Papers by Francisco Villatoro
International Conference on Applied Mathematics, Dec 16, 2005
The numerical simulation of the propagation of nonlinear waves may present numericallyinduced rad... more The numerical simulation of the propagation of nonlinear waves may present numericallyinduced radiation. Compactons, solitary waves with compact support, are no exception. The numerical radiation generated by compactons of the Rosenau-Hyman K(2, 2) equation calculated by means of a fourth-order finite element method is illustrated. Small-amplitude forward and backward radiation are shown in the simulations, both having self-similar envelope profiles and high frequency carriers. The amplitude and velocity of the envelope of both radiations are determined.
The numerical simulation of colliding solitary waves with compact support arising from the Rosena... more The numerical simulation of colliding solitary waves with compact support arising from the Rosenau-Hyman K(n,n) equation requires the addition of artificial dissipation for stability. The price to pay is the appearance of trailing tails, amplitude damping, and delays as the solution evolves. These undesirable effects can be corrected by properly counterbalancing two sources of artificial dissipation; this procedure is designed by using the slow time evolution of the parameters of the solitary waves under the presence of the dissipation determined by means of adiabatic perturbation methods. The tails are completely removed leaving only a small compact ripple at the original position of their front. This tail removal methodology preserves the numerical stability of the scheme under compacton collisions, as shown by extensive numerical experiments for several values of n.
arXiv (Cornell University), Oct 27, 2009
Fractal sets, by definition, are non-differentiable, however their dimension can be continuous, d... more Fractal sets, by definition, are non-differentiable, however their dimension can be continuous, differentiable, and arithmetically manipulable as function of their construction parameters. A new arithmetic for fractal dimension of polyadic Cantor sets is introduced by means of properly defining operators for the addition, subtraction, multiplication, and division. The new operators have the usual properties of the corresponding operations with real numbers. The combination of an infinitesimal change of fractal dimension with these arithmetic operators allows the manipulation of fractal dimension with the tools of calculus.
arXiv (Cornell University), Feb 22, 2008
The Parallel Virtual Machine (PVM) tool has been used for a distributed implementation of Greg Wa... more The Parallel Virtual Machine (PVM) tool has been used for a distributed implementation of Greg Ward's Radiance code. In order to generate exactly the same primary rays with both the sequential and the parallel codes, the quincunx sampling technique used in Radiance for the reduction of the number of primary rays by interpolation, must be left untouched in the parallel implementation. The octree of local ambient values used in Radiance for the indirect illumination has been shared among all the processors. Both static and dynamic image partitioning techniques which replicate the octree of the complete scene in all the processors and have load-balancing, have been developed for one frame rendering. Speedups larger than 7.5 have been achieved in a network of 8 workstations. For animation sequences, a new dynamic partitioning distribution technique with superlinear speedups has also been developed.
The European Symposium on Artificial Neural Networks, 2005
Abstract. The exponential stability of continuous-time Hopfield neural networks is not preserved ... more Abstract. The exponential stability of continuous-time Hopfield neural networks is not preserved when implemented on digital computers by means of explicit numerical methods, whereas the implicit (or backward) Euler method preserves this exponential stability under exactly the same sufficient conditions as those previously obtained for the continuous model. The proof is based on the nonlinear measure approach, here extended to discrete-time systems. This approach also allows the estimation of the exponential convergence rate of ...
International Journal of Computer Mathematics, Apr 1, 2010
A new class of weighted, homogeneous, differentiable means is introduced, referred to as θ-means,... more A new class of weighted, homogeneous, differentiable means is introduced, referred to as θ-means, which may be extended to the whole plane. These means are applied to Evans–Sanugi nonlinear one-step methods for initial value problems of scalar differential equations. Their general local truncation error is obtained, showing the first order of these methods for θ≠ 1/2 and second order for θ= 1/2. Numerical results for scalar DETEST problems using Arithmetic, Harmonic, Contraharmonic, Quadratic, Geometric, Heronian, Centroidal and ...
