▪ Abstract An overview of preconditioning for the steady-state compressible inviscid fluid dynamic equations is presented. Extensions to the Navier-Stokes equations are also considered. These preconditioners are necessary for many algorithms in order to have the correct behavior at low speeds and to converge to the solution of the incompressible equations as the Mach number goes to zero. In addition, the preconditioning accelerates the convergence to a steady state for problems in which a significant portion of the flow is low speed. This low speed preconditioner can be combined with Jacobi and line preconditioners to damp high frequencies at all speeds. This is necessary for use with multigrid methods. Such combined methods are also better at accelerating problems with high aspect ratios. Details of the implementation are presented including several different variants for the preconditioning matrix.

Preconditioning techniques in computational fluid dynamics

https://www-sop.inria.fr/members/Herve.Guillard/Mywebpage/MESPUBLIS/A2004/A2004Guillard_Murrone_CF.pdf

On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes

A Unified Method for Computing Incompressible and Compressible Flows in Boundary-Fitted Coordinates

Assessment of Preconditioning Methods for Multidimensional Aerodynamics

Improved convergence rates for a unified grid framework are achieved by combining several convergence acceleration strategies, which include local implicit time-stepping, low-speed preconditioning, and relaxation methods. It is demonstrated that good convergence can be achieved on various grid types and topologies, all speed regimes, and for both inviscid and viscous flows.

Convergence acceleration for unified-grid formulation using preconditioned implicit relaxation

The use of preconditioning methods to accelerate the convergence to a steady state for both the incompressible and compressible fluid dynamic equations are considered. The relation between them for both the continuous problem and the finite difference approximation is also considered. The analysis relies on the inviscid equations. The preconditioning consists of a matrix multiplying the time derivatives. Hence, the steady state of the preconditioned system is the same as the steady state of the original system. For finite difference methods the preconditioning can change and improve the steady state solutions. An application to flow around an airfoil is presented.

B. Van Leer

Papers

Preconditioning and the Limit to the Incompressible Flow Equations