Some continuous and discontinuous Galerkin methods and structure preservation for incompressible flows

TL;DR

This paper introduces new discontinuous Galerkin methods for incompressible flows that ensure energy stability and conservation of physical quantities, demonstrating their effectiveness through numerical experiments.

Contribution

It develops consistent and inconsistent DG schemes with proven energy stability and conservation properties for incompressible Euler and Navier-Stokes equations.

Findings

Proved semi- and fully discrete energy stability.

Achieved conservation of total energy, momentum, and angular momentum.

Numerical experiments show competitive performance and physical accuracy.

Abstract

In this paper, we present consistent and inconsistent discontinuous Galerkin methods for incompressible Euler and Navier-Stokes equations with the kinematic pressure, Bernoulli function and EMAC function. Semi- and fully discrete energy stability of the proposed dG methods are proved in a unified fashion. Conservation of total energy, linear and angular momentum is discussed with both central and upwind fluxes. Numerical experiments are presented to demonstrate our findings and compare our schemes with conventional schemes in the literature in both unsteady and steady problems. Numerical results show that global conservation of the physical quantities may not be enough to demonstrate the performance of the schemes, and our schemes are competitive and able to capture essential physical features in several benchmark problems.