Palindromic 3-stage splitting integrators, a roadmap

The implementation of multi-stage splitting integrators is essentially the same as the implementation of the familiar Strang/Verlet method. Therefore multi-stage formulas may be easily incorporated into software that now uses the Strang/Verlet integrator. We study in detail the two-parameter family of palindromic, three-stage splitting formulas and identify choices of parameters that may outperform the Strang/Verlet method. One of these choices leads to a method of effective order four suitable to integrate in time some partial differential equations. Other choices may be seen as perturbations of the Strang method that increase efficiency in molecular dynamics simulations and in Hybrid Monte Carlo sampling.Comment: 20 pages, 8 figures, 2 table

A stroboscopic averaging algorithm for highly oscillatory delay problems

We propose and analyze a heterogenous multiscale method for the efficient integration of constant-delay differential equations subject to fast periodic forcing. The stroboscopic averaging method (SAM) suggested here may provide approximations with \(\mathcal{O}(H^2+1/\Omega^2)\) errors with a computational effort that grows like \(H^{-1}\) (the inverse of the stepsize), uniformly in the forcing frequency Omega

From ergodic to non-ergodic chaos in Rosenzweig-Porter model

The Rosenzweig-Porter model is a one-parameter family of random matrices with three different phases: ergodic, extended non-ergodic and localized. We characterize numerically each of these phases and the transitions between them. We focus on several quantities that exhibit non-analytical behaviour and show that they obey the scaling hypothesis. Based on this, we argue that non-ergodic chaotic and ergodic regimes are separated by a continuous phase transition, similarly to the transition between non-ergodic chaotic and localized phases.Comment: 12 page

Convergence of Scalar-Tensor theories toward General Relativity and Primordial Nucleosynthesis

In this paper, we analyze the conditions for convergence toward General Relativity of scalar-tensor gravity theories defined by an arbitrary coupling function $\alpha$ (in the Einstein frame). We show that, in general, the evolution of the scalar field $(\phi)$ is governed by two opposite mechanisms: an attraction mechanism which tends to drive scalar-tensor models toward Einstein's theory, and a repulsion mechanism which has the contrary effect. The attraction mechanism dominates the recent epochs of the universe evolution if, and only if, the scalar field and its derivative satisfy certain boundary conditions. Since these conditions for convergence toward general relativity depend on the particular scalar-tensor theory used to describe the universe evolution, the nucleosynthesis bounds on the present value of the coupling function, $\alpha_0$, strongly differ from some theories to others. For example, in theories defined by $\alpha \propto \mid\phi\mid$ analytical estimates lead to very stringent nucleosynthesis bounds on $\alpha_0$ ($\lesssim 10^{-19}$). By contrast, in scalar-tensor theories defined by $\alpha \propto \phi$ much larger limits on $\alpha_0$ ($\lesssim 10^{-7}$) are found.Comment: 20 Pages, 3 Figures, accepted for publication in Class. and Quantum Gravit