3 Finite-Volume Variables and Coupled Solution System

This chapter explains the variable ordering, residual sign convention and coupled Jacobian used by NeuralFlow. These conventions matter to users when interpreting residuals, nonlinear corrections, physical-time terms and NeuralFlowML physics gradients.

3.1 Primitive columns and conservative equation rows

NeuralFlow forms the coupled Jacobian directly with respect to the primitive correction variables. Therefore the matrix columns correspond to derivatives with respect to , while the equation rows retain their physical conservation meaning.

The row slots use the same indices for storage convenience, but their physical meaning is different:

When turbulence, particles, or species are active, their equations and primitive corrections are appended according to the active equation layout selected by the coupled equation layout.

3.2 Residual and matrix sign

Define the finite-volume residual by Eq. (2.8). The current assembly accumulates a face contribution as A physical source has the opposite residual sign and is therefore accumulated as Thus (3.2) and the linear correction satisfies (3.3) After relaxation/limiting, the primitive variables are updated by addition of the solved increment.

3.3 3.3 Face-local Jacobian blocks

NeuralFlow evaluates the face Jacobians and consistently with the numerical flux used in the residual.

In this manual, an “exact Jacobian” means exact differentiation of the active residual expression for the coefficients held fixed during that nonlinear assembly. Coefficients refreshed outside that assembly are updated on the next nonlinear iteration.

3.4 Stored state and pseudo-time state

NeuralFlow maintains primitive variables and conservative or conservative-like histories. The time derivative need not be the identity in primitive variables. Let A local pseudo-time contribution has the generic form For the compressible solver, may be modified by low-Mach time-derivative preconditioning. For the constant-density incompressible solver, the pressure row contains the artificial-compressibility factor while the stored pressure component remains .

3.5 Spatial versus physical-time residuals

In steady calculations, pseudo time is a convergence device: it stabilizes the Newton-like iteration but does not add a physical unsteady residual. In transient calculations, a physical BDF residual is added to the spatial residual and dual-time iterations converge that physical step.

For a stored physical state , (3.4) with For artificial-compressibility incompressible flow, the pressure equation has no physical pressure-storage term; its pressure time derivative remains a pseudo-time coupling device.

3.6 Why the conventions matter

Several common formulation errors can be avoided by keeping these identities explicit:

• a pressure-index row is continuity, not a pressure conservation equation;

• already contains the Newton minus sign because ;

• the artificial-compressibility coefficient belongs to the pseudo-time mass matrix and must not be confused with the stored pressure state;

• a coarse-grid or machine-learning path must use the same row/column ordering and sign convention as the main residual if it reuses the NeuralFlow matrix.