1,720,976 research outputs found
Trigonometrically fitted-IMEX discretization of advection-reaction-diffusion problems
No full textWe propose an adapted numerical approximation of advection-reaction-diffusion problems on a bi-dimensional spatial domain and generating periodic wavefronts [3]. The adaptation is carried out by merging into the numerical scheme information known in advance concerning the qualitative behaviour of the exact solution and the structure
of the problem. Classic numerical methods based on finite difference formulae could
strongly reduce the stepsize in order to accurately reproduce the prescribed oscillations
of the exact solution because they are constructed to be exact (within round-off error)
on algebraic polynomials up to a certain degree. Broadening the investigation presented
in [1, 2], we devise an adapted method of lines combined with trigonometrically-fitted finite differences. However, the coefficients of these formulae rely on unknown parameters, related to the exact solution, which we propose to estimate through an appropriate treatment of the local truncation error. The resulting system of ordinary differential
equations exhibits a vector field consisting of stiff and nonstiff terms, so we adopt an
implicit-explicit (IMEX) time solver, which implicitly integrates only stiff constituents
and explicitly integrates the others, gaining benefits in terms of efficiency and accuracy.
A rigorous analysis on the stability and accuracy properties of the overall method is
presented, joint with some numerical tests, in order to highlight its effectiveness.
References
[1] D’Ambrosio, R., Moccaldi, M. and Paternoster, B. 2017 Adapted numerical methods
for advection-reaction-diffusion problems generating periodic wavefronts. Comput.
Math. Appl. 74(5), 1029–1042.
[2] D’Ambrosio, R., Moccaldi, M. and Paternoster, B. 2018 Parameter estimation
in IMEX-trigonometrically fitted methods for the numerical solution of reaction-
diffusion problems. Comput. Phys. Commun. 226, 55–66.
[3] Hundsdorfer, W. and Verwer, J. 2003 Numerical Solution of Time-Dependent
Advection-Diffusion-Reaction Equations. Springer-Verlag
Adapted time-integration of partial differential equations generating periodic wavefronts
No full textThe talk focuses on the numerical solution of advection-reaction-diffusion problems
by adapted finite difference schemes. In other terms, the numerical scheme is developed
in order to exploit the a-priori knowledge of the qualitative behaviour of the solution,
gaining advantages in terms of efficiency and accuracy with respect to classical schemes
already known in literature, which mostly rely on algebraic polynomials. The adaptation
is carried out by the so-called trigonometrical fitting technique for the space-discretization,
giving rise to a system of ODEs whose vector field contains both stiff and non-stiff terms.
Due to this mixed nature of the vector field, an Implicit-Explicit (IMEX) method is employed for the time-integration. The coefficients of the introduced numerical scheme depend on unknown parameters which have to be properly estimated: such an estimate is performed by an efficient offline minimization of the leading term of the local truncation error. The effectiveness of this problem-oriented approach is shown through a rigorous theoretical analysis and selected numerical experiments.
References
[1] D’Ambrosio, R., Moccaldi, M. and Paternoster, B. 2017 Adapted numerical methods for advection-reaction-diffusion problems generating periodic wavefronts. Comput. Math. Appl. 74(5), 1029–1042.
[2] D’Ambrosio, R., Moccaldi, M. and Paternoster, B. 2018 Parameter estimation in IMEX-trigonometrically fitted methods for the numerical solution of reaction-diffusion problems., Comput. Phys. Commun. 226, 55–66.
[3] Perumpanani, A.J., Sherratt, J.A., Maini, P.K. 1995 Phase differences in reaction–diffusion–advection systems and applications to morphogenesis, J. Appl. Math. 55, 19–33
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
Numerical solution of partial differential equations by IMEX methods based on non-polynomial fitting
No full textThis talk deals with the numerical solution of partial differential equations,
with special attention to problems that, after a spatial semi-discretization of the
operator, reduce to a system of ODEs whose vector field can be split into two
terms: one generating stiff components in the solution and one giving rise to nonstiff
ones. Such problems can be efficiently treated by suitable IMEX numerical
methods, as clearly visible from the existing literature.
We will mainly focus our attention on systems of partial differential equations
represented in terms of two coupled reaction-diffusion equations profitably
solvable by IMEX methods and which are known to generate traveling waves
as fundamental solutions [4,5]. Such problems have been typically used as models
for life science phenomena exhibiting the generation of periodic waves along
their dynamics (e.g. cell cycles [3], frequently behaving if they are driven by an
autonomous biochemical oscillator; intracellular calcium signalling [1,5], since
calcium shows many differrent types of oscillations in time and space, in response
to various extracellular signals).
