Two- and multilevel parallel-in-time integration of the shallow water equations on the rotating sphere



14/03/2023 - 13:00 Joao Steinstraesser (University of Sao Paulo) IMAG 106

Parallel-in-time (PinT) methods have gained increasing interest over the past two decades as an approach to accelerate numerical simulations, by replacing the traditional serial time integration by the simultaneous computation of several time steps. Atmospheric modeling is one of the several application domains aiming to take advantage of PinT and reduce the computational times required to provide accurate and stable solutions. However, PinT methods usually suffer from stability and convergence issues when applied to hyperbolic problems, such as those arising in atmospheric models.
In this work, we study the temporal parallelization of the shallow water equations on the rotating sphere using two- and multilevel PinT schemes, respectively Parareal and MGRIT. In these methods, the solution is iteratively improved by the simultaneous use of a fine and one or more coarse discretizations of the problem. The choice of coarse discretization has an important impact on the performance of the temporal parallelization, since it needs to provide sufficiently accurate solutions in order to ensure stability and convergence of the PinT method. We consider four alternatives of coarse method, two of them being well-established schemes in atmospheric modeling, namely an implicit-explicit discretization and the semi-Lagrangian semi-implicit SL-SI-SETTLS, and the other two, the exponential integration scheme ETDRK and its semi-Lagrangian version SL-ETDRK, being more recent approaches in this framework. We compare the use of these methods in PinT both via a stability analysis on a linearized ODE and numerical simulations of standard benchmarks. Also, we study the influence of the use of artificial viscosity in each discretization level on the performance of the temporal parallelization.