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

We analyze the behavior of an ensemble of time integrators applied to the semi discrete problem resulting from the spectral discretization of the equations describing Boussinesq thermal convection in a cylindrical annulus. The equations are cast in their 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 whose theoretical order varies from 2 to 5 are assessed for 11 different physical setups that cover laminar and turbulent regimes. Multistep and order 2 IMEXRK 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 at hand 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, dissipation-based limit of stability yields a lower error. In our most turbulent setup, where the behavior of the methods is almost entirely dictated by their explicit component, 13 IMEX-RK integrators outperform CNAB2 in terms of accuracy and efficiency.(c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This article analyzes the behavior of time integrators applied to the semi discrete problem resulting from the spectral discretization of Boussinesq thermal convection equations. Different schemes of various orders are assessed for laminar and turbulent regimes, and it is found that some high-order IMEX-RK methods exhibit occasional order reduction. Comparing with the reference integrator CNAB2, six schemes consistently outperform CNAB2 in terms of accuracy and performance. In the most turbulent setup, 13 IMEX-RK integrators outperform CNAB2 in terms of both accuracy and efficiency.