arXiv subjects

Amelie Porfetye

Publications and source records attributed to Amelie Porfetye.

4 recordsLinked to original sources

arXiv2026

Fully Discrete Active Flux Methods for the Euler Equations: Exploring Different Reconstructions and New Multi-Dimensional Evolution Operators

We present new fully discrete Cartesian grid Active Flux methods for the two-dimensional Euler equations of gas dynamics. A key component of these methods is the evolution operator used to evolve point-value degrees of freedom. Therefore, we derive several new, third-order accurate, multidimensional evolution operators for point-value updates of the linearized Euler equations using the method of bicharacteristics. We also investigate how the reconstruction and choice of approximate evolution operator affect the accuracy and stability of the resulting methods.

math.NA↗

arXiv2026

A fully discrete Active Flux method for the Euler equations comparing different truly multi-dimensional evolution operators

This article builds upon our recently published fully discrete Cartesian grid Active Flux method for the Euler equations by introducing a new evolution operator for the linearized Euler equations. The evolution of the point value degrees of freedom located at grid cell boundaries and used to compute numerical fluxes is a key component of fully discrete Active Flux methods. Our methods are based on local linearizations of the Euler equations and the use of truly multi-dimensional evolution operators for these linearized problems. Here, we propose to solve the linearized Euler equations in moving coordinates, reducing them to acoustics, which can be solved exactly. We compare this approach with our previous one, in which we used approximate evolution operators derived using the method of bicharacteristics. The moving-grid approach is closely related to recent methods proposed by Barsukow as well as Duraisamy. Our choice of local linearization, as well as correction of linearization errors, enables us to construct third-order accurate methods for smooth solutions of the nonlinear Euler equations. Numerical results illustrate the performance of these methods for different flow regimes, ranging from discontinuous solution structures with shock waves to vortex structures in the low Mach number regime.

math.NA↗

arXiv2025

New Fully Discrete Active Flux Methods with Truly Multi-Dimensional Evolution Operators and WENO Reconstruction

We propose new fully discrete third-order accurate Active Flux and WENO methods based on truly multidimensional evolution operators for the two-dimensional acoustic equations. Building on the method of bicharacteristics, several approximate evolution operators are derived that yield an improved stability of the resulting schemes. A linear stability analysis is applied to determine the maximal CFL number. The schemes are tested extensively on both continuous and discontinuous problems, confirming their robustness and accurate approximation even on coarse grids.

math.NA↗

arXiv2025

A Fully Discrete Truly Multidimensional Active Flux Method For The Two-Dimensional Euler Equations

The Active Flux method is a finite volume method for hyperbolic conservation laws that uses both cell averages and point values as degrees of freedom. Several versions of such methods are currently under development. We focus on third order accurate, fully discrete Active Flux methods with compact stencil in space and time. These methods require exact or approximate evolution operators for the update of the point value degrees of freedom which are provided by the method of bicharacteristics. Here we propose new limiting strategies that guarantee positivity of pressure and density and furthermore discuss the implementation of reflecting boundary conditions. Numerical results show that the method leads to accurate approximates on coarse grids.

math.NA↗