540 research outputs found
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)
Equation dependent numerical methods for FDEs
No full textIn this talk we describe techniques that allow to enlarge the absolute stability regions of classical explicit numerical methods for Ordinary Differential Equations (ODEs). The basic idea of these techniques consists in the modification of method coefficients, which result in depending on the Jacobian of the ODE to be solved. We then analyze the possibility to apply these methodologies to numerical methods for Fractional Differential Equations (FDEs), in order to obtain an improvement in terms of accuracy and stability properties. This is a joint work with Prof. Beatrice Paternoster and Dajana Conte
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)
Adapted numerical approximation of advection-reaction-diffusion problems
No full textWe present an adapted numerical method for the approximate solution of advection-reaction-diffusion problems on a bidimensional spatial domain and generating periodic wavefronts [3]. In particular, we propose to merge into the numerical scheme the a-priori known information about the qualitative behaviour of the exact solution and the structure of the problem. Traditional finite difference methods could impose a severe reduction of the stepsize in order to accurately follow the oscillations because they are developed in order to achieve exactness (within round-off error) on algebraic polynomials up to a certain degree. Extending the ideas described in [1,2], we develop an adapted method of lines based on trigonometrically fitted finite differences, whose coefficients depend on unknown parameters characterising the exact solution. We deal with the more challenging issue of estimating these parameters by properly manipulating the leading term of the local truncation error a-priori.
The vector field of the resulting system of ordinary differential equations is composed by stiff and non-stiff terms, so we suggest to employ an implicit-explicit (IMEX) time method, which implicitly integrates only stiff components and explicitly integrates the others, obtaining advantages in terms of efficiency and stability. The stability and accuracy properties of the overall scheme are rigorously investigated and some numerical tests are presented to show its effectiveness.
This is a joint work with Raffaele D’Ambrosio from University of L’Aquila and Beatrice Paternoster from University of Salerno.
[1] D’Ambrosio, R., Moccaldi, M., Paternoster, B., Adapted numerical methods for advection-reaction-
diffusion problems generating periodic wavefronts, Comput. Math. Appl. 74(5), 1029–1042, 2017.
[2] D’Ambrosio, R., Moccaldi, M., Paternoster, B., Parameter estimation in IMEX-trigonometrically
fitted methods for the numerical solution of reaction-diffusion problems, Comput. Phys. Commun.
226, 55–66, 2018.
[3] Hundsdorfer, W., Verwer, J., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction
Equations, Springer-Verlag, 2003
Raffaele D’Ambrosio, Martina Moccaldi, Beatrice Paternoster, Adapted IMEX Numerical Methods for Reaction-Diffusion Problems
No full textAbstract—The treatise is focused on the numerical solution of λ-ω reaction-diffusion problems, by means of a suitably adapted method of lines. Due to the non linearity of the vector field and the oscillatory behaviour of the solution, we propose to combine a spatial semidiscretization of the operator through trigonometrically fitted finite differences with an IMEX integration in time. Accuracy and stability properties of the overall numerical scheme are proved and experiments confirming the effectiveness of the approach are also provided
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
Metodi numerici impliciti-espliciti adattati per problemi di reazione-diffusione semidiscretizzati
No full textOggetto della comunicazione `e il trattamento numerico di equazioni differenziali
ordinarie derivanti da equazioni alle derivate parziali semi-discretizzate
rispetto alla variabile spaziale, il cui campo vettoriale sia decomponibile
nella somma di due termini da cui originano, contestualmente, sia componenti
stiff che non-stiff nella soluzione. Tali problemi vengono usualmente
trattati in maniera efficiente mediante l’impiego di schemi numerici implicitiespliciti.
L’attenzione verr`a concentrata su problemi che hanno origine dalle applicazioni
e di cui `e noto a priori il comportamento qualitativo della soluzione,
con particolare enfasi al caso di equazioni di reazione-diffusione che, come
noto, generano soluzioni ondulatorie lungo la loro dinamica [2, 3]. Il carattere
periodico delle soluzioni suggerisce l’impiego di tecniche numeriche che
seguano il comportamento oscillante in maniera accurata ed efficiente, evitando
riduzioni troppo severe del passo di integrazione.
