Multistep variable methods for exact integration of perturbed stiff linear systems

A family of real and analytical functions with values within the ring of M(m, R) is introduced. The solution for linear systems of differential equations is expressed as a series of Φ-functions. This new multistep method is defined for variable-step and variable-order, maintains the good properties of the Φ-function series method. It incorporates to compute the coefficients of the algorithm a recurrent algebraic procedure, based in the existing relation between the divided differences and the elemental and complete symmetrical functions. In addition, under certain hypotheses, the new multistep method calculates the exact solution of the perturbed problem. The new method is implemented in a computational algorithm which enables us to resolve in a general manner some physics and engineering IVP’s modeled by means systems of differential equations. The good behaviour and precision of the method is evidenced by contrasting the results with other-reputed algorithms and even with methods based on Scheifele’s G-functions.This work has been supported by GRE09-13 project of the University of Alicante and the project of the Generalitat Valenciana GV/2011/032

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit

Seismic model analysis by means of a series method

La respuesta desde la Mecánica Estructural a los fenómenos sísmicos, hace necesario mejorar el cálculo de las estructuras así como su análisis. Para ello los métodos especialmente basados en el análisis estático no lineal necesitan tener una mayor precisión. El análisis no lineal se puede abordar mediante modelos discretos o continuos. Los modelos discretos representan la estructura a través de un número finito de grados de libertad; en este caso las ecuaciones de movimiento son ecuaciones diferenciales ordinarias que se resuelven por métodos numéricos. En este trabajo se muestra una aplicación del método de series ɸ-funciones para calcular la respuesta ante un terremoto de las estructuras modeladas mediante sistemas SDOF (Single Degree Of Freedom system) y 2DOF (Two Degree Of Freedom systems). Además, en el caso de SDOF, el método se ha aplicado tomando como la frecuencia forzada la frecuencia natural de vibración. La solución de los modelos sísmicos se ha obtenido mediante la generación de un algoritmo numérico y su implementación computacional. El método de series ɸ-funciones integra osciladores forzados y es una adaptación de los métodos de Scheifele, con la ventaja de integrar, sin error de truncamiento, el problema perturbado con sólo las dos primeras ɸ-funciones. El cálculo de coeficientes de la serie se efectúa por recurrencias algebraicas sencillas en las que se implica la función de perturbación. El buen comportamiento y precisión del método de series ɸ-funciones se ilustra cuando se contrasta con otros métodos de integración ya conocidos e implementados en MAPLE, comparándose también con los métodos clásicos de Ingeniería de Estructuras.The seismic events have attracted interest and the need to improve the structures and their analysis to sustain this type of oscillation. To do this, new methods especially those based on static non-linear analysis need to have increased accuracy. The non-linear analysis can be approached by means of discrete or continuous models. The discrete models represent the structure through a finite number of degrees of freedom; in this case the equations of motion are ordinary differential equations which are solved by numerical methods. This paper shows an application of the ɸ-functions series method to calculate the response of structures, modeled as both SDOF(Single Degree Of Freedom system) and 2DOF (Two Degree Of Freedom systems) systems, to an earthquake. Furthermore, in the case of SDOF, the method has been applied taking as the forcing frequency the natural frequency of vibration. The solution of the seismic models has been obtained by the generation of the numerical algorithm and its computational implementation. The ɸ-functions series method integrates forced oscillators and it is an adaptation of Scheifele's methods, with the advantage of integrating, without truncation error, the perturbed problem with just the first two ɸ-functions. The calculation of series coefficients is effected by simple algebraic recurrences in which the perturbation function is takes part. The good precision of ɸ-functions series method is illustrated when contrasted with other methods of integration already known and implemented in MAPLE and even with classic methods of Structural Engineering

Research Achievements Review Series no. 20 - Mathematics and computation research

Computational mathematics, perturbed orbit three-body problem, and periodic trajectories solutions through computer method

Delay differential equations in a nonlinear cochlear model.

