Exponential versus IMEX high-order time integrators for thermal convection in rotating spherical shells

Abstract

We assess the accuracy and efficiency of several exponential time integration methods coupled to a spectral discretization of the three-dimensional Boussinesq thermal convection equations in rotating spherical shells. Exponential methods are compared to implicitexplicit (IMEX) multi-step methods already studied previously in [1]. The results of a wide range of numerical simulations highlight the superior accuracy of exponential methods for a given time step, especially when employed with large time steps and at low Ekman number. However, presently available implementations of exponential methods appear to be in general computationally more expensive than those of IMEX methods and further research is needed to reduce their computational cost per time step. A physically justified extrapolation argument suggests that some exponential methods could be the most efficient option for integrating flows near Earth's outer core conditions.

Radial collocation methods for the onset of convection in rotating spheres

“…Similar problems occur in time evolution codes in spheres or spherical shells when the nonlinear terms are truncated. This was noticed in [52] and by the authors in their implementations of codes in the case of spherical shells [34,35,37,38]. In some cases the numerical instability shows up only after long time integrations.…”

“…These methods have been used successfully in the case of spherical shells [33][34][35]37,38]. They are very efficient due to the possibility of using optimized matrix-matrix products to evaluate the action by the operators in order to compute the nonlinear terms in the time evolution of the full non-linear equations.…”

“…The authors have developed several time integration and stability analysis codes, using this approach, for the convection in spherical shells of pure and binary fluids with very good results [33][34][35][36][37][38]. The main reasons for using collocation methods were simplicity and that changing boundary conditions was easy.…”

A parallel time integrator for solving the linearized shallow water equations on the rotating sphere

“…However, it turns out to be challenging to find a computationally efficient formulation including the Coriolis effect, which allows large time steps. An application of exponential integrators in the context of atmospheric simulations can be found, for example, in the work of Garcia et al Here, significant improvements in accuracy were observed; however, performance issues under realistic test conditions were found to be a severe issue.…”

Energy conserving discontinuous Galerkin spectral element method for the Vlasov–Poisson system

“…Notice that a centered finite difference could be also considered, although at the price of doubling the evaluations of R(f ); this solution has been considered in [GBNS14]. A second consequence is that a stopping criterion must be provided: a large number of iterations results in an accurate approximation of φ (∆t J n R ), but also leads to a higher computational cost.…”