A tal fine, verr`a proposto un possibile adattamento dei classici schemi implicitiespliciti
basati su differenze finite, che tengano conto del comportamento
qualitativo delle soluzioni, estendendo le idee in [1]. L’adattamento avverr`a
lungo tre livelli differenti: lungo spazio, mediante differenze finite su basi
non polinomiali; lungo il tempo, mediante l’impiego di opportuni metodi
numerici per l’integrazione temporale; lungo il problema, sfruttando le peculiarit`a
del suo campo vettoriale nella formulazione dello schema implicitoesplicito.
Verranno presentati aspetti legati alla costruzione dello schema
numerico, alla sua accuratezza, alla stima dei parametri da cui esso dipende,
unitamente ad alcuni test numerici che mostrino l’efficacia dell’approccio introdotto.
Questo lavoro `e frutto della ricerca svolta in collaborazione con
Raffaele D’Ambrosio e Beatrice Paternoster (Univ. di Salerno).
Bibliografia
[1] R. D’Ambrosio, B. Paternoster, Numerical solution of a diffusion problem
by exponentially fitted finite difference methods, Springer Plus 3, 425-
431 (2014).
[2] N. Kopell, L.N. Howard, Plane wave solutions to reaction-diffusion equations,
Stud. Appl. Math. 52, 291-328 (1973).
[3] J.A. Sherratt, Periodic waves in reaction-diffusion models of oscillatory
biological systems, FORMA 11, 61-80 (1996)
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
Exponentially-fitted direct quadrature methods for Volterra integral equations with periodic solution
No full textWe consider Volterra integral equations with periodic solution of the type:
y(x) = f(x) + int_0^x k(x - s)y(s)ds; x in [0; xend]
y(x) = fi(x); -inf < x <= 0;
where k in L_1(R+), f is continuous and periodic on [0; xend], fi is continuous and bounded on R+.
These equations model periodic phenomena with memory, such as the spread of seasonal epidemics and the response of nonlinear circuits to a periodic input. General purpose methods usually require a high computational cost to follows the oscillations of the solution, thus numerical methods specially
tuned on the problem should be applied. With this aim, we propose direct quadrature (DQ) methods based on exponential fitting theory [2], which exploit the knowledge on the qualitative behavior of the solution, to improve accuracy without increasing the computational cost. In [1] we introduced a DQ method based on an exponentially-fitted Simpson-like formula. Now we go one step further, introducing a Gaussian DQ method, to increase the order of convergence. Here we illustrate the construction of these methods, analyze convergence and stability and furnish some numerical experiments of comparison with other numerical methods. This is a joint work with Beatrice Paternoster from Università di Salerno
[1] Cardone, A.; Ixaru, L. Gr.; Paternoster, B. 2010 Exponential fitting direct quadrature methods for Volterra integral equations. Numer. Algorithms, vol. 55, no. 4, pp. 467-480.
[2] Ixaru, L.Gr.; Vanden Berghe, G. 2004 Exponential fitting. Kluwer Academic Publishers, Dordrecht
ALGEBRAICALLY STABLE TWO-STEP RUNGE-KUTTA METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS
No full textWe investigate algebraic stability of two-step Runge-Kutta methods [2] for ordinary differential equations using the criterion proposed by Hewitt and Hill [1] for general linear methods. This criterion is based on suitable transformations on the coefficient matrices of the methods under consideration, in such a way that the G-matrix of algebraically stable formulae is the identity matrix. This gives a remarkable improvement, since the determination of the G-matrix is, in general, a nontrivial task. Examples of algebraically stable two-step Runge-Kutta methods possessing the above feature are presented. This work is in collaboration with Zdzislaw Jackiewicz (Arizona State University), Beatrice Paternoster (University of Salerno) and Dajana Conte (University of Salerno).
REFERENCES [1] L. L. Hewitt, A. T. Hill. Algebraically stable diagonally implicit general linear methods. Appl. Numer. Math., 60 (6):629–636, 2010.
[2] Z. Jackiewicz. General linear methods for ordinary differential equations. John Wiley & Sons, 2009
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