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Hugo Jiménez-Pérez

Publications and source records attributed to Hugo Jiménez-Pérez.

10 recordsLinked to original sources
arXiv2020

Towards exact symplectic integrators from Liouvillian forms

In this article we introduce a low order implicit symplectic integrator designed to follow the Hamiltonian flow as close as possible. This integrator is obtained by the method of Liouvillian forms and does not require particular hypotheses on the Hamiltonian. The numerical scheme introduced in this paper is a modification of the symplectic mid-point rule, it is symmetric and it is obtained by an isotopy of the deformation of the exact Hamiltonian flow to the straight line passing by two consecutive points of the discretized flow. This isotopy generates an alternative vector field on the flow lines transversal to the Hamiltonian vector field. We consider only the line arising from the mid-point to construct the symplectic integrator.

math.SG↗
arXiv2019

On the Poincaré's generating function and the symplectic mid-point rule

The use of Liouvillian forms to obtain symplectic maps for constructing numerical integrators is a natural alternative to the method of generating functions, and provides a deeper understanding of the geometry of this procedure. Using Liouvillian forms we study the generating function introduced by Poincaré (1899) and its associated symplectic map. We show that in this framework, Poincaré's generating function does not correspond to the symplectic mid-point rule, but to the identity map. We give an interpretation of this result based on the original framework constructed by Poincaré.

math-ph↗
arXiv2017

Symplectic maps: from generating functions to Liouvillian forms

In this article we introduce a new method for constructing implicit symplectic maps using special symplectic manifolds and Liouvillian forms. This method extends, in a natural way, the method of generating functions to 1-forms which are globally defined on the symplectic manifold. The maps constructed by this method, are related to the symplectic Cayley's transformation and belong to a continuous space of dimension n(2n+1). Applying the implicit map to the discrete Hamilton equations we obtain the generalized symplectic Euler scheme. We show the relations of the elements of this family with other discrete symplectic mapping, in particular 1) with the mappings obtained by generating functions of type I, II, and III and IV; 2) with the symplectic Euler methods A and B; and 3) with the mid-point rule. Moreover, we show the corresponding symplectic diffeomorphisms and their Liouvillian forms on the product symplectic manifold. We illustrate the details of the method in constructing two different families of implicit symplectic maps for $n=1$. This is a geometrical method which overcomes the difficulties of the Hamilton-Jacobi theory and generating functions.

math.SG↗
arXiv2015

Geometrization of symplecticity conditions for implicit schemes

In this note we give simple symplecticity conditions for implicit schemes in the linear case. We consider implicit maps on generic symplectic manifold and we introduce the concept of consistent implicit maps, to generalize the symplecticity conditions to symplectic manifolds. Additionally, we give a preliminary geometrical interpretation of those conditions.

math.SG↗
arXiv2015

New insights on numerical error in symplectic integration

We implement and investigate the numerical properties of a new family of integrators connecting both variants of the symplectic Euler schemes, and including an alternative to the classical symplectic mid-point scheme, with some additional terms. This family is derived from a new method, introduced in a previous study, for generating symplectic integrators based on the concept of special symplectic manifold. The use of symplectic rotations and a particular type of projection keeps the whole procedure within the symplectic framework. We show that it is possible to define a set of parameters that control the additional terms providing a way of "tuning" these new symplectic schemes. We test the "tuned" symplectic integrators with the perturbed pendulum and we compare its behavior with an explicit scheme for perturbed systems. Remarkably, for the given examples, the error in the energy integral can be reduced considerably. There is a natural geometrical explanation, sketched at the end of this paper. This is the subject of a parallel article where a finer analysis is performed. Numerical results obtained in this paper open a new point of view on symplectic integrators and Hamiltonian error.

math.NA↗
arXiv2011

Symplectic integrators in the realm of Hofer's geometry

Symplectic integrators constructed from Hamiltonian and Lie formalisms are obtained as symplectic maps whose flow follows the exact solution of a "sourrounded" Hamiltonian K = H + h^k H_1. Those modified Hamiltonians depends virtually on the time by h. When the numerical integration of a Hamiltonian system involves more than one symplectic scheme as in the parallel-in-time algorithms, there are not a simple way to control the dynamical behavior of the error Hamiltonian. The interplay of to different symplectic integrators can degenerate their behavior if both have different dynamical properties, reflected in the number of iterations to approximate the sequential solution. Considered as flows of time-dependent Hamiltonians we use the Hofer's geometry to search for the optimal coupling of symplectic schemes. As a result we obtain the constraints in the Parareal method to have a good behavior for Hamiltonian dynamics.

math.SG↗
arXiv2011

On the self-CPG curves and the Björling problem

Schwartz's solution to the Björling problem leads to an equivalence class of spatial strips S(t)=(c(t),n(t)) which produce equivalent minimal surfaces. For the particular case when the generating strip S(t) belongs to some plane E and c(t) is symmetric with respect to some straight line in E, the symmetries of the minimal surface permit us to identify another planar curve ~c(t) that we call the CPG curve to c(t). A simple symmetric argument shows that self-CPG curves produce minimal surfaces whose adjoint surface contains another self-CPG curve. We ask for minimal surfaces generated by self-CPG curves which are self-adjoints.

math.DG↗
arXiv2011

A time-parallel algorithm for almost integrable Hamiltonian systems

We introduce a time-parallel algorithm for solving numerically almost integrable Hamiltonian systems in action-angle coordinates. This algorithm is a refinement of that introduced by Saha, Stadel and Tremaine in 1997 (SST97) for the same type of problems. Our refined algorithm has a better convergence obtained from the use of derivatives of the perturbing term not considered in the original SST97 algorithm. An advantage of this algorithm is its independence of the step-size for the parallelized procedures which can be consider as a particular case of the parareal scheme.

math.NA↗
arXiv2011

On Alfred Gray's Elliptical Catenoid

We give a parameterization of Alfred Gray's Elliptical Catenoid and Elliptical Hellicoid using Jacobi's elliptic functions. This parameterization avoids some problems present in the original depiction of these surfaces.

math.DG↗