Local Preconditioning

An Overview

Local preconditioning is a technique to remove stiffness from a system of equations. In the context of the CFD community, typically the set of time-dependent, Reynolds-averaged, ``Navier-Stokes'' equations are solved in an iterative fashion to achieve a steady-state solution. Although only the steady-state solution is desired, the time-dependent equations are employed so that the system retains the desirable property of hyperbolicity (with respect to time) for all Mach numbers. Thus, the system has real eigenvalues and is comprised of wave-like solutions.

However, convergence to steady-state is impaired for low Mach number regions due to the spread in the characteristic wave speeds, e.g., q, q+a, and q-a, where q is the total flow speed and a is the speed of sound. The ratio of the largest to smallest wave speed is termed the condition number, K, and serves as a measure of "stiffness". In the plot of condition number versus Mach number shown at the right, the solid line represents the Euler equations while the dashed line represents the Euler equations preconditioned with the Van Leer-Lee-Roe preconditioner. As shown by the solid line, stiffness occurs in both the subsonic and transonic regimes; and, as Mach increases, the condition number approaches the ideal condition number: unity. The dashed line shows that is it is possible to completely eliminate the subsonic stiffness region, greatly reduce the transonic stiffness region, and, in general, substantially reduce the condition number for the system of equations.

Benefits

• It removes the stiffness of the system of equations caused by the spread in wave speeds, thus improving the convergence rate of any marching scheme.
• It causes a system of equations to behave more like a scalar equation, facilitating the design of concomitant techniques. In addition, some preconditioners decouple the system of equations into purely elliptic and hyperbolic parts.
• It improves accuracy for low Mach number flows.

Detriments

• Without special considerations, time accuracy is lost.

Caveats

• Lack of robustness, especially in the vicinity of stagnation points.
• Proper incorporation of Viscous effects is still unclear.

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