#O04 Exponential time differencing and parallel implementation

Abstract

MPAS-Ocean utilizes a multi-resolution mesh that resolves sensitive scales with a locally refined mesh. This allows for a reduction in computation time in the coarse regions. However, due to the CFL-condition, the global time-step of explicit methods is restricted by the smallest grid cell. To address this, we combine two approaches: exponential time differencing (ETD) and domain decomposition (DD) methods.

On the one hand, ETD methods allow for large time-steps, while still retaining key properties of explicit integrators. We investigate a modified ETD-Rosenbrock scheme applied to the rotating shallow water equations, discretized by the TRiSK scheme. We prove conservation of mass up to machine precision and demonstrate stability for large time-steps (orders of magnitude above the CFL-compliant one), and conservation of energy up to a time-truncation error. The main difficulty in the implementation arises in the computation of the $\varphi$-functions of the ETD-method, which are usually resolved by Krylov-subspace methods. Here, we exploit the energy conserving properties of the underlying spatial scheme and replace the Arnoldi process by a skew-Lanczos process with respect to a carefully chosen inner product, which significantly reduces computation time. Results are shown for double gyre and Gaussian pulse test cases.

On the other hand, DD methods allow to partition the domain into regions with similar mesh size and adapt the time step locally. We apply overlapping domain decomposition in combination with ETD. A semi-discrete multidomain problem with Dirichlet transmission conditions on the interfaces is derived, from which two localized ETD methods are proposed. The first method is obtained by using ETD for time integration at each time step and then applying Schwarz iteration to solve problems in the subdomains. The second method is based on the Schwarz waveform relaxation algorithm, in which time-dependent subdomain problems are solved at each iteration. Note that the subdomain problems are solved in parallel in each case. Convergence of the associated iterative solutions to the fully discrete multidomain solution and to the exact semi-discrete solution is proved for the diffusion problem. Numerical results for the one-dimensional shallow water equation are presented, which show significant speed-ups obtained by the localized approach compared to a single domain solver.