December 4, 2025Open Access

On the Relaxation Technique Applied to Linearly Implicit Rosenbrock Schemes for a Fully-Discrete Entropy Conserving/Stable dG Method

Key Points

Robustness is enhanced through relaxation techniques in high-order schemes for solving Euler equations.
Validation involves assessing accuracy using multiple numerical experiments, including the isentropic vortex and Kelvin-Helmholtz instability.
Discontinuous Galerkin methods ensure entropy conservation and stability at both spatial and temporal levels with careful flux design.
The approach may enable better predictive capabilities for unsteady flows in complex fluid dynamics scenarios.

Abstract

In this work, a high-order modal discontinuous Galerkin (dG) method is employed to solve the Euler equations using entropy variables. Entropy conservation and stability are ensured at the spatial semi-discrete level through entropy-conserving/stable numerical fluxes and the over-integration technique. For time integration, linearly implicit Rosenbrock-type Runge–Kutta schemes are used. However, since these schemes are not provably entropy-conserving/stable, their use to predict unsteady flows may lead to solutions that lack the desired entropy properties. To address this issue, a relaxation technique is applied to enforce entropy conservation or stability at the fully discrete level. The accuracy, conservation/stability properties and robustness of the fully-discrete scheme equipped with the relaxation technique are assessed through the following numerical experiments: (1) the isentropic vortex, (2) the Kelvin-Helmholtz instability, (3) the Taylor–Green vortex.

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