Nonlinear Extension of Multiproduct Expansion Schemes and Applications to Rigid Bodies

In this paper we discuss time integrators for nonlinear differential equations. In recent years, splitting approaches have become an important tool for reducing the computational time needed to solve differential equations. Moreover, nonlinearity is a challenge to splitting schemes, while one has to extend the exp-functions in terms of a nonlinear Magnus expansion. Here we discuss a novel extension of the so-called multiproduct expansion methods, which is used to improve the standard Strang splitting schemes as to their nonlinearity. We present an extension of linear splitting schemes and concentrate on nonlinear systems of differential equations and generalise in this respect the recent MPE method; see (Chin and Geiser, 2011). Some first numerical examples, of rigid body problems, are given as benchmarks. 1. Introduction In recent years, applications to nonlinear differential equations of multiscale problems, for example, the rigid body (see [1]), have arisen and become important. Here, splitting schemes are important for decoupling the different nonlinear scales, for example, a Hamiltonian with kinetic and potential operators, and treating them with the best solver schemes; see [2, 3]. Theoretically, we deal with subproblems which can be solved independently, but we have taken into account their nonlinearity, so that standard fundamental solutions, for example, -functions, of the subproblems have to be extended in respect to their nonlinearity. Here we apply the nonlinear Magnus expansion; see [4, 5]. Our contributions are to use the first and second order schemes and extend them with an extrapolation scheme, in our case with a multiproduct expansion, to gain more accurate and efficient higher order schemes; see [6]. In this paper we concentrate on approximate solutions of the nonlinear evolution equation; for example, with unbounded operators and . We have further , . We assume there are suitable chosen subspaces of the underlying Banach space such that . For such equations, we concentrate on applying nonlinear splitting schemes to extrapolation ideas to obtain higher order schemes. Here, we deal with Suzuki's methods and apply factorised symplectic algorithms with forward derivatives; see [6, 7]. The exact solution of the evolution problem (1) is with the evolution operator depending on the actual time and the initial value . We use a formal notation: Here the evolution operator and the Lie derivative associated with are for any unbounded nonlinear operator with Frechet’s derivative . The paper is outlined as follows. In Section 2, we discuss the