Splitting and composition methods in the numerical integration of differential equations

We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable. This class of integrators are explicit, simple to implement and preserve structural properties of the system. In consequence, they are specially useful in geometric numerical integration. In addition, the numerical solution obtained by splitting schemes can be seen as the exact solution to a perturbed system of ODEs possessing the same geometric properties as the original system. This backward error interpretation has direct implications for the qualitative behavior of the numerical solution as well as for the error propagation along time. Closely connected with splitting integrators are composition methods. We analyze the order conditions required by a method to achieve a given order and summarize the different families of schemes one can find in the literature. Finally, we illustrate the main features of splitting and composition methods on several numerical examples arising from applications.Comment: Review paper; 56 pages, 6 figures, 8 table

A general formula for the Magnus expansion in terms of iterated integrals of right-nested commutators

We present a general expression for any term of the Magnus series as an iterated integral of a linear combination of independent right-nested commutators with given coefficients. The relation with the Malvenuto--Reutenauer Hopf algebra of permutations is also discussed.Comment: 16 page

On time-dependent perturbation theory in matrix mechanics and time averaging

The time-dependent quantum perturbation theory developed by Born, Heisenberg and Jordan in 1926 is revisited. We show that it not only reproduces the standard theory formulated in the interaction picture, but also allows one to construct more accurate approximations if time averaging techniques are employed. The theory can be rendered unitary even if the expansion is truncated by using a transformation previously suggested by Heisenberg. We illustrate the main features of the procedure on a simple example which clearly shows its advantages in comparison with the standard perturbation theory

Strategic Alliances Of Small And Medium Entrepreneurs A Challenge For The Colective Modernization Of Transport In The City Of Bogota

The public collective transportation of the city of Bogotá is facing a new reality, caused by a significant advance in the urban development, as well as, making progress of the mobility which expressed by the implementation of in a new system of massive transport for the city called TRANSMILENIO. In response to the challenge that this new urban dynamics has generated, the entrepreneurs of the collective transport of the city of Bogotá have made them aware of the urgent need to modernize their sector, in order to, succeed in being competitive within the new concept of public transport. This complex process of modernization of the collective transport has raised the need to create new paradigms of business management which include the construction of strategic alliances of small and medium entrepreneurs of the collective transportation of the city. This process has relied on the participation of the academy, local authorities and the support of programs that Inter American Development Bank has been implementing in the area of managerial partnership. The article on consideration shows the process of modernization that is given inside the collective companies of transports in the city of Bogotá, taking as a main referring the experience of “ Alliance Tránsfer” which considers the construction of new paradigms of business management, quality of the service, social managerial responsibility and the construction of citizenship. The principles on which the “Alliance Tránsfer” is getting built are based on a new conception of the urban collective public transport of passengers. “ALIANZA TRANSFER” not only includes the implementation of high standards of planning, operation, management, and quality in the service, but also the commitment in rebuilding values and the managerial social responsibility. At the same time, “Alliance Tránsfer” attempts to contribute trough the development of the collective public service of transport in the process of the social inclusion using it as a tool to fortify processes of construction and formation of the civilian population. Finally, “Alliance Tránsfer” seeks to agglutinate entrepreneurs through the sum of synergies of the transporters of the collective public sector to constitute themselves in to authentic agents of the social and economic development.Institute of Transport and Logistics Studies. Faculty of Economics and Business. The University of Sydne

A Lie-Deprit perturbation algorithm for linear differential equations with periodic coefficients

A perturbative procedure based on the Lie-Deprit algorithm of classical mechanics is proposed to compute analytic approximations to the fundamental matrix of linear di erential equations with periodic coe cients. These approximations reproduce the structure assured by the Floquet theorem. Alternatively, the algorithm provides explicit approximations to the Lyapunov transformation reducing the original periodic problem to an autonomous sys- tem and also to its characteristic exponents. The procedure is computationally well adapted and converges for su ciently small values of the perturbation pa- rameter. Moreover, when the system evolves in a Lie group, the approximations also belong to the same Lie group, thus preserving qualitative properties of the exact solution

High order integrators obtained by linear combinations of symmetric-conjugate compositions

A new family of methods involving complex coefficients for the numerical integration of differential equations is presented and analyzed. They are constructed as linear combinations of symmetric-conjugate compositions obtained from a basic time-symmetric integrator of order (). The new integrators are of order , , and preserve time-symmetry up to order when applied to differential equations with real vector fields. If in addition the system is Hamiltonian and the basic scheme is symplectic, then they also preserve symplecticity up to order . We show that these integrators are well suited for a parallel implementation, thus improving their efficiency. Methods up to order 10 based on a 4th-order integrator are built and tested in comparison with other standard procedures to increase the order of a basic scheme.This work has been funded by Ministerio de Ciencia e Innovación (Spain) through project PID2019-104927GB-C21 (AEI/FEDER, UE) and by Universitat Jaume I (UJI-B2019-17). A.E.-T. has been additionally supported by the predoctoral contract BES-2017-079697 (Spain)

Supporting End-User Development through a New Composition Model: An Empirical Study

End-user development (EUD) is much hyped, and its impact has outstripped even the most optimistic forecasts. Even so, the vision of end users programming their own solutions has not yet materialized. This will continue to be so unless we in both industry and the research community set ourselves the ambitious challenge of devising end to end an end-user application development model for developing a new age of EUD tools. We have embarked on this venture, and this paper presents the main insights and outcomes of our research and development efforts as part of a number of successful EU research projects. Our proposal not only aims to reshape software engineering to meet the needs of EUD but also to refashion its components as solution building blocks instead of programs and software developments. This way, end users will really be empowered to build solutions based on artefacts akin to their expertise and understanding of ideal solution

Composition Methods for Dynamical Systems Separable into Three Parts

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

On the Linear Stability of Splitting Methods

A comprehensive linear stability analysis of splitting methods is carried out by means of a 2 × 2 matrix K(x) with polynomial entries (the stability matrix) and the stability polynomial p(x) (the trace of K(x) divided by two). An algorithm is provided for determining the coefficients of all possible time- reversible splitting schemes for a prescribed stability polynomial. It is shown that p(x) carries essentially all the information needed to construct processed splitting methods for numerically approximating the evolution of linear systems. By selecting conveniently the stability polynomial, new integrators with processing for linear equations are built which are orders of magnitude more efficient than other algorithms previously available