Pseudo-transient continuation (Ψtc) is a Newton-like iterative method commonly used to compute steady-state solutions of differential equations when the initial guess is far from the final steady state. The method introduces an artificial time-derivative term into the original steady-state equation to construct a pseudo-transient evolution process, allowing the solution to gradually approach the steady state. As the iteration proceeds, the pseudo-time step is progressively increased, eventually degenerating into a Newton iteration for the original nonlinear equation. Thus, Ψtc can be viewed as an evolution-based strategy to accelerate steady-state convergence via nonlinear path-following.

In computational fluid dynamics (CFD), the coupled solution of incompressible Navier–Stokes equations often results in a highly nonlinear algebraic system due to the strong coupling between pressure and velocity, making convergence challenging for fully Coupled algorithms. Ψtc introduces an artificial time derivative term, which significantly enhances the diagonal dominance of the system matrix, thereby improving the condition number of the linear system and increasing the robustness of the nonlinear iteration.

As noted by F. Moukalled, in The Finite Volume Method in Computational Fluid Dynamics[1], under-relaxation factors play a critical role in nonlinear convergence, yet no universally optimal values exist. Different physical equations, spatial regions, and solution phases may all require different under-relaxation factors. In contrast, Ψtc offers a more general and automated nonlinear under-relaxation mechanism: the pseudo-time step not only governs the convergence rate but also affects the nonlinear stability of the solution. Therefore, in pseudo-transient methods, selecting a reasonable initial estimate of the time scale and adaptively adjusting it based on the evolution of nonlinear residuals are key to improving the efficiency of steady-state computations.In practice, physical time scales (such as convective or diffusive scales) are often used to estimate the initial time step, followed by adaptive enlargement strategies based on residual ratios and amplification factors to accelerate convergence while maintaining numerical stability.

This paper, based on the Finite Volume Method(FVM), proposes an initial time scale estimation strategy for the pseudo-transient continuation method under incompressible natural convection governed by the Boussinesq approximation. By nondimensionalizing the discretized Navier–Stokes equations, the characteristic velocities associated with viscous, convective, and buoyancy terms are analyzed, leading to the derivation of a pseudo-transient characteristic time scale expression specific to the Boussinesq assumption (see Equation 1).

\(\Delta \tau = \sqrt{\frac{L_0}{\beta g (T_{\text{max}} - T_{\text{min}})}} \)                                                                                                                                                                                                                                     (1)

This time scale is further applied to a fully Coupled solver, and the annular natural convection problem is used as a test case(grid division and results are shown in Fig. 1), with benchmark data from both experiments and validated simulations for comparative analysis. By comparing velocity residual convergence curves under different time scale estimation strategies (as shown in Fig. 2), numerical experiments demonstrate that the characteristic time scale under Boussinesq approximation enables the velocity residuals to converge to 10⁻⁶ in approximately one-tenth the iterations required by standard convective/diffusive time scale methods.

In conclusion, pseudo-transient continuation serves as a robust Newton-like framework for accelerating steady-state convergence in nonlinear CFD problems. By introducing a physically motivated pseudo-time term, it enhances numerical stability and alleviates the need for manual under-relaxation tuning. This study proposes a characteristic time scale tailored to natural convection under the Boussinesq approximation, derived through nondimensional analysis of the discretized Navier–Stokes equations. Application to a benchmark annular cavity problem demonstrates that the proposed time scale significantly outperforms conventional convection–diffusion-based estimates, offering an order-of-magnitude improvement in convergence speed. These results highlight the importance of problem-specific time scale estimation in the efficient implementation of pseudo-transient continuation methods.
（The image can be found in the abstract file）