Numerical Methods for All Mach Number flows for Gas Dynamics

Abstract

An original numerical method to solve the all-Mach number flow for the Euler equations of gas dynamics on staggered grid is presented in this thesis. The system is discretized to second order in space on staggered grid, in a fashion similar to the Nessyahu-Tadmor central scheme for 1D model and Jang-Tadmor central scheme for 2D model, thus simplifying the flux computation. This approach turns out to be extremely simple, since it requires no equation splitting. We consider the isentropic case and the general case. For simplicity we assume a gamma-law gas in both cases.

Both approaches are based on IMEX strategy, in which some term is treated explicitly, while other terms are treated implicitly, thus avoiding the classical CFL restriction due to acoustic waves.

- In Isentropic Euler Case: The core if the implicit term contains a non-linear elliptic equation for the pressure, which has to be treated by a fully implicit technique. Because of the non-linearity, it is necessary to adopt an iterative method to compute the pressure. In our numerical experiments Newton's method worked with few iterations.

- General Euler Case: In this case the implicit term is treated in a semi-implicit fashion, thus avoiding the use of Newton's iterations.

In both cases the schemes are implemented to second order accuracy in time. Suitably well-prepared initial conditions are considered, which depend on the Mach number. In one space dimension we obtain the same profiles found in the literature for the isentropic case and for the general Euler system for all Mach numbers.

In particular, the schemes have been shown to be AP, in the sense that they become a consistent discretizzation of the incompressible Euler equation as the Mach number approaches zero. Numerical evidence of such AP property is provided on a two dimensional test case.

The last chapter deals with the piston problem in Lagrangian coordinates treated by a semi-implicit scheme. The implicit treatment of the boundary conditions is originally developed in the thesis. It is shown that for very low Mach number the scheme is able to recover the adiabatic solution with very large CFL numbers. For moderate Mach numbers, or in presence of an initial acoustic wave, loss of accuracy is observerd if the CFL is too large. This drawback can be cured by using a suitable time step control, which will be subject of future investigation. Current work is also related on the development of higher order accurate schemes for 1D and 2D problems.