321 research outputs found

    Symmetric integrators with improved uniform error bounds and long-time conservations for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

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    In this paper, we are concerned with symmetric integrators for the nonlinear relativistic Klein--Gordon (NRKG) equation with a dimensionless parameter 0<ε10<\varepsilon\ll 1, which is inversely proportional to the speed of light. The highly oscillatory property in time of this model corresponds to the parameter ε\varepsilon and the equation has strong nonlinearity when \eps is small. There two aspects bring significantly numerical burdens in designing numerical methods. We propose and analyze a novel class of symmetric integrators which is based on some formulation approaches to the problem, Fourier pseudo-spectral method and exponential integrators. Two practical integrators up to order four are constructed by using the proposed symmetric property and stiff order conditions of implicit exponential integrators. The convergence of the obtained integrators is rigorously studied, and it is shown that the accuracy in time is improved to be \mathcal{O}(\varepsilon^{3} \hh^2) and \mathcal{O}(\varepsilon^{4} \hh^4) for the time stepsize \hh. The near energy conservation over long times is established for the multi-stage integrators by using modulated Fourier expansions. These theoretical results are achievable even if large stepsizes are utilized in the schemes. Numerical results on a NRKG equation show that the proposed integrators have improved uniform error bounds, excellent long time energy conservation and competitive efficiency

    Efficient computation of the sinc matrix function for the integration of second-order differential equations

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    This work deals with the numerical solution of systems of oscillatory second-order differential equations which often arise from the semi-discretization in space of partial differential equations. Since these differential equations exhibit pronounced or highly) oscillatory behavior, standard numerical methods are known to perform poorly. Our approach consists in directly discretizing the problem by means of Gautschi-type integrators based on sinc\operatorname{sinc} matrix functions. The novelty contained here is that of using a suitable rational approximation formula for the sinc\operatorname{sinc} matrix function to apply a rational Krylov-like approximation method with suitable choices of poles. In particular, we discuss the application of the whole strategy to a finite element discretization of the wave equation

    Exponential integrators: tensor structured problems and applications

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    The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed

    Product integration rules by the constrained mock-Chebyshev least squares operator

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    In this paper we consider the problem of the approximation of definite integrals on finite intervals for integrand functions showing some kind of "pathological" behavior, e.g. "nearly" singular functions, highly oscillating functions, weakly singular functions, etc. In particular, we introduce and study a product rule based on equally spaced nodes and on the constrained mock-Chebyshev least squares operator. Like other polynomial or rational approximation methods, this operator was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. Unlike methods based on piecewise approximation functions, mainly used in the case of equally spaced nodes, our product rule offers a high efficiency, with performances slightly lower than those of global methods based on orthogonal polynomials in the same spaces of functions. We study the convergence of the product rule and provide error estimates in subspaces of continuous functions. We test the effectiveness of the formula by means of several examples, which confirm the theoretical estimates

    Au delà de la dissipation markovienne à l’échelle nanométrique : vers la découverte de règles quantiques pour la conception de nano-dispositifs bio-organiques

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    A better understanding of dissipation is crucial for understanding real-world quantum systems. Indeed, all quantum systems experience interactions with an (often) uncontrollable outside environment that can lead to a decay of excited state populations and a loss of quantum coherences. The study of dissipation is timely as the development of next-generation nanoscale quantum technologies is on its way, and the existence of non-trivial quantum effects in biological systems is being seriously investigated. However, descriptions of dissipation in quantum systems are reduced (most of the time) to time-local approaches and (everywhere) to space-local independent environments. These simplifying assumptions do render analytic and numerical calculations possible, yet they get rid of a breadth of physical processes that can alter radically the quantum systems' dynamics. In this thesis, building on a numerically exact tensor networks method, we developed a technique able to handle spatio-temporal correlations between a quantum system and bosonic (i.e. vibrational, electromagnetic, magnons, etc.) environments. With this method we studied the signalling process - a form of information backflow - in quantum systems, and uncovered how it can induce non-trivial dynamics, and be leveraged to populate otherwise inaccessible excited states. We also evidenced the ability of 'non-local' environment reorganisation, induced by system-environment interactions, to radically change the nature of the thermodynamically favoured system ground state. The new phenomenology of physical processes, resulting from considering quantum systems interacting with a common environment, has important consequences for the design of nanodevices as it gives access to new control, sensing and cross-talk mechanisms. In another vein, these results might also give us a new framework to study and interpret (quantum?) effects in the biological realm

    A Lotka-Volterra type model analyzed through different techniques

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    We consider a modified Lotka-Volterra model applied to the predator-prey system that can also be applied to other areas, for instance the bank system. We show that the model is well-posed (non-negativity of solutions and conservation law) and study the local stability using different methods. Firstly we consider the continuous model, after which the numerical schemes of Euler and Mickens are investigated. Finally, the model is described using Caputo fractional derivatives. For the fractional model, besides well-posedness and local stability, we prove the existence and uniqueness of solution. Throughout the work we compare the results graphically and present our conclusions. To represent graphically the solutions of the fractional model we use the modified trapezoidal method that involves the modified Euler method.Comment: Accepted on June 22, 2023 for publicatio

    Is polynomial interpolation in the monomial basis unstable?

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    In this paper, we show that the monomial basis is generally as good as a well-conditioned polynomial basis for interpolation, provided that the condition number of the Vandermonde matrix is smaller than the reciprocal of machine epsilon. We also show that the monomial basis is more advantageous than other polynomial bases in a number of applications.Comment: 30 pages, 12 figure
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