Why do coupled iterations converge slowly?

Each physics affects the other, whose result feeds back. In Picard iteration (alternating substitution) error is multiplied by q per round trip, so convergence slows as q approaches 1. In the heater example q = I²R₀α/G of 0.2 needed 12 iterations, 0.78 needed 78 and 0.94 needed 335; above 1 it diverges.

Does Newton's method solve it?

Convergence is much faster: two iterations at every current in the heater example. But Newton can return a mathematical answer even where no physical solution exists. At 11.5 A (runaway) it returned −8,364 °C, below absolute zero. Convergence alone doesn't mean correctness, so always check results make physical sense.

What is thermal runaway?

Positive feedback where higher temperature increases heating, raising temperature further, beyond what cooling can contain. For this wire, above I = √(G/(R₀α)) ≈ 11.3 A heating grows faster than cooling, temperature rises without limit and no equilibrium exists. When coupled iterations diverge, it matters whether the cause is numerical or the absence of a physical solution.

Convergence in Coupled Problems

Category: Coupled Analysis – Multiphysics | Revised 2026-10-01

CAE visualization for convergence coupled theory - technical simulation diagram

Convergence of Coupled Problems

Theory: iteration convergence is set by coupling strength

Overview

🙋

Our electro-thermal coupled analysis converges very slowly. Is it a settings problem?

🎓

More often it's the nature of the problem than settings. Current heats the conductor, raising temperature; higher temperature raises resistance and heating further. Solving electrical and thermal parts alternately, each round trip leaves error reduced only by a fixed factor q. Small q converges quickly; q near 1 takes hundreds of iterations; above 1 it diverges. And q above 1 may not be a numerical failure at all but the physical absence of an equilibrium temperature (thermal runaway). So when convergence fails, estimate coupling strength first and tell numerical from physical problems. Let's check with a simple heater-wire model.

Heater-wire balance

$$ G\,(T - T_0) = I^2 R_0\left[1 + \alpha (T - T_0)\right]\ \Rightarrow\ T - T_0 = \frac{I^2 R_0}{G - I^2 R_0\alpha},\qquad I_{run} = \sqrt{\frac{G}{R_0\alpha}} $$

G is cooling conductance (0.05 W/K), R₀ resistance at 20 °C (0.1 Ω), α temperature coefficient of resistance (0.0039 /K for copper), T₀ = 20 °C. The runaway current is about 11.3 A.

Picard and Newton iterations

$$ \text{Picard: } T_{k+1} = T_0 + \frac{I^2 R(T_k)}{G},\quad e_{k+1} = q\,e_k,\ q = \frac{I^2 R_0\alpha}{G};\qquad \text{Newton: } T_{k+1} = T_k - \frac{F(T_k)}{F'(T_k)} $$

Coffee Break Trivia

Light-bulb filaments and thermal runaway

A metal heater whose resistance rises with temperature, run at constant voltage, stabilises naturally: hotter means less current. That's why incandescent bulbs glow steadily. Driven at constant current, though, hotter means more heating, and once that outpaces cooling it can't stop. Some semiconductor devices draw more current as they heat, and thermal runaway, where current concentrates in one of several parallel devices and destroys it, has long been known. Coupled analysis is also used to find such boundaries between stability and instability.

Worked examples

Example: current and convergence speed (start at 20 °C, converged at 10⁻⁶ K change)

Current	Coupling q	Equilibrium temperature	Picard	Newton
5 A	0.195	82 ℃	12	2
8 A	0.499	276 ℃	28	2
10 A	0.780	929 ℃	78	2
11 A	0.944	4,326 ℃	335	2
11.5 A	1.032	None (runaway)	Diverges	−8,364 °C (unphysical)

Temperatures at 10 A and above would actually melt the wire, and the temperature coefficient isn't constant, but the same formula is kept to show convergence behaviour. Picard with relaxation 0.5 was slower still: 160 iterations at 10 A and 654 at 11 A.

🙋

Newton converged in two iterations, yet still gives an answer under runaway conditions.

🎓

The equation here is linear in temperature, so Newton lands on the exact solution in one step (the second confirms). At 11.5 A, however, the mathematical solution is a negative temperature — algebraically correct but physically nonexistent. Newton reports convergence, so without physical checks such as temperature range, the answer would be used as is. Picard's divergence, by contrast, honestly reflects the problem's nature (no solution). Smaller relaxation slowed convergence because here error decreases monotonically (positive q), so relaxation only acts as a brake; it helps when error oscillates. Whatever the method, make a habit of checking that results are physically reasonable and that a solution exists.

Steps toward convergence

Estimate coupling strength (error factor per round trip).
Use alternating iteration for weak coupling, Newton or monolithic solvers for strong.
Judge convergence by both change and residual.
Add physical checks such as temperature and pressure ranges.
When it won't converge, suspect absence of a solution (runaway, buckling etc.).

Coffee Break Trivia

“A converged analysis that disagreed with tests”

A heating analysis of power-board traces converged without trouble at high current and showed trace temperatures within limits, yet the trace burned out at that current in testing. During Newton iterations temperature had exceeded the material data range, resistance was extrapolated too low, and the solve 'converged' at an unnaturally low temperature. Extending material data to high temperature and stopping when an upper temperature limit was exceeded revealed that no equilibrium existed at that current (thermal runaway).

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Assuming converged means correct	Unphysical solutions used	Physical checks
Blaming divergence on numerics	Runaway missed	Check existence
Always heavier relaxation	Slower convergence	Watch error behaviour
Extrapolating material data	Wrong properties	High-temperature data
Change-only convergence test	Slow drift mistaken for convergence	Check residual too

🙋

I'd like to learn related topics.

🎓

Rate this article

Thank you for your feedback!

Helpful

More details

Report error