Composition Methods for Dynamical Systems Separable into Three Parts

Abstract

New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a way that each part is explicitly solvable. The methods are obtained by applying different optimization criteria and preserve geometric properties of the continuous problem by construction. Different numerical examples exhibit their improved performance with respect to previous splitting methods in the literature. used in GNI because the composition of symplectic (or volume preserving, orthogonal, etc.) transformations is again symplectic (volume preserving, orthogonal, etc., respectively). In composition methods the numerical scheme is constructed as the composition of several simpler integrators for the problem at hand, so as to improve their accuracy.When in (1) can be separated into two parts, very efficient splitting schemes have been designed and applied to solve a wide variety of problems arising in several fields, ranging from Hamiltonian Monte Carlo techniques to the evolution of the -body gravitational problem in Celestial Mechanics (see [3,4] and references therein).There are, however, relevant problems in applications where has to be decomposed into three or more parts in order to have subproblems that are explicitly solvable. Examples include the disordered discrete nonlinear Schrödinger equation [7], Vlasov-Maxwell equations in plasma physics [8], the motion of a charged particle in an electromagnetic field according with the Lorentz force law [9] and problems in molecular dynamics [10]. In that case, although in principle methods of any order of accuracy can be built, the resulting algorithms involve such a large number of maps that they are not competitive in practice. It is the purpose of this paper to present an alternative class of efficient methods for the problem at hand and compare their performance on some non-trivial physical examples than can be split into three parts.The paper is structured as follows. We first review how splitting methods can be directly applied to get numerical solutions (Section 2). Then the attention is turned to the application of composition methods, and we get a family of 4th-order schemes obtained by applying a standard optimization procedure (Section 3). In Section 4 we show how standard splitting methods, when formulated as a composition scheme, lead to very competitive integrators, and also propose a different optimization criterion for systems possessing invariant quantities. This allows us to get a new family of 4th-order schemes. All these integration algorithms are subsequently tested in Section 5 on a pair of numerical examples. Finally, Section 6 contains some concluding remarks.

Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow

“…Such adapted weights are initially set to zero. Therefore, it is easy to prove that {∂e(xk)/∂wk} has a positive definite value, and consequently, (14) has been directly utilized.…”

“…One of the common methods in geometric integrators is the composition method [14]. It is based on the composition of several simpler integrators of the problem in order to increase the degree of accuracy of the ODE solver.…”

“…It is based on the composition of several simpler integrators of the problem in order to increase the degree of accuracy of the ODE solver. Fourthorder composition methods for the numerical integration of IVPs defined by ODEs for dynamical systems were proposed [14]. Some other analytical techniques were proposed such as the Differential Transform Method (DTM).…”

Comparison of high-order Eulerian methods for electron hybrid model

“…Let us recall that the Strang method is second order accurate whereas the Suzuki method is fourth order. We refer to [29] for other composition methods designed from a three terms decomposition. We observe that the number of stages increases dramatically when high order are considered which is, as we will see, a drawback of these methods.…”

Semi-Implicit and Semi-Explicit Adams-Bashforth-Moulton Methods

“…These algorithms were initially designed to preserve the geometric properties of the simulated continuous system in the discrete model. However, later it was discovered that even in the case of non-Hamiltonian systems, extrapolation [14] and composition [15] ODE solvers based on symmetric and semi-implicit methods possess lower solution error and computational costs than their explicit and implicit counterparts [16][17][18][19]. The hypothesis of this study is that introducing semi-implicit integration can improve the performance of multistep ODE solvers because the semi-implicit Euler method possesses better precision and stability than an explicit Euler integrator.…”