A Conservative Finite Element Method for the Incompressible Euler Equations with Variable Density

TL;DR

This paper introduces a novel finite element method for the incompressible Euler equations with variable density that exactly conserves key physical quantities and demonstrates stability and convergence through numerical tests.

Contribution

The paper develops a conservative finite element discretization using Raviart-Thomas or BDM elements with a unique weak formulation and second-order time-stepping for variable density Euler equations.

Findings

Exact conservation of mass, energy, and density squared achieved.

Method demonstrates stability and convergence in numerical experiments.

Second-order accuracy confirmed through numerical examples.

Abstract

We construct a finite element discretization and time-stepping scheme for the incompressible Euler equations with variable density that exactly preserves total mass, total squared density, total energy, and pointwise incompressibility. The method uses Raviart-Thomas or Brezzi-Douglas-Marini finite elements to approximate the velocity and discontinuous polynomials to approximate the density and pressure. To achieve exact preservation of the aforementioned conserved quantities, we exploit a seldom-used weak formulation of the momentum equation and a second-order time-stepping scheme that is similar, but not identical, to the midpoint rule. We also describe and prove stability of an upwinded version of the method. We present numerical examples that demonstrate the order of convergence of the method.