Running simulations on high-performance computers faces new challenges due to the stagnating or even decreasing per-core speed. This poses new restrictions on solving PDEs within a particular time frame. Here, disruptive mathematical reformulations which exploit additional degrees of parallelism also in the time dimension gained increasing interest over the last two decades.
This talk will focus on linear hyperbolic operators with a time integration formulated with a rational approximation of exponential integrators (REXI). Such a rational approximation replaces a CFL-limited and therefore sequential time integration for linear oscillatory and diffusive systems by a sum of solutions of decoupled systems. With this, the CFL condition can be entirely eliminated, allowing e.g. time step sizes of 1.5 days for a horizontal wave propagation on the rotating sphere (linearized shallow-water equations) if only coping with the linear operator. Additionally, they yield a significantly improved accuracy compared to traditional time integration methods, free of phase shifts. Regarding the parallelization, each of these terms in the rational approximation can then be solved independently, hence massively parallel.
Alltogether, these time integration methods motivate a further investigation and combination of them for operational weather/climate systems in order to cope with the hardware imposed restrictions of future computer architectures.
Workshop website: https://www.rcgi.poli.usp.br/stmi-workshop-2019/