1,561 research outputs found
Spline collocation methods for fractional differential equations: theoretical and computational aspects
No full textSpline collocation methods are a powerful tool to discretize fractional differential equations, since they have a high order of convergence and strong stability properties. In this talk, we illustrate the convergence and stability analysis of one and two step spline collocation methods. We pay attention also to the efficient implementation of these methods, which requires the evaluation of fractional integrals. Some numerical experiments are provided to confirm theoretical results and to compare one and two step collocation methods. This is a joint work with B. Paternoster and D. Conte
Adapted discretization of evolutionary problems by non-polynomially fitted numerical methods
No full textThe talk is devoted to the discretization of selected evolutionary problems generating periodic
wavefronts [5] and aims to explain the benefits gained by adapting the numerical scheme to the
problem. Such an adaptation is carried out by merging the a-priori known qualitative information
on the problem, as well as the structure of the vector field itself, into the numerical scheme.
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Particular emphasis will be given to advection-reaction-diffusion problems, for which the adaptation
in space is developed by means of a finite difference scheme based on trigonometrical basis
functions [3], rather than on algebraic polynomials which could strongly reduce the stepsize in
order to accurately reproduce the prescribed oscillations of the exact solution. The adaptation
in time takes into account that the spatially discretized problem is characterized by a vector
field consisting in stiff and nonstiff terms, hence it makes sense to adopt an implicit-explicit
(IMEX) time integration, which implicitly integrate only the stiff constituents, while the nonstiff
part is computed explicitly. Clearly, the employ of non-polynomial basis functions makes
the coefficients of the numerical method dependent on unknown parameters (i.e. the frequency
of the oscillations), which need to be properly estimated [4]; the proposed estimation relies
on a minimization procedure of the local truncation error that is carried out a-priori, without
affecting the computational cost of the integration. A rigorous analysis on the stability and
accuracy properties of the overall method is presented, together with some numerical tests, in
order to highlight the effectiveness of the approach. The introduced technique also covers the
case of periodic dynamics generated by evolutionary problems with memory [1, 2], discretized
in terms of non-polynomially fitted quadrature methods able to accurately reproduce the oscillatory
behavior with a reduced computational cost with respect to their analogous polynomial
version, when a good estimate of the unknown frequency is provided. Stability issues for such
a discretization are also addressed. References
[1] Cardone, A., Ixaru, L.Gr. and Paternoster, B. Exponential fitting direct quadrature methods
for Volterra integral equations, Numer. Algorithms 55(4), 467–480 (2010).
[2] Cardone, A., Ixaru, L.Gr., Paternoster, B. and Santomauro, G. Ef-gaussian direct quadrature
methods for Volterra integral equations with periodic solution, Math. Comput. Simul.,
110, 125–143 (2015).
[3] D’Ambrosio, R., Moccaldi, M. and Paternoster, B. Adapted numerical methods for
advection-reaction-diffusion problems generating periodic wavefronts. Comput. Math. Appl.
74(5), 1029–1042 (2017).
[4] D’Ambrosio, R., Moccaldi, M. and Paternoster, B. Parameter estimation in IMEXtrigonometrically
fitted methods for the numerical solution of reaction-diffusion problems.,
Comput. Phys. Commun. 226, 55–66 (2018).
[5] Perumpanani, A.J., Sherratt, J.A. and Maini, P.K. Phase differences in reaction-diffusionadvection
systems and applications to morphogenesis, J. Appl. Math. 55, 19–33 (1995)
A MATLAB Code for Fractional Differential Equations Based on Two-Step Spline Collocation Methods
No full textWe illustrate a MATLAB implementation of two-step spline collocation methods for the numerical solution of fractional differential equations, introduced by Cardone, Conte and Paternoster in (Discrete Dyn. Syst. Ser. B 23(7) 2709--2725 (2018)). The computational tasks include the evaluation of fractional integrals, a suitable starting procedure, and the efficient computation of the coefficients of certain polynomials. The whole algorithm is described in detail. Some numerical experiments show the performances of the proposed algorithm
STABILITY OF COLLOCATION METHODS FOR FRACTIONAL DIFFERENTIAL EQUATIONS
No full textCollocation methods for fractional differential equations have been introduced by Blank [1] and later on rigorously analyzed by Pedas and Tamme [3,4]. Recently Cardone, Conte and Paternoster [2] introduced two step collocation methods, which raise the order of convergence, by using additional information from the past, without increasing the computational cost. Here
we study the stability of both classes of methods, in order to nd methods with unbounded stability regions.
