Well-balanced nodal discontinuous Galerkin method for Euler equations with gravity

TL;DR

This paper introduces a well-balanced nodal discontinuous Galerkin scheme for Euler equations with gravity, accurately preserving stationary solutions and improving the resolution of small perturbations.

Contribution

It develops a novel DG scheme that maintains hydrostatic equilibrium solutions exactly using specific source term discretization and GLL nodes.

Findings

Preserves isothermal and polytropic stationary solutions up to machine precision.

Effectively captures small perturbations around stationary solutions.

Demonstrates robustness on various mesh configurations.

Abstract

We present a well-balanced nodal discontinuous Galerkin (DG) scheme for compressible Euler equations with gravity. The DG scheme makes use of discontinuous Lagrange basis functions supported at Gauss-Lobatto-Legendre (GLL) nodes together with GLL quadrature using the same nodes. The well-balanced property is achieved by a specific form of source term discretization that depends on the nature of the hydrostatic solution, together with the GLL nodes for quadrature of the source term. The scheme is able to preserve isothermal and polytropic stationary solutions upto machine precision on any mesh composed of quadrilateral cells and for any gravitational potential. It is applied on several examples to demonstrate its well-balanced property and the improved resolution of small perturbations around the stationary solution.