1,721,214 research outputs found
General Two-Step Runge-Kutta methods based on algebraic and trigonometric polynomials
No full textThe general family of two-step Runge-Kutta methods introduced by Jackiewicz and Tracogna is investigated in this paper. Conditions for obtaining two-step Runge-Kutta methods which integrate algebraic polynomials exactly are derived. Some simplifying conditions on the parameters of the method, which form the collocation methods within this family of general linear methods, are presented. Two-step Runge-Kutta methods trigonometrically-fitted for ODEs having periodic or oscillatory solution are also considered. The author assumes that the dominant frequency w can be estimated in advance. Based on this assumption the resulting methods depend on the parameter wh, where h is the stepsize. The one-stage case is also investigated
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.
53
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)
Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70-th anniversary
No full textThe standard monograph in this area is the book Exponential fitting by Ixaru and Vanden Berghe (Kluwer, Boston
- Dordrecht - London, 2004) but a fresh look on the things is necessary because many new contributions have been
accumulated meantime. With no claim that our investigation is exhaustive we consider various directions of interest,
try to integrate the new contributions in a natural, easy to follow way, and also detect some open problems of acute
interest.
Keywords: Exponential fitting, oscillatory functions, frequency-dependent coefficients
Two step Runge-Kutta-Nystrom methods based on algebraic polynomials
No full textWe consider the new family of two step Runge-Kutta-Nystr ̈om methods for the numerical integration of y"=f(x,y).
We derive the conditions to obtain two step Runge-Kutta-Nystr ̈om methods which integrate algebraic polynomials exactly and analyze the one-stage case
Trattamento numerico conservativo di equazioni differenziali: sviluppi recenti e prospettive future
Full text linkScopo di questa comunicazione è illustrare alcuni sviluppi recenti nell’ambito del trattamento numerico di equazioni differenziali, offrendo possibili prospettive future e problemi aperti che rendono questa tematica di vivo interesse nella letteratura scientica.
Particolare enfasi verrà posta su alcune famiglie di equazioni dierenziali di interesse in numerosi contesti applicativi, ove si ritiene utile fornire unulteriore potenziamento degli aspetti numerici. Nello specifico, verranno illustrati avanzamenti recenti nell’ambito dei problemi Hamiltoniani, problemi stiff e sistemi di equazioni dffierenziali ordinarie che nascono dalla semidiscretizzazione spaziale di equazioni alle derivate parziali con soluzione periodica. L’approccio proposto sarà generalmente di tipo structure-preserving, orientato al problema e teso a conservarne numericamente le proprietà qualitative. Nello specifico, verranno presentati solutori conservativi non standard per problemi Hamiltoniani che nascono da schemi numerici multi-value, accurati nel preservare a lungo termine la struttura simplettica dello spazio delle fasi; tecniche di collocazione modificata per problemi altamente stiff che, al contrario dei solutori classici, non soffrono del problema di riduzione dell’ordine; tecniche numeriche di fitting non polinomiale per equazioni con soluzione periodica che nascono da problemi di reazione-diffusione, integrati mediante differenze finite non standard.
Unitamente agli aspetti di analisi teorica, verranno fornite evidenze sperimentali a supporto dell’efficacia degli approcci introdotti
Runge-Kutta(-Nystrom) methods for ODEs with periodic solutions based on trigonometric polynomials
No full textWe consider the construction of Runge-Kutta(-Nystrom) methods for ordinary differential equations whose solutions are known to be periodic. We assume that the frequency omega can be estimated in advance. The resulting methods depend on the parameter nu = omega h, where h is the stepsize. Using the linear stage representation of a Runge-Kutta method given in Albrecht's approach, we derive Runge-Kutta and Runge-Kutta-Nystrom methods which integrate trigonometric polynomials exactly
Composite symplectic Runge-Kutta(-Nystrom) methods
No full textNon reducible composite Runge--Kutta methods can't be symplectic. For non reducible composite symplectic Runge--Kutta--Nystr\"om methods the order barrier is shown to be tw
A phase-fitted collocation-base Runge-Kutta-Nystrom method
No full textWe consider collocation-based Runge-Kutta-Nystrom methods with symmetric points and derive a three-stage method which results exact in phase for second order linear ODEs having a periodic or an oscillatory solution. We assume that the dominant frequency k can be estimated in advance; then the resulting method depends on the parameter theta = kh, where h is the stepsize. Due to its stability properties, the method is suitable to integrate systems which exhibits a moderate stiffness. The procedure can be easily generalized to derive phase-fitted RKN methods with an arbitrary high order
Order bound for a family of parallel Runge-Kutta-Nystrom methods through computer algebra
No full textWe prove a bound on the obtainable order of consistency for a family of parallel Runge-Kutta-Nystrom methods, using computer algebra
General two step collocation methods for special second order Ordinary Differential Equations
No full text- …
