Fully Discrete Active Flux Method based on Transported Acoustic Increments for the Compressible Euler Equations

TL;DR

This paper introduces a fully discrete Active Flux method for 2D compressible Euler equations that improves accuracy and preserves acoustic evolution, validated through various numerical experiments.

Contribution

It develops a novel acoustic increment reconstruction and evolution approach that enhances accuracy and stability over existing methods.

Findings

Third-order point accuracy demonstrated with Fourier wave packets.

Confirmed third-order convergence and reduced error constants in vortex convection.

Preserves radial symmetry and exhibits low entropy dissipation in acoustic pulse evolution.

Abstract

A fully discrete Active Flux method is proposed for the 2D compressible Euler equations. The method builds on the evolution-operator formulation proposed by Roe in which conservative cell averages are updated by unsplit flux quadrature while primitive point values are evolved by acoustic and advective subsolvers. The proposed method reconstructs the acoustic increment as a cellwise Q2 field and evaluates this field at the convective foot of the target point. For constant frozen coefficients, the resulting point update reduces to the transported composition, eliminating the additive split defect and yielding the exact unsplit frozen evolution when the acoustic and advective generators commute. The resulting method preserves the exact locally linearized acoustic evolution operator of Barsukow (2025), the compact stencil, and the conservative one-stage average update. Numerical experiments probe several facets of the numerical method. A mixed Fourier wave packet isolates the split error and shows third-order point accuracy for the transported update, compared with second-order behavior for the additive update. Isentropic vortex convection confirms third-order convergence for the full nonlinear scheme, reduced error constants, and an enlarged empirical CFL range. Nonlinear Gaussian acoustic pulse evolution demonstrates preservation of radial symmetry and near-third-order decay of the symmetry error. Low-Mach shear layer tests show coherent vorticity evolution, ultra-low entropy dissipation, and absence of the coarse-grid secondary vortices seen in displayed DG/CG comparisons. Finally, a compressible under-resolved Kelvin-Helmholtz test demonstrates robust no-limiter evolution to late time with consistent entropy dissipation. Fourier diagnostics of the vertical-edge point operator support the observed improvements in acoustic phase and amplification behavior.