CMPS Theory Manual › Chapter 19

19 Schur-Complement Pressure–Velocity Coupling

The coupled CMPS Jacobian already contains the cross-derivatives that connect continuity, pressure, and momentum. A Schur complement is therefore a useful way to interpret or precondition the block system, but it should not automatically be described as an additional pressure Laplacian inserted into the production residual unless such an augmentation is explicitly implemented.

19.1 Coupled block system

Partition a linearized aerodynamic system into a continuity/pressure-index block and the remaining velocity block, \[\begin{bmatrix} A_{pp} & A_{pu}\\ A_{up} & A_{uu} \end{bmatrix} \begin{bmatrix} \delta p\\ \delta\mathbf{u} \end{bmatrix} = \begin{bmatrix} b_p\\ \mathbf{b}_u \end{bmatrix}. \](19.1) Here the subscript \(p\) labels the pressure column and the continuity row slot; it does not imply a pressure-conservation equation.

Eliminating the velocity correction gives \[\delta\mathbf{u} =A_{uu}^{-1}(\mathbf{b}_u-A_{up}\delta p)\] and therefore \[\underbrace{\left( A_{pp}-A_{pu}A_{uu}^{-1}A_{up} \right)}_{S_p} \delta p =b_p-A_{pu}A_{uu}^{-1}\mathbf{b}_u. \](19.2) \(S_p\) is the exact pressure Schur complement of the selected block partition.

Partition the unknown correction into a momentum-like block \(\delta\mathbf u\) and pressure-like block \(\delta p\):

\[\begin{bmatrix}\mathbf A_{uu}&\mathbf A_{up}\\\mathbf A_{pu}&\mathbf A_{pp}\end{bmatrix}\begin{bmatrix}\delta\mathbf u\\\delta p\end{bmatrix}=\begin{bmatrix}\mathbf b_u\\\mathbf b_p\end{bmatrix}.\]

Eliminating the momentum correction gives

\[\delta\mathbf u=\mathbf A_{uu}^{-1}(\mathbf b_u-\mathbf A_{up}\delta p),\]

and the exact pressure Schur system

\[\underbrace{(\mathbf A_{pp}-\mathbf A_{pu}\mathbf A_{uu}^{-1}\mathbf A_{up})}_{\mathbf S_p}\delta p=\mathbf b_p-\mathbf A_{pu}\mathbf A_{uu}^{-1}\mathbf b_u.\]

19.2 Physical meaning

\(A_{up}\) contains the response of momentum to pressure, dominated by pressure-gradient/pressure-flux terms. \(A_{pu}\) contains the response of continuity/mass flux to velocity. Their product through \(A_{uu}^{-1}\) therefore creates a pressure operator whose long-wavelength behaviour resembles an elliptic pressure-correction equation.

This is the same algebraic mechanism by which pressure-based methods obtain a pressure Poisson-like equation, but in a fully coupled primitive-variable method it arises from eliminating variables from the complete Jacobian rather than from deriving a separate segregated pressure equation.

19.3 A local approximation

If the velocity block is approximated by a local diagonal/pseudo-time term, \[A_{uu}\approx \frac{\rho_P\Omega_P}{\Delta\tau_P}\mathbf{I},\] then \[A_{uu}^{-1}\approx \frac{\Delta\tau_P}{\rho_P\Omega_P}\mathbf{I}.\] Substitution in Eq. (19.2) shows why stronger pseudo-time stabilization changes pressure correction propagation: the approximate pressure operator is scaled by the local momentum inverse.

On a two-point orthogonal stencil, pressure-gradient and continuity-velocity couplings produce terms proportional to \[\frac{A_f}{d_{PN}},\] which is the familiar geometric structure of a finite-volume Laplacian. The precise coefficient, however, depends on the actual numerical mass flux, pressure flux, preconditioning, reconstruction, and the state-dependent Jacobian. It should be obtained from the implemented derivatives rather than asserted from one simplified low-Mach formula.

A practical pressure-oriented preconditioner replaces the expensive inverse by a local or easily inverted approximation \(\widetilde{\mathbf A}_{uu}^{-1}\),

\[\widetilde{\mathbf S}_p=\mathbf A_{pp}-\mathbf A_{pu}\widetilde{\mathbf A}_{uu}^{-1}\mathbf A_{up}.\]

The quality of this approximation controls how effectively long-range pressure error is reduced without destroying the fully coupled matrix used by the outer solve.

19.4 AUSM⁺-up coupling

For AUSM⁺-up, continuity depends on pressure through the pressure-difference term in the numerical mass flux, while momentum depends on pressure through the split mechanical pressure. Therefore \[A_{pp}\neq0,\qquad A_{pu}\neq0,\qquad A_{up}\neq0\] even before a Schur complement is formed. Forward AD evaluates these derivatives from the same face expression used for the residual.

At low Mach number, time-derivative preconditioning changes the pseudo-time relation between pressure and the remaining variables (Weiss and Smith 1995). In the constant-density incompressible regime, artificial compressibility instead contributes the explicit pressure-row pseudo-time coefficient \(1/c_{ac}^{2}\) and the artificial-compressibility AUSM⁺-up mass-flux pressure derivative.

19.5 Use as a preconditioning concept

Equation (19.2) suggests several possible linear preconditioners:

approximate block factorization;
pressure Schur-complement approximation;
block ILU that preserves pressure–velocity couplings;
multigrid/coarse operators that retain the low-frequency pressure mode.

These approaches can accelerate a Krylov solve without changing the nonlinear residual. This distinction is important: a preconditioner may be approximate as long as it improves the linear solve, whereas a new term inserted into the residual changes the discrete governing equations and requires a separate consistency derivation.

19.6 Implementation statement

The implementation-aligned conclusion is therefore:

CMPS obtains pressure–velocity coupling directly from the fully coupled numerical-flux/source Jacobian. Schur-complement algebra explains the elliptic pressure correction embedded in that block system and motivates preconditioning, but this guide does not claim a separate hand-added continuity Laplacian or a fixed iteration-speedup factor unless those are demonstrated by a specific implementation and benchmark.