International Journal of Computer Mathematics, 2010
The linear stability analysis of non-linear one-step methods based on means is studied by means o... more The linear stability analysis of non-linear one-step methods based on means is studied by means of the concept of stability regions and order stars. Concretely, non-linear θ-methods based on harmonic, contraharmonic, quadratic, geometric, Heronian, centroidal and logarithmic means are considered. Their stability diagrams and order stars show their A-stability for θ≥ 1/2, and L-stability in some cases. Order stars in the Riemann surface are a requirement for non-linear one-step methods. The advantages and disadvantages of this ...
Applied Mathematics and Computation, 2012
Imperfections in the geometry of photonic crystal wires (PCWs) are always a result of current man... more Imperfections in the geometry of photonic crystal wires (PCWs) are always a result of current manufacture techniques. The influence of such random defects on the propagating modes in PCWs is studied computationally by means of a full vectorial finite-element method using a perfectly matched layer (PML) as absorbing boundary condition, and the numerical results show a large sensitivity of the birefringence to the geometrical defects in the transversal section of the fibre. It has also been found that imperfections in PCWs have a greater effect on wave propagation than those in photonic crystal fibres (PCFs).
Applied Mathematics and Computation, Oct 1, 2019
The sine-Gordon equation turn up in several problems in science and engineering. Although it is i... more The sine-Gordon equation turn up in several problems in science and engineering. Although it is integrable, in practical applications, its numerical solution is powerful and versatile. Four novel implicit finite difference methods based on (q , s) Padé approximations with (q + s) th order in space have been developed and analyzed for this equation; all share the same treatment for the nonlinearity and integration in time. Concretely, (0,4), (2,2), (2,4), and (4,4) Padé methods; additionally, the energy conserving, Strauss-Vázquez scheme has been considered in a (0,2) Padé implementation. These methods have been compared among them for both the kink-antikink and breather solutions in terms of global error, computational cost and energy conservation. The (0,4) and (2,4) Padé methods are the most cost-effective ones for small and large global error, respectively. Our results indicate that spatial order of accuracy is more relevant to effectiveness of a method than energy conservation even in very long time integrations.
Understanding complex systems, 2018
Gravitational wave astronomy is born in 2016. The laser interferometers of Advanced LIGO have det... more Gravitational wave astronomy is born in 2016. The laser interferometers of Advanced LIGO have detected two gravitational waves, each one generated by two black hole pairs. The observed wave profiles result from the fusion of two stellar-mass black holes into a single rotating black hole, with the emission of gravitational radiation with energy in the solar-mass scale. Indeed, they are the most violent astrophysical events recorded to date. Since gravitational waves solve the weak-field approximation of the Einstein equations in vacuum, in this limit, they evolve as linear waves. However, gravitational waves are intrinsically nonlinear waves; in fact, the chirp of the signal, the change in frequency observed by Advanced LIGO detectors, is due to the nonlinearity at the sources, even being negligible far away from them. Both cylindrical and planar nonlinear gravitational waves can be interpreted as soliton solutions of Einstein’s equations outside the sources. Actually, even black holes, the main sources of gravitational radiation, are two-soliton solutions of Einstein’s equations in vacuum. Gravitational solitons differ from standard nonlinear solitons in several aspects, including new phenomena such as multi-soliton coalescence, a phenomenon that emits low-amplitude radiation. Indeed, the pair-of-pants solution for the fusion of two black holes can be interpreted in such a way. In conclusion, although gravitational waves propagate in spacetime like linear waves, at their sources they are nonlinear gravitational waves and, even, gravitational solitons.
