This work investigates the Discontinuous Galerkin (DG) method in this regime. Rather than tracking the cell averages as in Godunov methods, DG schemes track a finite number of moments of the system in each cell and connect them through finite volume-like flux functions. They are especially attractive because they allow for high orders of accuracy while remaining compact, which allows for easy parallelization. Under the right conditions, numerical experiments with DG methods appear to have eliminated the numerical dissipation problems suffered by Godunov schemes by driving the variables to be continuous at leading order at the cell interfaces, which relaxes the spatial resolution requirements.

In this talk, I will present an asymptotic analysis of the DG method for the incompressible limit of the one-dimensional isentropic Euler equations. The analysis confirms that the method eliminates the need for high spatial resolution, and shows that the limiting method is consistent with the incompressible equations.