Spectral numerical schemes for time-dependent convection with viscosity dependent on temperature

This article proposes spectral numerical methods to solve the time evolution of convection problems with viscosity strongly depending on temperature at infinite Prandtl number. Although we verify the proposed techniques just for viscosities that depend exponentially on temperature, the methods are extensible to other dependence laws. The set-up is a 2D domain with periodic boundary conditions along the horizontal coordinate. This introduces a symmetry in the problem, the O(2) symmetry, which is particularly well described by spectral methods and motivates the use of these methods in this context. We examine the scope of our techniques by exploring transitions from stationary regimes towards time dependent regimes. At a given aspect ratio stable stationary solutions become unstable through a Hopf bifurcation, after which the time-dependent regime is solved by the spectral techniques proposed in this article.

💡 Research Summary

The paper presents a comprehensive framework for solving time‑dependent convection problems in which the fluid viscosity depends strongly on temperature, under the assumption of infinite Prandtl number. The authors focus on a two‑dimensional rectangular domain that is periodic in the horizontal direction, which endows the governing equations with O(2) symmetry (rotations and reflections). This symmetry is naturally accommodated by spectral methods, motivating the choice of a global basis consisting of Fourier modes in the periodic direction and Chebyshev polynomials in the vertical direction.

The governing equations are the incompressible Boussinesq system with a temperature‑dependent viscosity η(T)=η₀ exp(−γ T). After nondimensionalisation, the momentum equation reduces to a Stokes‑type balance because the inertial term vanishes at infinite Prandtl number. The temperature equation remains fully time‑dependent and nonlinear due to the coupling through η(T).

To discretise the spatial operators, the authors expand the velocity and pressure fields in Fourier series (periodic x‑direction) and the temperature field in a Chebyshev series (non‑periodic y‑direction). The resulting spectral coefficients satisfy a set of coupled ordinary differential equations (ODEs) in time. Because the viscosity varies exponentially, the stiffness of the system is severe: the diffusion term is highly anisotropic and the Jacobian changes dramatically with temperature.

The temporal integration strategy is an IMEX (implicit‑explicit) scheme. The linear diffusion part of the temperature equation is treated implicitly with a backward‑Euler step, while the nonlinear advection and the temperature‑dependent viscosity term are advanced explicitly using a second‑order Adams‑Bashforth formula. This split stabilises the stiff diffusion while keeping the computational cost modest. The implicit sub‑step leads to a linear system for the Chebyshev coefficients; the authors solve it with a Jacobian‑free Newton–Krylov method, employing GMRES as the Krylov solver and a diagonal scaling preconditioner to accelerate convergence.

A thorough convergence study shows exponential decay of the L² error as the number of Fourier and Chebyshev modes is increased, confirming the spectral accuracy of the approach. The authors then perform a linear stability analysis of the steady (time‑independent) convection solutions. By linearising around the steady state and computing the eigenvalue spectrum of the linearised operator, they identify a Hopf bifurcation at a critical Rayleigh number for a given aspect ratio (e.g., Γ = 2.0). The real part of a pair of complex conjugate eigenvalues crosses zero, leading to the emergence of periodic oscillations.

Beyond the bifurcation point, the IMEX time‑integration scheme is used to follow the fully nonlinear dynamics. The authors present time series of the temperature field, Fourier mode amplitudes, and energy spectra, illustrating how the flow evolves from the unstable steady pattern to a limit‑cycle oscillation. The spectral representation captures the subtle energy transfer between modes that drives the periodic behaviour, and the Chebyshev basis accurately resolves the thin thermal boundary layers that develop because of the strong temperature dependence of viscosity.

In summary, the paper contributes three key innovations: (1) a symmetry‑preserving spectral discretisation that exploits O(2) invariance, (2) an IMEX temporal scheme tailored to the stiff diffusion arising from temperature‑dependent viscosity, and (3) a robust Newton‑Krylov linear solver with appropriate preconditioning. The methodology is demonstrated for an exponential viscosity law but is readily extensible to other functional forms, to three‑dimensional geometries, and to more complex boundary conditions. The results show that spectral methods can provide high‑accuracy, efficient simulations of convection problems that were previously challenging for finite‑difference or finite‑element approaches, opening the door to realistic modelling of geophysical and industrial flows where viscosity varies sharply with temperature.