Exploration of 1-D Preconditioning Methods for the Euler Equations
by
Marco R. Zaccanti
MSU/ERC for Computational Field Simulation
Coauthors: Pasquale Cinnella

Preconditioning of a system of equations at the differential level represents the newest area of research in convergence acceleration of discrete schemes for the fluid dynamics equations. It attempts to remove the intrinsic stiffness of the equations caused by the different time scales present. The understanding of one-dimensional preconditioning for the Euler equations is essential to the subsequent exploration of the more practical 2-D and 3-D cases. Specifically, the analysis and design of 1-D preconditioning methods bring to light certain issues that are less obvious in the multi-dimensional cases, and at the same time allow to clarify the inherent limitations of preconditioning. In this study, after a discussion of the intimate relationship existing between symmetrizability of a preconditioned system and its eigenvector structure (two concepts always treated as separate in all the former work on this subject), we identify the intrinsic problems of preconditioning and clarify the role of positivity. Then, through an exhaustive exploration of a general preconditioner, we show how to find a unique perfect matrix for the supersonic regime, where `perfect' refers to the possibility of satisfying all of the properties or design criteria we established before starting the investigation. For subsonic flow, we first present a new key criterion a preconditioner should satisfy, then show that it is possible to find a one-parameter family of perfect preconditioners, proving the non-uniqueness of the subsonic ideal solution. The one-dimensional analysis also allows us to understand an intrinsic deficiency of every symmetric preconditioner: we prove that symmetry and accuracy preservation of a preconditioner are incompatible with symmetrizability in the incompressible limit. We also clarify the matter of sonic transition, and present two new optimal matrices, which connect the unique perfect supersonic matrix with two simple members of the perfect subsonic family through the sonic point in a continuous manner.

Date received: March 29, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacr-29.