws with a compact stencil including only immediate neighboring finite elements. We extend cRKFR methods to handle hyperbolic equations with stiff source terms and non-conservative products. To handle stiff source terms, we use IMplicit EXplicit (IMEX) time integration schemes such that the implicitness is local to each solution point, and thus does not increase inter-element communication. Although non-conservative products do not correspond to a physical flux, we formulate the scheme using numerical fluxes at element interfaces. We use similar numerical fluxes for a lower order finite volume scheme on subcells of each element, which is then blended with the high order cRKFR scheme to obtain a robust scheme for problems with non-smooth solutions. Combined with a flux limiter at the element interfaces, the subcell based blending scheme preserves the physical admissibility of the solution, e.g., positivity of density and pressure for compressible Euler equations. The procedure thus leads to an admissibility preserving IMEX cRKFR scheme for hyperbolic equations with stiff source terms and non-conservative products. The capability of the scheme to handle stiff terms is shown through numerical tests involving Burgers' equations, reactive Euler's equations, and the ten moment problem. The non-conservative treatment is tested using variable advection equations, shear shallow water equations, the GLM-MHD, and the multi-ion MHD equations.

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arXiv2025

Automatic differentiation for performing the Cauchy-Kovalevskaya procedure in Lax-Wendroff type discretizations

Lax-Wendroff methods combined with discontinuous Galerkin/flux reconstruction spatial discretization provide a high-order, single-stage, quadrature-free method for solving hyperbolic conservation laws. In this work, we introduce automatic differentiation (AD) for performing the Cauchy-Kowalewski procedure used in the element-local time average flux computation step (the predictor step) of Lax-Wendroff methods. The application of AD is similar for methods of any order and does not need positivity corrections during the predictor step. This contrasts with the approximate Lax-Wendroff procedure, which requires different finite difference formulas for different orders of the method and positivity corrections in the predictor step for fluxes that can only be computed on admissible states. The method is Jacobian-free and problem-independent, allowing direct application to any physical flux function. Numerical experiments demonstrate the order and positivity preservation of the method. Additionally, performance comparisons indicate that the wall-clock time of automatic differentiation is always on par with the approximate Lax-Wendroff method.

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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.

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arXiv2025

Compact Runge-Kutta Flux Reconstruction for Hyperbolic Conservation Laws with admissibility preservation

Compact Runge-Kutta (cRK) Discontinuous Galerkin (DG) methods, recently introduced in [Q. Chen, Z. Sun, and Y. Xing, SIAM J. Sci. Comput. SIAM J. Sci. Comput., 46: A1327-A1351, 2024], are a variant of RKDG methods for solving hyperbolic conservation laws and are characterized by their compact stencil including only immediate neighboring finite elements. This article proposes a cRK Flux Reconstruction (FR) method by interpreting cRK as a procedure to approximate time-averaged fluxes, which requires computing only a single numerical flux for each time step and further reduces data communication. The numerical flux is carefully constructed to maintain the same Courant-Friedrichs-Lewy (CFL) numbers as cRKDG methods and achieve optimal accuracy uniformly across all polynomial degrees, even for problems with sonic points. A subcell-based blending limiter is then applied for problems with nonsmooth solutions, which uses Gauss-Legendre solution points and performs MUSCL-Hancock reconstruction on subcells to mitigate the additional dissipation errors. Additionally, to achieve a fully admissibility-preserving cRKFR scheme, a flux limiter is applied to the time-averaged numerical flux to ensure admissibility preservation in the means, combined with a positivity-preserving scaling limiter. The method is further extended to handle source terms by incorporating their contributions as additional time averages. Numerical experiments including Euler equations and the ten-moment problem are provided to validate the claims regarding the method's accuracy, robustness, and admissibility preservation.

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arXiv2024

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.

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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.

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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.

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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.

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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.

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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.

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arXiv2023

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.

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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.

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