A. Murua

A. Murua, J. M. Sanz-Serna

In the first part of the present work we consider periodically or quasiperiodically forced systems of the form $(d/dt)x = εf(x,t ω)$, where $ε\ll 1$, $ω\in\mathbb{R}^d$ is a nonresonant vector of frequencies and $f(x,θ)$ is $2π$-periodic in each of the $d$ components of $θ$ (i.e.\ $θ\in\mathbb{T}^d$). We describe in detail a technique for explicitly finding a change of variables $x = u(X,θ;ε)$ and an (autonomous) averaged system $(d/dt) X = εF(X;ε)$ so that, formally, the solutions of the given system may be expressed in terms of the solutions of the averaged system by means of the relation $x(t) = u(X(t),tω;ε)$. Here $u$ and $F$ are found as series whose terms consist of vector-valued maps weighted by suitable scalar coefficients. The maps are easily written down by combining the Fourier coefficients of $f$ and the coefficients are found with the help of simple recursions. Furthermore these coefficients are {\em universal} in the sense that they do not depend on the particular $f$ under consideration. In the second part of the contribution, we study problems of the form $(d/dt) x = g(x)+f(x)$, where one knows how to integrate the "unperturbed" problem $(d/dt)x = g(x)$ and $f$ is a perturbation satisfying appropriate hypotheses. It is shown how to explicitly rewrite the system in the "normal form" $(d/dt) x = \bar g(x)+\bar f(x)$, where $\bar g$ and $\bar f$ are {\em commuting} vector fields and the flow of $(d/dt) x = \bar g(x)$ is conjugate to that of the unperturbed $(d/dt)x = g(x)$. In Hamiltonian problems the normal form directly leads to the explicit construction of formal invariants of motion. Again, $\bar g$, $\bar f$ and the invariants are written as series consisting of known vector-valued maps and universal scalar coefficients that may be found recursively.

S. Blanes, F. Casas, A. Murua

We present a practical algorithm based on symplectic splitting methods to integrate numerically in time the Schrödinger equation. When discretized in space, the Schrödinger equation can be recast as a classical Hamiltonian system corresponding to a generalized high-dimensional separable harmonic oscillator. The particular structure of this system combined with previously obtained stability and error analyses allows us to construct a set of highly efficient symplectic integrators with sharp error bounds and optimized for different tolerances and time integration intervals. They can be considered, in this setting, as polynomial approximations to the matrix exponential in a similar way as methods based on Chebyshev and Taylor polynomials. The theoretical analysis, supported by numerical experiments, indicates that the new methods are more efficient than schemes based on Chebyshev polynomials for all tolerances and time intervals. The algorithm we present incorporates the new splitting methods and automatically selects the most efficient scheme given a tolerance, a time integration interval and an estimate on the spectral radius of the Hamiltonian.

A. Murua, J. M. Sanz-Serna

In a series of papers the present authors and their coworkers have developed a family of algebraic techniques to solve a number of problems in the theory of discrete or continuous dynamical systems and to analyze numerical integrators. Given a specific problem, those techniques construct an abstract, {\em universal} version of it which is solved algebraically; then, the results are tranferred to the original problem with the help of a suitable morphism. In earlier contributions, the abstract problem is formulated either in the dual of the shuffle Hopf algebra or in the dual of the Connes-Kreimer Hopf algebra. In the present contribution we extend these techniques to more general Hopf algebras, which in some cases lead to more efficient computations.