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.
The reproducibility repository is available on GitHub and Zenodo.
As usual, you can find the preprint on arXiv.
]]>We study a hyperbolic approximation (“hyperbolization”) of the Cahn-Hilliard (CH) equation, originally proposed by Dhaouadi, Dumbser, and Gavrilyuk (2025, DOI: 10.1098/rspa.2024.0606) and study its convergence towards the CH model in a relaxation limit both via formal asymptotic expansions and, for a slightly modified approximation, via the relative energy framework. Moreover, we develop energy-stable semidiscretizations of the CH equation and of this hyperbolization using upwind summation-by-parts operators in space. Subsequently, we combine them with (additive) implicit-explicit (IMEX) Runge-Kutta methods based on a convex-concave splitting. We show that the resulting method is asymptotic preserving, i.e., it converges in the limit of the relaxation parameter to a stable discretization of the original CH equation. The choice of the necessary parameters is guided by the a priori error estimate based on the relative energy framework.
The reproducibility repository is available on Zenodo.
]]>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.
The reproducibility repository is available on GitHub and Zenodo.
As usual, you can find the preprint on arXiv.
]]>Saurav Samantaray joined our group in February 2025. Before, he was a visiting faculty member at the Department of Mathematics, IIT Madras.
]]>Arpit Babbar joined our group in August 2024. Before, he was a PhD student of Praveen Chandrashekar at the Tata Institute of Fundamental Research—Centre for Applicable Mathematics.
]]>We present a fully discrete stability analysis of the domain-of-dependence stabilization for hyperbolic problems. The method aims to address issues caused by small cut cells by redistributing mass around the neighborhood of a small cut cell at a semi-discrete level. Our analysis is conducted for the linear advection model problem in one spatial dimension. We demonstrate that fully discrete stability can be achieved under a time step restriction that does not depend on the arbitrarily small cells, using an operator norm estimate. Additionally, this analysis offers a detailed understanding of the stability mechanism and highlights some challenges associated with higher-order polynomials. We also propose a way to mitigate these issues to derive a feasible CFL-like condition. The analytical findings, as well as the proposed solution are verified numerically in one- and two-dimensional simulations.
The reproducibility repository is available on GitHub and Zenodo.
As usual, you can find the preprint on arXiv.
]]>The ultra–relativistic Euler equations describe gases in the relativistic case when the thermal energy dominates. These equations for an ideal gas are given in terms of the pressure, the spatial part of the dimensionless four-velocity, and the particle density. Kunik et al. (2024, https://doi.org/10.1016/j.jcp.2024.113330) proposed genuine multi–dimensional benchmark problems for the ultra–relativistic Euler equations. In particular, they compared full two-dimensional discontinuous Galerkin simulations for radially symmetric problems with solutions computed using a specific one-dimensional scheme. Of particular interest in the solutions are the formation of shock waves and a pressure blow-up. In the present work we derive an entropy-stable flux for the ultra–relativistic Euler equations. Therefore, we derive the main field (or entropy variables) and the corresponding potentials. We then present the entropy-stable flux and conclude with simulation results for different test cases both in 2D and in 3D.
The reproducibility repository is available on GitHub and Zenodo.
As usual, you can find the preprint on arXiv.
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