18.5.2 Preconditioning

Time-derivative preconditioning modifies the time-derivative term in Equation  18.5-1 by pre-multiplying it with a preconditioning matrix. This has the effect of re-scaling the acoustic speed (eigenvalue) of the system of equations being solved in order to alleviate the numerical stiffness encountered in low Mach numbers and incompressible flow.

Derivation of the preconditioning matrix begins by transforming the dependent variable in Equation  18.5-1 from conserved quantities ${\mbox{\boldmath$W$}}$ to primitive variables ${\mbox{\boldmath$Q$}}$ using the chain-rule as follows:

where ${\mbox{\boldmath$Q$}}$ is the vector $\left\{p, u, v, w, T \right\}^T$ and the Jacobian ${\partial {\mbox{\boldmath$W$}}}/{\partial {\mbox{\boldmath$Q$}}}$ is given by

where

and $\delta$ = 1 for an ideal gas and $\delta$ = 0 for an incompressible fluid.

The choice of primitive variables ${\mbox{\boldmath$Q$}}$ as dependent variables is desirable for several reasons. First, it is a natural choice when solving incompressible flows. Second, when we use second-order accuracy we need to reconstruct ${\mbox{\boldmath$Q$}}$ rather than ${\mbox{\boldmath$W$}}$ in order to obtain more accurate velocity and temperature gradients in viscous fluxes, and pressure gradients in inviscid fluxes. And finally, the choice of pressure as a dependent variable allows the propagation of acoustic waves in the system to be singled out [ 356].

We precondition the system by replacing the Jacobian matrix ${\partial {\mbox{\boldmath$W$}}}/{\partial {\mbox{\boldmath$Q$}}}$ (Equation  18.5-5) with the preconditioning matrix ${\Gamma}$ so that the preconditioned system in conservation form becomes

where

The parameter $\Theta$ is given by

The reference velocity $U_r$ appearing in Equation  18.5-8 is chosen locally such that the eigenvalues of the system remain well conditioned with respect to the convective and diffusive time scales [ 372].

The resultant eigenvalues of the preconditioned system (Equation  18.5-6) are given by

where

$$\begin{eqnarray*} u &=& {\mbox{\boldmath$v$}}\cdot \hat{n} \\ u' &=& u \, \left... ...)/2 \\ \beta &=& \left(\rho_p + \frac{\rho_T}{\rho C_p} \right) \end{eqnarray*}$$

For an ideal gas, $\beta = (\gamma R T)^{-1} = 1/c^2$. Thus, when $U_r = c$ (at sonic speeds and above), $\alpha = 0$ and the eigenvalues of the preconditioned system take their traditional form, $u \pm c$. At low speed, however, as $U_r \rightarrow 0$, $\alpha \rightarrow 1/2$ and all eigenvalues become of the same order as $u$. For constant-density flows, $\beta = 0$ and $\alpha = 1/2$ regardless of the values of $U_r$. As long as the reference velocity is of the same order as the local velocity, all eigenvalues remain of the order $u$. Thus, the eigenvalues of the preconditioned system remain well conditioned at all speeds.

Note that the non-preconditioned Navier-Stokes equations are recovered exactly from Equation  18.5-6 by setting $1/U_r^2$ to $\rho_p$, the derivative of density with respect to pressure. In this case ${\Gamma}$ reduces exactly to the Jacobian $\partial{\mbox{\boldmath$W$}}/\partial{\mbox{\boldmath$Q$}}$.

Although Equation  18.5-6 is conservative in the steady state, it is not, in a strict sense, conservative for time-dependent flows. This is not a problem, however, since the preconditioning has already destroyed the time accuracy of the equations and we will not employ them in this form for unsteady calculations.

For unsteady calculations, an unsteady preconditioning is available when the dual-time stepping method is used (Section  18.5.5). The unsteady preconditioning enhances the solution accuracy by improving the scaling of the artificial dissipation and maximizes the efficiency by optimizing the number of sub-iterations required at each physical time step [ 263]. For low Mach number flows in particular, for both low frequency problems (large time steps) and high frequency problems (small time step), significant savings in computational time are possible when compared with the non-preconditioned case.

The unsteady preconditioning adapts the level of preconditioning based on the user specified time-step and on the local advective and acoustic time scales of the flow. For acoustic problems, the physical time-step size is small as it is based on the acoustic CFL number. In this case the preconditioning parameter $U_r^2$ will approach $c^2$, which in effect will turn off the low-Mach preconditioning almost completely. For advection dominated problems, like the transport of turbulent vortical structures, etc., the physical time-step is large as it is based on the particle CFL number. The corresponding unsteady preconditioning parameter $U_r^2$ will then approach $u^2$, which corresponds to the steady preconditioning choice. For intermediate physical time-step sizes, the unsteady preconditioning parameter will be adapted to provide optimum convergence efficiency of the pseudo-time iterations and accurate scaling of the artificial dissipation terms, regardless of the choice of the physical time step.