On the use of exponential time integration methods in atmospheric models

Abstract

A B S T R A C T Exponential integration methods offer a highly accurate approach to the time integration of large systems of differential equations. In recent years, they have attracted increased attention in a number of diverse fields due to advances in their computational efficiency. This has been as a result of the use of Krylov subspace methods for the approximation of the matrix exponentials which typically arise. In this work, we investigate the potential of exponential integration methods for use in atmospheric models. Two schemes are implemented in a shallow water model and tested against reference explicit and semi-implicit methods. In a number of experiments with standard test cases, the exponential methods are found to yield very accurate solutions with time-steps far longer than even the semi-implicit method allows. The relative efficiency of the exponential integrators, which depends mainly on the choice of the specific algorithm used for the calculation of the matrix exponent, is also discussed. The future work aimed at further improvements of the proposed methodology is outlined.

Exponential integrators with parallel-in-time rational approximations for the shallow-water equations on the rotating sphere

“…EIs also represent one of the main building blocks in the asymptotic PinT approach (Haut et al [19] who used an analytical solution of the EI for the linear parts of the shallow-water equations on the plane). All these formulations require solving exponentials of matrices which can be accomplished with a Rational Approximation of Exponential Integrators (REXI) and its variants [9,11,20]. For exponentials of linear operators, such rational approximations can be used to approximate time-integrating functions with a sum over solutions for a complex-valued system of linear equations.…”

“…EIs have also been applied with Krylov subspace solvers (see [10] for an overview). Clancy et al [11] used such a solver strategy, but lost the possibility to exploit additional parallelism due to the purely sequential iterations over the Krylov subspaces. Bonaventura [12] studied local EIs by exploiting space-time locality properties of physical phenomena.…”

Improving the Laplace transform integration method

“…The analytic inversion represents the N → ∞ limit. In this case we have exact treatment of linear modes and the algorithm is among a class of exponential integration methods (Beylkin et al, 1998;Clancy and Pudykiewicz, 2013b).…”

Numerical modeling of mixing downstream of a mass slug injection using exponential Runge-Kutta integrator

“…Mincheva and Wright (2005) define the exponential integrator as 'a numerical method which involves an exponential function of the Jacobian or an approximation of it' and traces the use of EPI in the past. Recently, a number of studies to investigate the EPI schemes are published, see Krogstad (2005), Tokman (2006), Celledoni and Kometa (2009), Hochbruck and Ostermann (2010) and Clancy and Pudykiewicz (2013).…”