In Search Of An Optimal Local Navier-Stokes Preconditioner

Local preconditioning for the Navier-Stokes equations may be called optimal if it equalizes all propagation and dissipation time-scales, for all combinations of Mach number and Reynolds number. Previously designed preconditioners are ineffective for certain combinations of low Reynolds number and low Mach number; in addition some of these create a growing mode, making the PDE-system unstable. (Users may regain stability through an implicit discretization.) In this paper we first review the forms and properties of all previously published N-S preconditioners on the basis of the 1-D N-S equations, then derive an optimal preconditioning matrix for these equations. We find again that it creates an unstable mode; a sensitivity analysis shows that optimal preconditioning and stability are mutually exclusive. Two possible remedies are suggested and briefly investigated: (1) to redefine the complex condition number in a way more appropriate for explicit discretizations; (2) to reformulate the N-S equations as a larger first-order system of hyperbolic-relaxation equations and base the preconditioner on this system. The latter approach appears most promising.