Fernando Vadillo

An exponential time differencing method for the nonlinear Schrödinger equation

Abstract The spectral methods offer very high spatial resolution for a wide range of nonlinear wave equations, so, for the best computational efficiency, it should be desirable to use also high order methods in time but without very strict restrictions on the step size by reason of numerical stability. In this paper we study the exponential time differencing fourth-order Runge–Kutta (ETDRK4) method; this scheme was derived by Cox and Matthews in [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, J. Comp. Phys. 176 (2002) 430–455] and was modified by Kassam and Trefethen in [A. Kassam, L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comp. 26 (2005) 1214–1233]. We compute its amplification factor and plot its stability region, which gives us an explanation of its good behavior for dissipative and dispersive problems. We apply this method to the Schrodinger equation, obtaining excellent results for the cubic equation and the critical exponent case and, later, as an experimental approach to describe the various possible asymptotic behaviors with two space variables.

The Number of Catalytic Elements Is Crucial for the Emergence of Metabolic Cores

Background Different studies show evidence that several unicellular organisms display a cellular metabolic structure characterized by a set of enzymes which are always in an active state (metabolic core), while the rest of the molecular catalytic reactions exhibit on-off changing states. This self-organized enzymatic configuration seems to be an intrinsic characteristic of metabolism, common to all living cellular organisms. In a recent analysis performed with dissipative metabolic networks (DMNs) we have shown that this global functional structure emerges in metabolic networks with a relatively high number of catalytic elements, under particular conditions of enzymatic covalent regulatory activity. Methodology/Principal Findings Here, to investigate the mechanism behind the emergence of this supramolecular organization of enzymes, we have performed extensive DMNs simulations (around 15,210,000 networks) taking into account the proportion of the allosterically regulated enzymes and covalent enzymes present in the networks, the variation in the number of substrate fluxes and regulatory signals per catalytic element, as well as the random selection of the catalytic elements that receive substrate fluxes from the exterior. The numerical approximations obtained show that the percentages of DMNs with metabolic cores grow with the number of catalytic elements, converging to 100% for all cases. Conclusions/Significance The results show evidence that the fundamental factor for the spontaneous emergence of this global self-organized enzymatic structure is the number of catalytic elements in the metabolic networks. Our analysis corroborates and expands on our previous studies illustrating a crucial property of the global structure of the cellular metabolism. These results also offer important insights into the mechanisms which ensure the robustness and stability of living cells.

A mean extinction-time estimate for a stochastic Lotka-Volterra predator-prey model

The investigation of interacting population models has long been and will continue to be one of the dominant subjects in mathematical ecology; moreover, the persistence and extinction of these models is one of the most interesting and important topics, because it provides insight into their behavior. The mean extinction-time depends on the initial population size and satisfies the backward Kolmogorov differential equation, a linear second-order partial differential equation with variable coefficients; hence, finding analytical solutions poses severe problems, except in a few simple cases, so we can only compute numerical approximations (an idea already mentioned in Sharp and Allen (1998) 1]).In this paper, we study a stochastic Lotka-Volterra model (Allen, 2007) 2, p. 149]; we prove the nonnegative character of its solutions for the corresponding backward Kolmogorov differential equation; we propose a finite element method, whose Matlab code is offered in the Appendix; and, finally, we make a direct comparison between predictions and numerical simulations of stochastic differential equations (SDEs).

An integrating factor for nonlinear Dirac equations

Abstract This paper presents an efficient integrating-factor method for solving a nonlinear Dirac equation (NLD). Starting with the simplest case of one space-variable, this method, unlike other approaches proposed in the bibliography, can be easily extended to problems with more space-variables. Our algorithm is implemented in Matlab© and the numerical experiments performed reveal its effectiveness and reliability.

