Parallel exponential Rosenbrock methods

Further development of efficient and accurate time integration schemes for meteorological models

“…These include a fourth-order twostage scheme, a fourth-order parallel stages scheme, and a fifth-order threestage scheme. Our main references in this section are [26,27,28,29,30].…”

“…We note that this uses only two stages, and is therefore considered as a superconvergent scheme. Second, we consider a fourth-order 3-stage scheme satisfying the stiff order conditions, named pexprb43 in [29]:…”

“…The primary purpose of this work is, therefore, to explore time integration schemes that can retain the positive characteristics of the methods outlined in [2], while offering a higher-accuracy approximation of the nonlinearity that remains after dynamic linearization. One of the most promising candidates to fulfill theese requirements are the exponential Rosenbrock methods [26,27,28,29,30]. This paper focuses on the use of such methods for solving the meteorological equations, including both additional optimizations to the phipm algorithm (which results in a new routine called phipm simul iom2) for exponential Rosenbrock methods, as well as investigations of their accuracy and efficiency on a range of applicable test problems.…”

Fourth-order two-stage explicit exponential integrators for time-dependent PDEs

“…In order to overcome the two mentioned issues of such classical explicit and implicit methods, exponential integrators has been introduced (see the review paper [3] for details). This field has grown significantly since 1998 and it has been shown that the integrators are highly competitive, see for example [4,5,6,7,8,9,10,11,12,13,14]. High-order exponential integrators for stiff problems have been proposed in [15].…”

“…The idea is, first to make a continuous linearization of the vector field F along the numerical solution u n of (1.1) (due to Pope [17]) leading to semilinear problems u ′ (t) = J n u(t) + g n (u(t)) (1.2) with the Jacobian J n and the nonlinearity g n are J n = ∂F ∂u (u n ), g n (u) = F (u) − J n u, (1.3) and then to apply exponential Runge-Kutta methods [6] to (1.2) which resulted in exponential Rosenbrock methods. They have been studied intensively in a series of papers [16,18,9,13]. Methods up to order 6 have been derived in [13] and the stiff order conditions for methods up to arbitrary order are given in [19].…”