Exponential multistep methods of Adams-type

Abstract

The paper is concerned with the construction, implementation and numerical analysis of exponential multistep methods. These methods are related to explicit Adams methods but, in contrast to the latter, make direct use of the exponential and related matrix functions of a (possibly rough) linearization of the vector field. This feature enables them to integrate stiff problems explicitly in time.A stiff error analysis is performed in an abstract framework of linear semigroups that includes semilinear evolution equations and their spatial discretizations. A possible implementation of the proposed methods, including the computation of starting values and the evaluation of the arising matrix functions by Krylov subspace methods is discussed. Moreover, an interesting connection between exponential Adams methods and a class of local time stepping schemes is established.Numerical examples that illustrate the methods' properties are included.

Convergence of rational multistep methods of Adams-Padé type

“…According to [4,10,11], exponential p-step methods of Adams type are defined in the following way: On an equidistant grid {t n = nh, n = 1, 2, . .…”

“…-Similarly as in [11] for exponential Adams methods, we develop a convergence theory for constant step size h. Here we are not aiming at formulating results in a very general setting, but we stick to the practically relevant finite-dimensional case and formulate our results under the standard Assumption 1.1. -We show that the rational Padé version of a multistep integrator shows the same convergence behavior as the exponential version, which is a nontrivial result already under standard assumptions about the problem at hand.…”

“…For this linearized version, the stability analysis may become significantly more involved, depending on the assumptions on the problem, see [11]. -Naturally, all these extensions and generalizations are of interest also in the context of rational integrators considered here.…”

“…In particular, exponential Adams methods have been considered in detail, cf., e.g., [2,4,10,11,15], and convergence results are available for this class of methods, cf. [10,11].…”

Multi-level explicit local time-stepping methods for second-order wave equations

“…In contrast, locally explicit time-stepping methods remain fully explicit by taking smaller time-steps in the "fine" region, that is precisely where the smaller elements are located. Also known as multirate or multiple time-stepping methods in the ODE literature [9], various LTS methods have been proposed for numerical wave propagation based on classical Adams-Bashforth multistep methods [10]; they can also be interpreted as particular approximations of exponentialAdams multi-step methods [11]. Recently, Runge-Kutta based explicit LTS of arbitrarily highorder were proposed for wave propagation in [12].…”