Extinction-time for stochastic population models

The analysis of interacting population models is the subject of much interest in mathematical ecology. Moreover, the persistence and extinction of these models is one of the most interesting and important topics, because it provides insight into their behavior. The mean extinction-time for stochastic population models considered in this paper depends on the initial population size and satisfies a stationary partial differential equation, related to the backward Kolmogorov differential equation, a linear second-order partial differential equation with variable coefficients. In this communication we review several papers where we have proposed some numerical techniques in order to estimate the mean extinction-time for stochastic population models. Besides, we will compare the theoretical predictions and numerical simulations for stochastic differential equations (SDEs). This work can be viewed as a unified review of the contributions de la Hoz and Vadillo (2012), de la Hoz et?al. (2014) and Doubova and Vadillo (2014).

Numerical simulation of the N-dimensional sine-Gordon equation via operational matrices

Abstract In this paper, we develop a numerical method for the N -dimensional sine-Gordon equation using differentiation matrices, in the theoretical frame of matrix differential equations. Our method avoids calculating exponential matrices, is very intuitive and easy to express, and can be implemented without toil in any number of spatial dimensions. Although there is currently a vast literature on the numerical treatment of the one-dimensional sine-Gordon equation, the references for the two-dimensional case are much sparser, and virtually nonexistent for higher dimensions. We apply it to a battery of two-dimensional problems taken from the literature, showing that it largely outperforms the previously existing algorithms; while for three-dimensional problems, the results seem very promising.

Numerical simulations of time-dependent partial differential equations

When a time-dependent partial differential equation (PDE) is discretized in space with a spectral approximation, the result is a coupled system of ordinary differential equations (ODEs) in time. This is the notion of the method of lines (MOL), and the resulting set of ODEs is stiff; the stiffness may be even exacerbated sometimes. The linear terms are the primarily responsible for the stiffness, with a rapid exponential decay of some modes (as in a dissipative PDE), or a rapid oscillation of some modes (as in a dispersive PDE). Therefore, for a time-dependent PDE which combines low-order nonlinear terms with higher-order linear terms, it is desirable to use a higher-order approximation both in space and in time.Along our research, we have focused on a particular case of spectral methods, the so-called pseudo-spectral methods, to solve numerically time-dependent PDEs using different techniques: an integrating factor, in de?la Hoz and Vadillo (2010); an exponential time differencing method, in de?la Hoz and Vadillo (2008); and differentiation matrices in the theoretical frame of matrix differential equations, in de?la Hoz and Vadillo (2012, 2013a,b). This paper, which is a unified review of those contributions, aims at providing a better understanding of those methods, by illustrating their variety and, more importantly, their power. Furthermore, we also give emphasis to choosing adequate schemes to advance in time.

A numerical simulation for the blow-up of semi-linear diffusion equations

Many mathematical models have the property of developing singularities at a finite time; in particular, the solution u(x, t) of the semi-linear parabolic Equation (1) may blow up at a finite time T. In this paper, we consider the numerical solution with blow-up. We discretize the space variables with a spectral method and the discrete method used to advance in time is an exponential time differencing scheme. This numerical simulation confirms the theoretical results of Herrero and Velzquez [M.A. Herrero and J.J.L. Velzquez, Blow-up behavior of one-dimensional semilinear parabolic equations, Ann. Inst. Henri Poincare 10 (1993), pp. 131–189.] in the one-dimensional problem. Later, we use this method as an experimental approach to describe the various possible asymptotic behaviours with two-space variables.

Modelling DNA Denaturation

In the recent years much attention has been devoted to find a mechanism for energy transfer and localization in the DNA molecule, in particular in the context of the DNA denaturation problem1, 2, 3. A first attempt to incorporate thermal effects in anharmonic models has been presented by Muto et al. 4, 3 who investigated the possibility of packets of energy (solitons) being generated thermally at physiological temperature and the lifetime of open states, that are the precursors of full denaturation. The models presented in these works were always dealing with a homogeneous DNA molecule. In reality DNA is a double helix built from two antiparallel linear polymers and the base in one side is complementary to the base in the other side. Guanine is associated to cytosine, adenine is associated to thymine, and they alternate in a random fashion. First sudies of energy transfer in an inhomogeneous DNA molecule have been presented by Techera et al. 5 and Muto6. In the present work, we are dealing with the local denaturation of an inhomogeneous DNA molecule.