This is a joint work with D. CONTE and B. PATERNOSTER from UNIVERSITY OF SALERNO.
[1 ] L. Blank, Numerical treatment of differential equations of fractional order, Nonlinear World, 4 (1997), 473-491.
[2 ] A. Cardone, D. Conte, B. Paternoster, Two-step collocation methods forfractional dierential equations, to appear in Discrete Cont.Dyn.-B. [3 ] A. Pedas and E. Tamme, On the convergence of spline collocation methods for solving fractional differential equations, J. Comput. Appl.Math., 235 (2011), 3502-3514.
[4 ] A. Pedas and E. Tamme, Numerical solution of nonlinear fractional differential equations by spline collocation methods, J. Comput. Appl.Math., 255 (2014), 216-230
Efficient methods for special second order ODEs
No full textOur purpose is the design of efficient methods for solving stiff and nonstiff initial value problems for second-order ordinary differential
equations of the special form y''=f(y).
We analyse approximate factorization methods for second-order stiff differential equations, we construct general linear methods with parallel stages for nonstiff equations with peridic solution, we propose both parallel and sequential methods tuned to the special form of the solution.
A number of convergence and stability results are derived and the performances of the methods are illustrated by means of a few examples from the literature
ADAPTED NUMERICAL METHODS FOR ADVECTION DIFFUSION PROBLEMS
No full textWe present exponentially fitted two step peer methods for the numerical solution of systems of ordinary differential equations having oscillatory solutions (2; 3). Such equations arise for example in the semi-discretization in space of advection-diffusion problems whose solution exhibits an oscillatory behaviour, such as the Boussinesq equation (1). Exponentially fitted methods are able to exploita-prioriknowninformationaboutthequalitativebehaviourofthesolutionin order to efficiently furnish an accurate solution. Moreover peer methods are very suitable for a parallel implementation, which may be necessary when the number ofspatialpointsincreases. Theeffectivenessofthisproblem-orientedapproachis shown through numerical tests on well-known problems.
References
[1] A. Cardone, R. D’Ambrosio, B. Paternoster. (2017). Exponentially fitted IMEX methods for advectiondiffusion problems, J. Comput. Appl. Math. (316), 100–108.
[2] D. Conte, R. D’Ambrosio, M. Moccaldi, B. Paternoster. (2018). Adapted explicit two-step peer methods, J. Numer. Math., in press.
[3] D. Conte, L. Moradi, B. Paternoster. (2017). Adapted implicit two-step peer methods, in preparation
Exponentially fitted numerical methods for differential systems with equation dependent coefficients
No full textThe derivation of special purpose numerical methods for differential systems, i.e. adapted to accurately solve problems whose qualitative behaviour is supposed to be known a-priori, is usually carried out by means of non-polynomial fitting techniques. Exponential fitting (compare [2] and references therein) is certainly one of the most spread out techniques to obtain special purpose formulae in many fields of numerical analysis.
In the context of numerical methods for ordinary differential equations, exponentially fitted Runge-Kutta formulae have been considered by many authors (see [3] for an updated state-of-art on the topic). The issue we want to revisit in this talk is the way of deriving the coefficients of such methods: we decide, indeed, to take into account the effect of the error inherited from the computation of the internal stages. Such contribution has always been neglected in previous version of exponentially fitted Runge-Kutta methods: on the contrary, we aim to make the propagation of the error along the stages visible. The revised technique is illustrated for hybrid methods and Runge-Kutta methods [1], for which we obtain new expressions of the coefficients, explicitly depending on the form of the system to be solved. The version obtained in this way is then compared for accuracy and stability with that achieved by means of the standard exponential fitting technique. Acknowledgments The authors express their gratitude to prof. Liviu Gr. Ixaru for the profitable discussions we had on the topic.
References
[1] R. D’Ambrosio, L. Gr. Ixaru, B. Paternoster, Construction of the EF-based Runge-Kutta methods revisited, Comp. Phys. Commun. 182, 322-329 (2011).
[2] L. Gr. Ixaru and G.Vanden Berghe, Exponential Fitting, Kluwer Academic Publishers, Dordrecht (2004).
