Contribution to the study of a numerical implicit method for solving the Euler equations

A class of numerical implicit schemes, which was introduced by A. Lerat, to solve hyperbolic systems of conservation laws is studied. These schemes are applied to the calculation of flows around wing profiles using the Euler equations. A linear stability analysis of these schemes taking the boundary conditions into account is then presented; it is based on the theory of Gustafsson, Kreiss and Sundstrom. The theoretical results are confirmed by numerical experiments and show the importance of the choice of the boundary conditions on the stability of the global approximation. Finally, it is shown that the rate of convergence of the implicit method to reach a steady solution is directly connected with the behavior of the amplification factor of the scheme when the time step is large. The numerical experiments show the efficiency of the scheme in the one dimensional case. Another way to extend the scheme to the case of two space variables is also presented.