Preconditioned Methods for Solving the Incompressible and Low Speed Compressible Equations

Acceleration methods are presented for solving the steady state incompressible equations. These systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state. We also consider the compressible equations in conservation form with slow flow. Two arbitrary functions α and β are introduced in the general preconditioning. An analysis of this system is presented and an optimal value for β is determined given a constant α. It is further sown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed. Several generalizations to the compressible equations are presented which extend previous results.