An assessment of implicit-explicit time integrators for the pseudo-spectral approximation of Boussinesq thermal convection in an annulus

TL;DR

This paper evaluates various implicit-explicit time integrators for simulating Boussinesq thermal convection in an annulus, highlighting their accuracy, stability, and efficiency across different flow regimes.

Contribution

It provides a comprehensive assessment of 28 time integrators, identifying those that outperform the standard CNAB2 method in accuracy and computational efficiency.

Findings

Multistep and order 2 IMEX-RK methods show expected convergence.

Higher-order IMEX-RK methods experience order reduction due to stiffness and DAE effects.

Several IMEX-RK schemes outperform CNAB2 in turbulent regimes.

Abstract

We analyze the behaviour of an ensemble of time integrators applied to the semi-discrete problem resulting from the spectral discretization of the equations describing Boussinesq convection in a cylindrical annulus. The equations are cast in a vorticity-streamfunction formulation that yields a differential algebraic equation (DAE). The ensemble comprises 28 members: 4 implicit-explicit multistep schemes, 22 implicit-explicit Runge-Kutta (IMEX-RK) schemes, and 2 fully explicit schemes used for reference. The schemes are assessed for 11 different physical setups that cover laminar and turbulent regimes. Multistep and order 2 IMEX-RK methods exhibit their expected order of convergence under all circumstances. IMEX-RK methods of higher-order show occasional order reduction that impacts both algebraic and differential field variables. We ascribe the order reduction to the stiffness of the problem and, to a larger extent, the presence of the DAE. Using the popular Crank-Nicolson Adams-Bashforth of order 2 (CNAB2) integrator as reference, performance is defined by the ratio of maximum admissible time step to the cost of performing one iteration; the maximum admissible time step is determined by inspection of the time series of viscous dissipation within the system, which guarantees a physically acceptable solution. Relative performance is bounded between 0.5 and 1.5 across all studied configurations. Considering accuracy jointly with performance, we find that 6 schemes consistently outperform CNAB2, meaning that in addition to allowing for a more efficient calculation, the accuracy that they achieve at their operational limit of stability yields a lower error. In our most turbulent setup, where the behaviour of the methods is almost entirely dictated by their explicit component, 13 IMEX-RK integrators outperform CNAB2 in terms of accuracy and efficiency.