1 Well-posedness of the mean-flow system

The variable choice of state and recovery, the fixed equation split of equation structure, the smooth upwinding of transport, and the artificial-resistance term of elements are not matters of taste. Each repairs a structural singularity that defeats a formulation which is physically correct yet numerically hopeless — a set of equations that state the right physics and still cannot be solved from a general starting point. This document collects those failure modes, because each is a property of compressible-network equations rather than a tuning problem, and understanding them is what justifies the design decisions taken earlier.

The organizing object is the Jacobian, the matrix of first-order sensitivities on which Newton’s method entirely relies (see the solver). When two rows of the system respond identically to every unknown the Jacobian is singular, and the solver receives contradictory or empty guidance that no step-size control or damping can repair — the missing information is simply absent.

1.1 What ill-posedness means here

A steady operating point is found by Newton iteration on the residual vector \mathbf{R}(\mathbf{x}), which advances by solving \mathbf{J}\,\Delta\mathbf{x} = -\mathbf{R} with \mathbf{J} = \partial\mathbf{R}/\partial\mathbf{x} the Jacobian. The step is well defined only where \mathbf{J} is nonsingular, so the relevant question for every formulation is whether its Jacobian stays full-rank across the whole path the solver must travel — in particular through the quiescent cold start, where all flows vanish, and through flow reversal, where a flow changes sign. It should be noted that two kinds of rank loss must be told apart. A formulation artifact is a singularity introduced by how the equations are written, curable by rewriting them; a physical indeterminacy is a singularity of the problem itself, where the physics genuinely fails to fix the unknowns, and the only sound response is to regularize the question rather than pretend it has a unique answer. The first two root causes below are artifacts, removed by construction in the earlier documents; the last two are physical, and are handled by continuation and damping.

1.2 Root cause A: flux-form energy balances degenerate at zero flow

The most intuitive and tempting way to impose energy conservation on a two-port element is the flux form R_E = \sum_i \sigma_i\,\dot m_i\,h_{t,i}, summed over its ports. Differentiating this row exhibits its defect, and its derivative is given as:

\frac{\partial R_E}{\partial \mathbf{x}} = \sum_i h_{t,i}\,\frac{\partial(\sigma_i\dot m_i)}{\partial \mathbf{x}} \;+\; \sum_i \sigma_i\dot m_i\,\frac{\partial h_{t,i}}{\partial \mathbf{x}},

where the first sum weights the mass-flux sensitivities by the port enthalpies and the second is proportional to the mass flows themselves. As the flows vanish, \dot m_i \to 0, the second sum disappears; and if the enthalpy field is still uniform, h_{t,i} = h_t — the natural state of a cold start — the first sum is exactly h_t times the derivative of the mass balance. The energy row is then h_t times the mass row, the Jacobian is singular, and no information about the temperature field survives.

Intuitively, this degeneracy is physical and not merely algebraic: at zero flow the statement “energy carried in equals energy carried out” is satisfied by any temperature field, so the equation genuinely says nothing there. The same collapse appears in characteristic variables, as it must, because the M \to 0 limits of the characteristic energy-flux coefficients are \pm\varrho c^2/(\gamma-1) = \pm\varrho h, again h times the mass-flux coefficients (see characteristics). Every chain of isentropic elements started from rest sits on this singularity, which is precisely why a flux-form energy balance fails on multi-element chains. The structural cure is to abandon the flux form for the transport form of transport: the edge equation h_{t,e} = \theta H_{\text{tail}} + (1-\theta)H_{\text{head}} always carries a coefficient of 1 on h_{t,e} itself, so it never loses its diagonal and cannot collapse onto the mass balance (test: test_long_serial_chain_cold_start, solving a long chain from exactly zero flow).

