We introduce a symmetric version of the Fer expansion for the solution of the ODE y 0 = (t; y)y. We show that the scheme are time symmetric for linear problems, and, because of the symmetry of the scheme, one gets order 2p + 2 whereby the classical Fer expansion would display order 2p + 1 only. This reduces signiicantly the computational cost of the classical Fer expansion. We prove the converg...

Operator splitting is a powerful concept used in many diversed elds of applied mathematics for the design of eeective numerical schemes. Following the success of the additive operator splitting (AOS) in performing an eecient nonlinear diiusion ltering on digital images, we analyze the possibility of using multiplicative operator splittings to process images from diierent perspectives. We start ...

We provide a new analytical approach to operator splitting for equations of the type ut = Au + B(u) where A is a linear operator and B is quadratic. A particular example is the Korteweg–de Vries (KdV) equation ut−uux +uxxx = 0. We show that the Godunov and Strang splitting methods converge with the expected rates if the initial data are sufficiently regular.

A time-split transport scheme has been developed for the high-order multiscale atmospheric model (HOMAM). The spacial discretization of HOMAM is based on the discontinuous Galerkin method, combining the 2D horizontal elements on the cubed-sphere surface and 1D vertical elements in a terrain-following height-based coordinate. The accuracy of the time-splitting scheme is tested with a set of new ...

Abstract It is well-known that the finite difference discretization of Laplacian eigenvalue problem ? ?u = ?u leads to a matrix (EVP) Ax ?x where A Toeplitz-plus-Hankel. Analytical solutions tridiagonal matrices with various boundary conditions are given in recent work Strang and MacNamara. We generalize results develop analytical certain generalized problems (GEVPs) ?Bx which arise from elemen...

A comparison of triple jump and Suzuki fractals for obtaining high order from an almost symmetric Strang splitting scheme Lukas Einkemmer, Alexander Ostermann We consider the time discretization of ordinary and partial differential equations. More specifically, we assume that the considered problem can be written as the following abstract Cauchy problem (1) u′ = Au+B(u), u(0) = u0. In this cont...

In this paper, we consider the nonlinear Schrödinger equation ut + i∆u− F (u) = 0 in two dimensions. We show, by an operator-theoretic proof, that the well-known Lie and Strang formulae (which are splitting methods) are approximations of the exact solution of order 1 and 2 in time.

In this paper we are concerned with the convergence analysis of splitting methods for nonautonomous abstract evolution equations. We introduce a framework that allows us to analyze the popular Lie, Peaceman– Rachford and Strang splittings for time dependent operators. Our framework is in particular suited for analyzing dimension splittings. The influence of boundary conditions is discussed.