# SC²S Colloquium - Apr 26, 2018

### From Sccswiki

Date: |
Apr 26, 2018 |

Room: |
01.11.018 |

Time: |
15:00 - 16:00 |

## Martin Schreiber: Massively parallel rational approximation of time integrations with Cauchy contour integral methods for Weather and Climate simulations

The ongoing trend in high-performance computing of increasing numbers of cores, increased vector lengths as well as stagnating or even declining processor speeds poses new challenges to solve PDEs within a limited time frame. Here, new mathematical reformulations are required and gained increasing interest over the last two decades.

One of such reformulations is a rational approximation of exponential integrators (REXI) which is the main focus of this talk. A REXI formulation allows to replace a purely sequential time integration (e.g. using Runge-Kutta) for linear oscillatory or diffusive systems by a sum of solvers for decoupled systems of equations. Each of the terms in the sum can then be solved independently, hence massively parallel for arbitrarily long time step sizes for linear operators. We will infer the coefficients for the REXI time integration using Cauchy contour integral methods. Here, a physical understanding of the diffusive and oscillatory behavior of the underlying system is mandatory to avoid numerical cancellation effects.

Studies are conducted with the linear and non-linear shallow-water equations on the rotating sphere. These equations represent a simplified model of the real atmosphere, putting the focus on horizontal time integration challenges. Results on timestepsize-to-error as well as wallclocktime-to-error will be discussed in detail, revealing sweetspots of exponential integrators. The results motivate further explorations of REXI for operational weather/climate systems.