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

Abstract

A fully discrete Active Flux method is proposed for the two-dimensional compressible Euler equations. The method builds on the evolution-operator formulation of multidimensional Active Flux 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. This has recently been extended to the Euler equations by Barsukow, who developed an efficient approach by adding acoustic increments at the Eulerian target point. The proposed method reconstructs the acoustic increment as a cellwise Q2 field, using vertex, edg--midpoint, and cell-center acoustic increments, 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, the compact stencil, and the conservative one-stage average update. Numerical experiments probe several facets of the numerical method. A mixed Fourier wave packet with the linearized Euler equations isolates the split error and shows third-order point accuracy for the transported update. Isentropic vortex convection confirms 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, low entropy dissipation, and absence of the coarse-grid secondary vortices seen in displayed DG/CG comparisons. A compressible under-resolved Kelvin-Helmholtz test demonstrates robust evolution with consistent entropy dissipation.