Error analysis of an asymptotic-preserving, energy-stable finite volume method for barotropic Euler equations
This work provides theoretical guarantees for a numerical scheme that accurately handles both compressible and incompressible regimes, which is important for computational fluid dynamics simulations.
The authors design an energy-stable and asymptotic-preserving finite volume scheme for the compressible Euler system and establish rigorous error estimates that yield convergence of numerical solutions in both fixed Mach number and low Mach number regimes, with estimates uniform in discretisation parameters and Mach number.
We design an energy-stable and asymptotic-preserving finite volume scheme for the compressible Euler system. Using the relative energy framework, we establish rigorous error estimates that yield convergence of the numerical solutions in two distinct regimes. For a fixed Mach number $\varepsilon>0$, we derive error estimates between the numerical solutions and a strong solution of the compressible Euler system that are uniform with respect to the discretisation parameters, ensuring convergence as the underlying mesh is refined. In the low Mach number regime, we analyse the error between the numerical solutions and a strong solution of the incompressible Euler system and obtain asymptotic error estimates that are uniform in $\varepsilon$ and the discretisation parameters. These results imply convergence of the numerical solutions toward a strong solution of the incompressible Euler system as $\varepsilon$, and the discretisation parameters simultaneously tend to zero. Numerical experiments are presented to validate the theoretical analysis.