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.
]]>We consider the compressible Euler system with anelastic scaling, modeling isentropic flows under the influence of gravity. In the zero-Mach-number limit, the solution of the compressible Euler system converges to a variable density anelastic incompressible limit system. In this work, we present the design and analysis of a class of higher-order linearly implicit IMEX Runge-Kutta schemes that are asymptotic preserving, i.e., they respect the transitory nature of the governing equations in the limit. The presence of gravitational potential warrants the incorporation of the well-balancing property. The scheme is developed as a novel combination of a penalization of a linear steady state, a finite-volume balance-preserving reconstruction, and a source term discretization preserving steady states. The penalization plays a crucial role in obtaining a linearly implicit scheme, and well-balanced flux-source discretization ensures accuracy in very low Mach number regimes. Some results of numerical case studies are presented to corroborate the theoretical assertions.
The reproducibility repository is available on GitHub and Zenodo.
]]>We combine Patankar-type methods with suitable relaxation procedures that are capable of ensuring correct dissipation or conservation of functionals such as entropy or energy while producing unconditionally positive and conservative approximations. To that end, we adapt the relaxation algorithm to enforce positivity by using either ideas from the dense output framework when a linear invariant must be preserved, or simply a geometric mean if the only constraint is positivity preservation. The latter merely requires the solution of a scalar nonlinear equation while former results in a coupled linear-nonlinear system of equations. We present sufficient conditions for the solvability of the respective equations. Several applications in the context of ordinary and partial differential equations are presented, and the theoretical findings are validated numerically.
The reproducibility repository is available on GitHub and Zenodo.
]]>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.
The reproducibility repository is available on GitHub and Zenodo.
]]>We introduce the concept of volume term adaptivity for high-order discontinuous Galerkin (DG) schemes solving time-dependent partial differential equations. Termed v-adaptivity, we present a novel general approach that exchanges the discretization of the volume contribution of the DG scheme at every Runge-Kutta stage based on suitable indicators. Depending on whether robustness or efficiency is the main concern, different adaptation strategies can be chosen. Precisely, the weak form volume term discretization is used instead of the entropy-conserving flux-differencing volume integral whenever the former produces more entropy than the latter, resulting in an entropy-stable scheme. Conversely, if increasing the efficiency is the main objective, the weak form volume integral may be employed as long as it does not increase entropy beyond a certain threshold or cause instabilities. Thus, depending on the choice of the indicator, the v-adaptive DG scheme improves robustness, efficiency and approximation quality compared to schemes with a uniform volume term discretization. We thoroughly verify the accuracy, linear stability, and entropy-admissibility of the v-adaptive DG scheme before applying it to various compressible flow problems in two and three dimensions.
The reproducibility repository is available on GitHub and Zenodo.
]]>Many entropy-conservative and entropy-stable (summarized as entropy-preserving) methods for hyperbolic conservation laws rely on Tadmor’s theory for two-point entropy-preserving numerical fluxes and its higher-order extension via flux differencing using summation-by-parts (SBP) operators, e.g., in discontinuous Galerkin spectral element methods (DGSEMs). The underlying two-point formulations have been extended to nonconservative systems using fluctuations by Castro et al. (2013, doi:10.1137/110845379) with follow-up generalizations to SBP methods. We propose specific forms of entropy-preserving fluctuations for nonconservative hyperbolic systems that are simple to interpret and allow an algorithmic construction of entropy-preserving methods. We analyze necessary and sufficient conditions, and obtain a full characterization of entropy-preserving three-point methods within the finite volume framework. This formulation is extended to SBP methods in multiple space dimensions on Cartesian and curvilinear meshes. Additional properties such as well-balancedness extend naturally from the underlying finite volume method to the SBP framework. We use the algorithmic construction enabled by the chosen formulation to derive several new entropy-preserving schemes for nonconservative hyperbolic systems, e.g., the compressible Euler equations of an ideal gas using the internal energy equation and a dispersive shallow-water model. Numerical experiments show the robustness and accuracy of the proposed schemes.
The reproducibility repository is available on GitHub and Zenodo.
]]>Jin-Xin relaxation is a method for approximating non-linear hyperbolic conservation laws by a linear system of hyperbolic equations with an ε dependent stiff source term. The system formally relaxes to the original conservation law as ε→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 ε. 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 ε value in each cell, so that we can use larger ε 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.
The reproducibility repository is available on GitHub and Zenodo.
]]>