The human auditory system performs its primary function in the cochlea, the main organ of the inner ear, where the spectral analysis of a sound signal and its transduction into a neural signal occur. It is filled with liquid and divided in two cavities by the basilar membrane (BM). A sound stimulus propagates in air as an acoustic pressure wave through the outer and the middle ear. The pressure of the stapes on the oval window (boundary between the middle and the inner ear) causes the cochlear fluid to flow between the two cavities through a hole at the end of the BM. A spatial partial differential equation of fluid-dynamics describes this physical process. As a consequence of the differential pressure between the two cavities, each micro-element of the BM oscillates as a forced damped harmonic oscillator. The BM displacement is amplified by the overlying outer hair cells (OHCs) through a nonlinear nonlocal active feedback mechanism. The latter can be modeled by means of various representations. Among them, the delayed stiffness model of Talmadge et al. (J. Acoust. Soc. Am. 104, 1998) has been considered in this thesis. Specifically, the cochlear nonlinearity is introduced as a quadratic function of the BM displacement in the passive linear damping function. Moreover, the active mechanism is described by two additional forces, each one proportional to the BM displacement delayed by a slow and a fast feedback constant time, respectively. According to this model, a time delay differential equation (DDE) of the second order describes the oscillating dynamics of the BM. A different formulation of the nonlinear active mechanism, driven by the OHCs, is expressed as a nonlinear function of the BM velocity by the anti-damping model of Moleti et al. (J. Acoust. Soc. Am. 133, 2013). In this case the model equations do not contain time delays. The numerical integration of the above mentioned models has been obtained by finite differencing with respect to the space variable in the state space, as introduced by Elliott et al. (J. Acoust. Soc. Am. 122, 2007), and then integrating in time with the adaptive package introduced by Bertaccini and Sisto as a modification of the popular Matlab ode15s package (J. Comput. Phys. 230, 2011). The semidiscrete formulation of the delayed stiffness model and the anti-damping model has a non trivial mass matrix, and eigenvalues of the system matrix with large negative real part and imaginary part. That is why an implicit solver with an infinite region of absolute stability should be used. Therefore, the customized Matlab ode15s package by Bertaccini and Sisto seems to be the convenient choice to integrate the problem at hand numerically. In particular, for the delayed stiffness model, an integrator for constant DDEs (the method of steps; Bellen and Zennaro, Oxford University Press 2003) has been formulated and based on the customized ode15s. All these topics have been discussed in this doctoral thesis, which is subdivided in the following chapters. Chapter 1 describes the anatomy of the human ear, with special regard to the cochlea. Some experimental evidences about the cochlear mechanisms are discussed, in order to support the cochlear modeling. Two physical models with one degree of freedom are shown: the anti-damping model of Sisto et al. (J. Acoust. Soc. Am. 128, 2010) and Moleti et al. (J. Acoust. Soc. Am. 133, 2013), and the delayed stiffness model of Talmadge et al. (J. Acoust. Soc. Am. 104, 1998). Chapter 2 discusses the general theory of DDEs, with greater reference to constant and time dependent DDEs from Bellen and Zennaro (Oxford University Press 2003). Existence and uniqueness of time dependent DDEs are briefly analyzed, while the method of steps is shown as a basic approach to find a numerical approximation of the DDEs solution. According to this method, IVPs of constant DDEs (as for the semidiscrete delayed stiffness model) are turned into IVPs of ODEs in a subinterval (of length less than or equal to the time delay) of the whole integration interval. Each IVP of ODEs can be integrated by means of any ODEs numerical method, and its convergence is then discussed. Chapter 3 describes the main tools used to find an approximate solution of the considered models. In particular, the discretization for spatial partial derivatives by means of finite differences is shown. Such a representation turns a model, which is continuous in the space-time domain, into a semidiscrete model to be integrated in time. The models considered in this thesis are stiff, so the phenomenon of stiffness is discussed and the ode15s package of Matlab for integrating stiff ODEs is described. Nevertheless, greater benefits can be obtained by using the ode15s package customized by Bertaccini and Sisto as a hybrid direct-iterative solver which exploits Krylov subspace methods. Chapter 4 shows the semidiscrete formulation of the continuous models (anti-damping model and delayed stiffness model) in the state space with respect to the spatial variable, as introduced by Elliott et al. (J. Acoust. Soc. Am. 122, 2007). The algebraic properties of the semidiscrete models are discussed in order to show why the customized ode15s package may perform a faster numerical integration of the semidiscrete models and how this solver can be used in an integration numerical technique for constant DDEs (the method of steps). Chapter 5 shows the results produced by the numerical experiments of the delayed stiffness model by supplying a sinusoidal tone, and compares them with the numerical results produced by the anti-damping model. Some considerations about the numerical approach of the time integration are also discussed, and a part of the simplified code used for integrating the semidiscrete delayed stiffness model, is reported. The results are comparable with those obtained by the anti-damping model, and then the numerical experimental evidences seem to justify the proposed integration technique for constant DDEs. Delayed model properties of tonotopicity, anti-damping and nonlinearity are verified, as well as the dependence of the approximate solution on some free parameters of the model. The cochlear response described by the delayed stiffness model shows a typical tall and broad BM activity pattern. This behavior is also found in the numerical results of a model with two degree of freedom produced by Neely and Kim (J. Acoust. Soc. Am. 79, 1986) and Elliott et al. (J. Acoust. Soc. Am. 122, 2007)

SciCADE 95: International conference on scientific computation and differential equations

This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included

An overview on deep learning-based approximation methods for partial differential equations

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research, we review some of the main ideas of deep learning-based approximation methods for PDEs, we revisit one of the central mathematical results for deep neural network approximations for PDEs, and we provide an overview of the recent literature in this area of research.Comment: 23 page