A preconditioning mass matrix to accelerate the convergence to the steady Euler solutions using explicit schemes

Published in Int. J. Numer. Meht. Engng. Vol. 34 (2), pp. 519-541, 1992
doi: 10.1002/nme.1620340210

Abstract

When explicit time marching algorithms are used to reach the steady state of problems governed by the Euler equations, the rate of convergence is strongly impaired both in the zones with low Mach number and in the zones with transonic flow, e.g. Mach $\leq \alpha$ and | Mach $-1$| $\leq \alpha$, with $\alpha \leq 0\cdot 2$. The rate of convergence becomes slower as α diminishes.

We show in this paper, with analytical and numerical results, how the use of a preconditioning mass matrix accelerates the convergence in the aforementioned ranges of Mach numbers.

The preconditioning mass matrix (PMM) we advocate in this paper can be applied to any FEM/FVM that uses an explicit time marching scheme to find the steady state. The method's rate of convergence to the steady state is studied, and results for the one‐ and two‐dimiensional cases are presented.

In Sections 1‐3, using the one‐dimensional Euler equations, we first explain why there exists a slow rate of convergence when the plain lumping of mass is used. Then the convergence rate to steady solutions is analysed from its two constituents, that is, convergence by absorption at the boundaries and by damping in the domain. Next we give the natural solution to this problem, and with several examples we show the effectiveness of the proposed mass matrix when compared with the plain scheme.

In Sections 4‐8 we give the multidimensional version of the preconditioning mass matrix. We make a stability analysis and compare the group velocities and damping with and without the new mass matrix. To finish, we show the velocity of convergence for a common test problem.