9.5.1 Derivation of Preconditioned Equations 预条件方程的推导[cfd-9-5-1]

We demonstrate preconditioning with the aid of 1-D Euler equations. They can be written in differential form as

where \(\vec{W} = [\rho,\ \rho u,\ \rho E]^T\) is the vector of conservative variables and \(\vec{F}_c\) denotes the convective fluxes. In order to see the effect of preconditioning on the spectral radii (important for the computation of the time step) and on the convective flux Jacobian (important for upwind dissipation), the Euler equations (9.41) are rewritten in the quasilinear form

with \(\bar{A}_c = \partial\vec{F}_c/\partial\vec{W}\) being the convective flux Jacobian (cf. Appendix A.2).

The idea behind low Mach-number preconditioning is to transform the governing equations (9.41) such that the new equations have more favourable properties at low Mach numbers (i.e. below \(M \approx 0.2\)). First of all, we want to equalise the convective eigenvalues in order to bound the condition number (see Eq. (9.40)) and thus to remove the stiffness at \(M \rightarrow 0\). We further want to change the scaling of the numerical dissipation in order to improve the accuracy. Finally, we want to couple the pressure and the velocity as it is done in the pressure-based schemes. The transformation replaces the conservative variables \(\vec{W}\) by a different set of flow variables. Various choices are possible [90], but the most often used are the pressure, the velocity components and the temperature. Transforming the quasilinear form in Eq. (9.42) into the new variables \(\vec{W}_p\), we obtain

In the above Eq. (9.43), \(\bar{P} = \partial\vec{W}/\partial\vec{W}_p\) represents the transformation matrix from the new variables \(\vec{W}_p\) into the conservative variables \(\vec{W}\). Introducing a new flux Jacobian

we can write Eq. (9.43) as

If we now replace \(\bar{P}\) in front of the time derivative by a suitable preconditioning matrix \(\bar{\Gamma}\) (to be specified later), the preconditioned Eq. (9.45) reads

or equivalently

It is now possible to solve Eq. (9.47) in the new variables \(\vec{W}_p\) using any standard spatial and temporal discretisation scheme. Another possibility is to transform the preconditioned equations (9.47) back into the conservative variables \(\vec{W}\). We have

where \(\bar{P}^{-1} = \partial\vec{W}_p/\partial\vec{W}\) is the transformation matrix from the conservative variables. Equation (9.48) can be for convenience written as

or

The term \(\bar{P}\bar{\Gamma}^{-1}\) in Eq. (9.49) and (9.50) is called the conservative variable preconditioning matrix.

We can see now that the preconditioned equations in the conservative variables (9.49), (9.50) have the following features:

• Spatial derivatives are multiplied by \(\bar{P}\bar{\Gamma}^{-1}\).
• Unsteady equations are different from the original form in Eq. (9.42) and hence the solution is no longer time accurate.
• Stationary solution (i.e. \(\partial\vec{W}/\partial t = 0\)) remains unchanged.
• Eigenvalues and eigenvectors of the preconditioned system correspond to those of the matrix \(\bar{P}\bar{\Gamma}^{-1}\bar{A}_c = \bar{P}\bar{\Gamma}^{-1}\bar{A}_{c,p}\bar{P}^{-1}\) and hence are different from those of the original system in Eq. (9.41).

The main task is now to find a suitable set of variables \(\vec{W}_p\) and the matrix \(\bar{\Gamma}\) such that the convective eigenvalues of the preconditioned system (9.50) are equalised as close as possible. But it is also important that the matrices remain defined for \(M \rightarrow 0\). Furthermore, it is desirable that the preconditioned system converts into the original system for higher Mach numbers.

Before we present the transformation and preconditioning matrices in Subsection 9.5.3, we shall discuss the implementation of the low Mach number preconditioning in a flow solver.