High-Order Isogeometric Methods for Compressible Flows. II. Compressible Euler Equations

TL;DR

This paper extends high-resolution isogeometric analysis to the compressible Euler equations, employing a stabilized group finite element formulation with flux correction to ensure positivity and accuracy in gas dynamics simulations.

Contribution

It introduces a stabilized high-order isogeometric method for compressible Euler equations, combining algebraic flux correction with Roe-averaged flux Jacobians for improved accuracy and stability.

Findings

Effective stabilization with artificial viscosities proportional to spectral radius.

Positivity-preserving flux limiting applied to high-order B-Spline discretizations.

Enhanced accuracy in simulating compressible gas flows.

Abstract

This work extends the high-resolution isogeometric analysis approach established for scalar transport equations to the equations of gas dynamics. The group finite element formulation is adopted to obtain an efficient assembly procedure for the standard Galerkin approximation, which is stabilized by adding artificial viscosities proportional to the spectral radius of the Roe-averaged flux-Jacobian matrix. Excess stabilization is removed in regions with smooth flow profiles with the aid of algebraic flux correction \cite{KBNII}. The underlying principles are reviewed and it is shown that linearized FCT-type flux limiting \cite{Kuzmin2009} originally derived for nodal low-order finite elements ensures positivity-preservation for high-order B-Spline discretizations.