[3] B. Paternoster, Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70-th anniversary, submitted
Funzioni, livelli e meccanismi. La spiegazione in scienza cognitiva e i suoi problemi
No full textQuesto lavoro ha due obiettivi: a)illustrare lo sviluppo delle scienze cognitive nell’arco dei circa cinquant’anni della loro storia; b)mettere in luce alcuni problemi che intersecano orizzontalmente tutte le aree delle scienze cognitive, problemi la cui soluzione è, a nostro giudizio, decisiva per il futuro degli studi sulla mente. Come vedremo, questi problemi sembrano implicare la necessità di contemplare una pluralità di livelli esplicativi le cui interrelazioni, tuttavia, non sono affatto semplici da comprendere. Particolarmente cruciale è la questione di quale statuto epistemologico si debba accordare a un livello di spiegazione del mentale, quello funzionale-computazionale, che si colloca all’intersezione fra l’immagine ordinaria di noi stessi in quanto persone (ossia in quanto soggetti di esperienze coscienti, stati intenzionali e agire deliberato) e la sfera subpersonale degli eventi cerebrali, oggetto delle neuroscienze
Adapted numerical methods for advection-reaction-diffusion problems generating periodic wavefronts
No full textWe present an adapted method of lines for advection-reaction-diffusion problems generating periodic wavefronts [3], by exploiting the a-priori known information about the qualitative behaviour of the solution.
Since the dynamics exhibits a non-polynomial character, classical finite difference methods could require a very small stepsize because they are constructed in order to be exact (within round-off error) on polynomials up to a certain degree. In our approach, the employ of non-polynomially fitted finite differences may guarantee a better balance between accuracy and efficiency requirements.
Once a advection-reaction-diffusion problem is discretized in space, the vector field of the resulting system of ordinary differential equations results to be split in two different terms, a stiff term and a nonlinear one. Hence, we propose an implicit-explicit (IMEX) method that implicitly integrates only stiff components and explicitly integrates the nonlinear part, with a significant benefit in terms of efficiency. For the overall numerical scheme, combining the non-polynomial fitting strategy with the IMEX time integration, accuracy and stability properties are rigorously studied, also in comparison with the classical polynomial case [1]. Moreover, since the adapted method has non-constant coefficients depending on unknown parameters linked to the solution, we propose an estimation strategy based on minimization of the leading term of the local discretization error [2].
This is a joint work with Raffaele D'Ambrosio and Beatrice Paternoster (University of Salerno).
[1] D'Ambrosio, R., Moccaldi, M., Paternoster, B., Adapted IMEX numerical methods for reaction-diffusion problems, Appl. Numer. Math. (submitted)
[2] D'Ambrosio, R., Moccaldi, M., Paternoster, B., Parameter estimation in adapted numerical methods for reaction-diffusion problems, J. Sci. Comput. (submitted)
[3] Perumpanani, A.J., Sherratt, J.A., Maini, P.K., Phase differences in reaction-diffusion-advection systems and applications to morphogenesis, IMA J. Appl. Math. 55, 19--33 (1995)
Numerical solution of Hamiltonian problems by G-symplectic integrators
No full textIt is the purpose of this talk to analyze the employ of General Linear Methods (GLMs) for the numerical integration of Hamiltonian problems. Indeed, even if the numerical flow generated by a GLM cannot be symplectic, Butcher recently introduced in [1] a concept of near conservation, denoted as G-symplecticity, for such methods which, properly combined with other desirable features (such as symmetry and boundedness of parasitic components),
allows to achieve a very accurate long time conservation of the Hamiltonian. We also focus our attention on the connections between the order of convergence of a GLM and the observable Hamiltonian deviation, by employing the theory of B-series [3]. Moreover, we derive a semi-implicit GLM [2] which results competitive with respect to symplectic Runge-Kutta methods. Numerical results on a selection of Hamiltonian problems are presented, confirming the structure-preserving capability of G-symplectic integrators. References [1] J.C. Butcher, Numerical methods for Ordinary Differential Equations, Second Edition, Wiley, Chichester, 2008. [2] R. D’Ambrosio, G. De Martino and B. Paternoster, Construction of nearly conservative multivalue numerical methods for Hamiltonian problems, Comm. Appl. Ind. Math, doi: 10.1685/journal.caim.412 (2013). [3] R. D’Ambrosio, G. De Martino and B. Paternoster, Numerical integration of Hamiltonian problems by G-symplectic integrators, submitted
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