Local preconditioning yields desirable effects such as convergence acceleration and preservation of accuracy in the incompressible limit, but these come at the expense of robust* Fellow AIAA. t ~ e m b e r AIAA. SMember AIAA. §Member AIAA. ness. Current research focuses on making the various preconditioned Euler discretizations that have been developed more reliable near flow singularities such as sonic and stagnation points. In this report we discuss the loss of stability in computing stagnating flow with the symmetric Van Leer-Lee-Roe preconditioning, and do an exhaustive search for modifications of the matrix (for low Mach numbers) by which the instability can be avoided. Some Copyright 01993 American Institute of Aeronwtiu and Astronautics, Inc. All rights m r v e d . numerical support regarding the effectiveness and v vanishes (but not its derivatives). The of these modifications is presented. Euler equations then read M I 0