Applied Mathematics and Computation, Nov 1, 1999
The modi®ed equation technique is extended to two-point, boundary-value problems, and a second-or... more The modi®ed equation technique is extended to two-point, boundary-value problems, and a second-order accurate, implicit, centered, ®nite dierence scheme for nonhomogeneous, second-order, ordinary dierential equations with linear boundary conditions is analyzed. The ®rst, second and third modi®ed equations, or equivalent, second equivalent and (simply) modi®ed equations, respectively, for this scheme and its boundary conditions are presented. It is shown that the three kinds of modi®ed equations are asymptotically equivalent when the equivalent equation is used for the boundary conditions, since an asymptotic analysis of these equations with the grid size as small parameter yields exactly the same results. For a linear problem, multiple scales and summed-up asymptotic techniques are used and the resulting uniform asymptotic expansions are shown to be equivalent to the solution of the original ®nite dierence scheme. Asymptotic successive-corrections techniques are also applied to the three kinds of modi®ed equations to obtain higher-order schemes. Higher-order boundary conditions are easily treated in the asymptotic successive-corrections technique, although these boundary conditions must be obtained by using the equivalent equation in order to obtain a correct estimate of the global error near the domain boundaries. The methods introduced in this paper are applied to homogeneous and non-homogeneous, second-order, linear and non-linear, ordinary dierential equations, and yield very accurate results.
Applied Mathematics and Computation, Oct 1, 2008
The analytical solution of a one-dimensional problem of heat transfer between an inert gas and an... more The analytical solution of a one-dimensional problem of heat transfer between an inert gas and an ultralow thermal conductivity porous medium is obtained by means of the Laplace transform method. The solution shows a sharp heat front propagation through the medium during its heating or cooling depending on its initial temperature.
Computer Physics Communications, Jul 1, 2009
Generalized finite difference methods require that a properly posed set of nodes exists around ea... more Generalized finite difference methods require that a properly posed set of nodes exists around each node in the mesh, so that the solution for the corresponding multivariate interpolation problem be unique. In this paper we first show that the construction of these meshes can be computerized using a relatively simple algorithm based on the concept of a Coatmèlec lattice. Then, we present a generalized finite difference method which provides a numerical solution of a partial differential equation over an arbitrary domain, using the generated meshes. The accuracy and mesh adaptivity of the method is evaluated using elliptical equations in several domains.
Applied Mathematics and Computation, Aug 1, 1999
The equivalent, second equivalent and (simply) modi®ed equations for the implicit midpoint rule a... more The equivalent, second equivalent and (simply) modi®ed equations for the implicit midpoint rule are shown to be asymptotically equivalent in the sense that an asymptotic analysis of these equations with the time step size as small parameter yields exactly the same results; for linear problems with constant coecients, they are also equivalent to the original ®nite dierence scheme. Straightforward (regular), multiple scales and summed-up asymptotic techniques are used for the analysis of the implicit midpoint rule dierence method, and the accuracy of the resulting asymptotic expansion is assessed for several ®rst-order, non-linear, autonomous ordinary dierential equations. It is shown that, when the resulting asymptotic expansion is uniformly valid, the asymptotic method yields very accurate results if the solution of the leading order equation is smooth and does not blow up. The modi®ed equation is also studied as a method for the development of new numerical schemes based on both direct-correction and asymptotic successive-correction techniques applied to the three kinds of modi®ed equations, the linear stability of these techniques is analyzed, and their results are compared with those of Runge±Kutta schemes for several autonomous and non-autonomous, ®rst-order, ordinary dierential equations.
Applied Mathematics and Computation, Aug 1, 1999
The modi®ed equation method is studied as a technique for the development of new numerical techni... more The modi®ed equation method is studied as a technique for the development of new numerical techniques for ordinary dierential schemes based on the third modi®ed or (simply) modi®ed equation of the explicit Euler forward method. Both direct-correction and successive-correction techniques based on the modi®ed equation are used to obtain higher-order schemes. The resulting numerical techniques are completely explicit, of order of accuracy as high as desired, and self-starting since the truncation error terms in the modi®ed equation have no derivatives. The methods introduced in this paper are applied to autonomous and non-autonomous, scalar and systems of ordinary dierential equations and compared with second-and fourth-order accurate Runge±Kutta schemes. It is shown that, for suciently small step sizes, the fourth-order direct-correction and successive-correction methods are as accurate as the fourth-order Runge± Kutta scheme.
