Compressible Flow Simulations

 The Discontinuous Galerkin method (DGM) that we outline in the next sections is a variant of the method proposed by Cockburn and Shu et al in a series of papers[68,69,70,67,71]. Lomtev implemented the method for triangular and tetrahedral spectral hp elements in [72,73,74]. We have extended it to the polymorphic family of spectral elements. Using the toolkit of element objects we have developed thus far this was a straightforward procedure because the DGM is a local method. At each stage of a computation, the only extra data that an element needs to access is from its neighbouring elements or domain boundary conditions.

We have applied the DGM to the Euler equations, the compressible Navier-Stokes equations and the Magnetohydrodynamic equations in two and three-dimensions using the array of spectral elements described in chapter 2. In this chapter we will consider the Euler and compressible Navier-Stokes equations, and in the next chapter we outline our progress with the compressible MHD equations.

We only demonstrate explicit methods in this chapter and the next chapter, which will force $\Delta t$ to be small if the problem is dominated by viscous effects. Oden, Babuska and Baumann presented a formulation for an implicit solver using DGM in a series of papers [75,76,77,78,79]. Their method shows promise if the iterative solver can be preconditioned well. Their method does not require the use of auxiliary methods but it does involve a large number of derivatives to be evaluated to compensate for this saving. It will be interesting to see if the implicit method is cheaper than the explicit method when the number of iterations per time step are factored into the total cost.