Rational Approximation of Exponential Integration (REXI) for dynamical cores

This presentation focuses on numerical solutions for Initial Value Problems (IVPs) involving linear PDEs dominating the time step size, as is the case for dynamical cores. We investigate using Rational Approximation of Exponential Integration (REXI). REXI replaces sequential time-stepping with a sum of rational terms, enabling parallelization and exploiting additional scalability on supercomputers for spatially limited problems.

We introduce the "unified REXI" method, showing its algebraic equivalence to methods developed decades ago, such as implicit Runge-Kutta methods, Cauchy-contour integration, and direct approximations. Our studies involve basic test cases for dynamical cores, offering a detailed numerical investigation, discussion, and comparisons of REXI methods. We address numerical issues and propose workarounds where feasible. Performance comparisons are conducted using nonlinear shallow-water equations on a rotating sphere on high-performance computing systems.

In addition to exposing more parallelism for faster solutions, we evaluate resource efficiency at prescribed accuracy. Our findings reveal that diagonalized lower-order Gauss Runge-Kutta methods (formulated as REXI) achieve a 64x reduction in computational resource requirements compared to prior work.