Journal of Mathematical Analysis and Applications, Feb 1, 2011
An approximate analytical solution for the one-dimensional problem of heat transfer between an in... more An approximate analytical solution for the one-dimensional problem of heat transfer between an inert gas and a porous semi-infinite medium is presented. Perturbation methods based on Laplace transforms have been applied using the solid thermal conductivity as small parameter. The leading order approximation is the solution of Nusselt (or Schumann) problem. Such solution is corrected by means of an outer approximation. The boundary condition at the origin has been taking into account using an inner approximation for a boundary layer. The gas temperature presents a discontinuous front (due to the incompatibility between initial and boundary conditions) which propagates at constant velocity. The solid temperature at the front has been smoothed out using an internal layer asymptotic approximation. The good accuracy of the resulting asymptotic expansion shows its usefulness in several engineering problems such as heat transfer in porous media, in exhausted chemical reactions, mass transfer in packed beds, or in the analysis of capillary electrochromatography techniques.
Journal of Computational Physics, Jun 1, 2009
Compacton propagation under dissipation shows amplitude damping and the generation of tails. The ... more Compacton propagation under dissipation shows amplitude damping and the generation of tails. The numerical simulation of compactons by means of dissipative schemes also show the same behaviors. The truncation error terms of a numerical method can be considered as a perturbation of the original partial differential equation and perturbation methods can be applied to its analysis. For dissipative schemes, or when artificial dissipation is added, the adiabatic perturbation method yields evolution equations for the amplitude loss in the numerical solution and the amplitude of the numerically-induced tails. In this paper, such methods are applied to the Kð2; 2Þ Rosenau-Hyman equation, showing a very good agreement between perturbative and numerical results.
arXiv (Cornell University), May 27, 2023
Deflation is an efficient numerical technique for identifying new branches of steady state soluti... more Deflation is an efficient numerical technique for identifying new branches of steady state solutions to nonlinear partial differential equations. Here, we demonstrate how to extend deflation to discover new periodic orbits in nonlinear dynamical lattices. We employ our extension to identify discrete breathers, which are generic exponentially localized, time-periodic solutions of such lattices. We compare different approaches to using deflation for periodic orbits, including ones based on a Fourier decomposition of the solution, as well as ones based on the solution's energy density profile. We demonstrate the ability of the method to obtain a wide variety of multibreather solutions without prior knowledge about their spatial profile.
Chaos, Solitons & Fractals, 2021
The graphene superlattice equation, a modified sine-Gordon equation, governs the propagation of s... more The graphene superlattice equation, a modified sine-Gordon equation, governs the propagation of solitary electromagnetic waves in a graphene superlattice. This equation has kink solutions without explicit analytical expression, requiring the use of quadrature methods. The inelastic collision of kinks and antikinks with the same but opposite speed is studied numerically for the first time; after their interaction they escape to infinity when its speed is either larger than a critical value or it is inside a series of resonance windows; otherwise, they form a breather-like state that slowly decays by radiating energy. Here, the fractal structure of these resonance windows is characterized by using a multiindex notation and their main features are compared with the predictions of the resonant energy exchange theory showing good agreement. Our results can be interpreted as new evidence in favour of this theory.
Communications in Nonlinear Science and Numerical Simulation, 2020
Four novel implicit finite difference methods with (q + s)-th order in space based on Padé approx... more Four novel implicit finite difference methods with (q + s)-th order in space based on Padé approximations have been analyzed and developed for the sine-Gordon equation. Specifically, (0, 4), (2, 2), (2, 4), and (4, 4) Padé methods. All of them share the treatment for the nonlinearity and integration in time, specifically, the one that results in an energy-conserving (0, 2) Padé scheme. The five methods have been developed with and without Richardson extrapolation in time. All the methods are linearly, unconditionally stable. A comparison among them for both the kink-antikink and breather solutions in terms of global error, computational cost and energy conservation is presented. Our results indicate that the (0, 4) and (4, 4) Padé methods without Richardson extrapolation are the most cost-effective ones for small and large global error, respectively; and the (4, 4) Padé methods in all the cases when Richardson extrapolation is used.
Uploads
Papers by Francisco Villatoro