1.3 Root cause B: hard switches destroy smoothness and rank

A second temptation is to select the upstream side of a convected quantity, or the sign of a loss, with a hard switch such as \operatorname{sign}(u) multiplying a residual row. Such a factor fails in two independent ways. It vanishes identically at u = 0, so the row it multiplies is null exactly at the quiescent state where the cold start lives, and the Jacobian loses a row there; and it jumps discontinuously across a flow reversal, whereas Newton’s method presumes a differentiable residual and a jump halts it. At a deeper structural level, the classical practice of reassigning which equation a port owns when the flow reverses changes the equation count on each side of the reversal, a discontinuity in the very shape of the system (see equation structure). Both disappear under the constructions adopted earlier: the fixed per-edge split of equation structure keeps the equation count invariant to flow direction, and the smooth upwind weight \theta = \operatorname{sstep}(\dot m;\varepsilon) of transport selects the upstream side continuously, so the residual and its Jacobian remain smooth through reversal.

1.4 Root cause C: zero flow is a stationary point of pressure-driven networks

The third failure is the subtlest because the residuals are perfectly smooth and full-rank in count, yet the solver still stalls at rest. The cause is that all pressure-type physics couples to the flow only through the dynamic head, whose leading behaviour is given as:

p_t - p \;=\; \frac{\gamma}{2}\,p\,M^2 + \mathcal{O}(M^4), \qquad M^2 \propto \dot m^2,

where M is the Mach number and the dynamic head is quadratic in the mass flow. Its sensitivity to the flow, \partial(p_t - p)/\partial\dot m \propto \dot m, therefore vanishes at \dot m = 0. In a network driven only by pressure boundary conditions — with no mass-flow specification anywhere — the pressure residuals thus have zero first-order sensitivity to every flow unknown at the quiescent state. Newton’s method, which moves entirely on first-order sensitivities, sees a flat landscape and no reason to set the flows in motion: zero flow is a stationary point, and descent-type iterations stall on or near it.

The cure must inject first-order flow sensitivity without altering the converged answer, which is exactly what the artificial-resistance continuation of elements does. With the fictitious friction \kappa active, each pressure relation acquires a term linear in the flow, so the network behaves like a pipe-resistance circuit in which pressure differences push directly on the flows — as a voltage drives a current through a resistor — and the solver locates the correct flow pattern readily. The friction is then reduced to zero over a short sequence of stages, the dimensionless schedule \kappa_s stepping through values such as (0.1,\ 0.01,\ 0) with each stage warm-started from the previous solution, so that the final stage solves the exact, friction-free equations (test: test_quiescent_cold_start_converges). An important remark is that this is a continuation in a physical parameter, not a numerical fudge: every intermediate problem is a well-posed resistive network, and only the limit \kappa \to 0 restores the original equations, whose solution the continuation reaches by a path that is nonsingular throughout.

1.5 A milder, physical indeterminacy

One further singularity is genuinely a property of the problem rather than of its formulation. In a perfectly symmetric branching network at rest, the split of the flow between identical branches is undetermined at first order — any split satisfies the loop balance when nothing flows — so the Jacobian is singular there even when a mass-flow inlet fixes the total. This is a physical indeterminacy: the symmetric problem truly has a one-parameter family of incipient splits, resolved only by whatever breaks the symmetry. Rather than rewrite the equations, the solver regularizes the step through Levenberg–Marquardt damping (see the solver), which replaces the singular \mathbf{J} by \mathbf{J} + \lambda\mathbf{I} near a rank deficiency and so takes a small, well-defined step that the following iterations refine (test: test_many_parallel_branches_converge).

Taken together, these repairs are why the mean-flow system is solvable from any admissible cold start: the transport form removes the energy degeneracy, the fixed split and smooth upwinding remove the switch discontinuities, the artificial-resistance continuation removes the pressure-driven stationarity, and the damping absorbs the residual symmetric indeterminacy. The wave language of characteristic variables — the natural coordinates of the acoustics, and the setting in which the same operating-point Jacobian is reused — is developed in characteristics.