Abstract

Solving partial differential equations is one of the most traditional tasks in scientific computing. In this work, we consider numerical solutions of initial value problems partly or entirely given by linear PDEs and how to compute solutions with a method we refer to as Rational Approximation of Exponential Integration (REXI). REXI replaces a typically sequential timestepping method with a sum of rational terms, leading to the possibility of parallelizing over this sum. Hence, this method can potentially exploit additional degrees of parallelization for scaling problems limited in their spatial scalability to large-scale supercomputers.

We present the “unified REXI” method in which we show algebraic equivalence to other methods developed up to five decades ago. Such methods cover, e.g., diagonalization of the Butcher table for implicit Runge–Kutta methods, Cauchy-contour integration-based methods, and direct approximations. In the present work, we target the hyperbolic problems considered to be a particularly challenging task. We provide for the first time a deep numerical investigation, discussion, and comparison of all these methods. In particular, we account for numerical problems and, if possible, workarounds for them. Finally, we demonstrate and compare the performance of REXI with off-the-shelf time integrators using the nonlinear shallow-water equations on the rotating sphere on a high-performance computing system.

While previous REXI studies have focused on exposing more parallelism to enable faster time to solution, we also consider computing resource efficiency at prescribed accuracy and find that diagonalized lower-order Gauss Runge–Kutta methods (formulated as REXI) are compelling highly efficient methods leading to a 64-fold reduction of the required computational resources compared to existing work.

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