The periodic character of the problem suggests to propose a numerical solution
which takes into account this oscillatory behavior, i.e. by tuning the numerical
solver to accurately and efficiently follow the oscillations appearing in the
solution, since classical numerical methods would require the employ of a very
small stepsize to accurately reproduce the dynamics.
For this reason, we propose an adaptation of classical IMEX schemes based
on finite differences which will take into account the qualitative nature of the
solutions. Extending the ideas in [2], we may say that a three-fold level of adaptation
to problem will be carried out: along time and space, by suitable semidiscretization
with problem-based finite differences and analog time solvers for
the semi-discrete problem, and along the problem by taking into account the
peculiarity of the vector field through the employ of IMEX schemes.
The approximant will be constructed in order to exactly integrate (within
round-off error) problems whose solution lies in a finite dimensional linear space
(the so-called fitting space) spanned by a set of functions other than polynomials,
properly chosen to achieve the desired level of tuning to the problem. The
corresponding numerical method will depend on variable coefficients, which are
functions of the parameters characterizing the solution (e.g. the frequency of the
oscillations). Thus, we handle two main aspects:
(i) choosing a fitting space which is as much as possible suitable to represent
the solution of the problem;
(ii) accurately computing/estimating the parameters on which the numerical
method depends.
We show how these aspects can be accurately approached by taking into
account the existing theoretical studies on the problem. Practical constructive
aspects and accuracy issues will be treated, as well as numerical experiments
showing the effectiveness of the approach will be provided.
References
[1] M.J. Berridge, Calcium oscillations, J. Biol. Chem. 265, 9583–9586 (1990).
[2] R. D’Ambrosio, B. Paternoster, Numerical solution of a diffusion problem by exponentially
fitted finite difference methods, Springer Plus 3, 425–431 (2014).
[3] J.E. Ferrell, T.Y. Tsai, Q. Yang, Modeling the cell cycle: why do certain circuits
oscillate?, Cell. 144(6), 874885 (2011).
[4] N. Kopell, L.N. Howard, Plane wave solutions to reaction-diffusion equations, Stud.
Appl. Math. 52, 291328 (1973).
[5] J.A. Sherratt, Periodic waves in reaction-diffusion models of oscillatory biological
systems, FORMA 11, 6180 (1996)
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)
Numerical preservation of long-term dynamics by stochastic two-step methods
No full textThe paper aims to explore the long-term behaviour of stochastic two-step methods applied to a class of second order stochastic differential equations. In particular, the treatment focuses on preserving long-term statistics related to the dynamics of a linear stochastic damped oscillator whose velocity, in the stationary regime, is distributed as a Gaussian variable and uncorrelated with the position. By computing the solution of a very simple matrix equality, we a-priori determine the long-term statistics characterizing the numerical dynamics and analyze the behaviour of a selection of methods
Implicit - explicit (IMEX) methods for reaction-diffusion systems with non-polynomial fitting
No full textThis work concerns the numerical solution of λ-ω reaction-diffusion systems by using a problemoriented
approach. Since these systems have travelling waves as fundamental solutions [1], they are
largely used to model life science phenomena that are characterized by the generation of periodic
waves along their dynamics. For example, they describe cell cycles that are driven by an
autonomous biochemical oscillator [2] and intracellular calcium signaling, since the concentration
of calcium oscillate in time and space according to various extracellular signals[3].
The goal of this work is developing a new numerical method by using information we have about
the nature of the problem and the expression of the exact solution. Indeed, Kopell and Howard in
[4] have proved that the considered problem has an one-parameter family of periodic wave
solutions. We have applied the method of lines to the problem and we have approximated the
spatial second derivatives with adaptedd finite differences extending the ideas in [5]. Such formulas
have been constructed in order to be exact (within round-off error) on functions belonging to a
finite-dimensional space (called fitting space). For this technique, two main problems deserve
careful treatment: the choice of a proper fitting space and the estimate of the parameters in the
variable coefficients of the fitted formulas.
The system of ODEs arising from spatial semi-discretization is characterized by a stiff component
and a non-linear one. So it has been solved by using an IMEX method, that implicitly integrate the
first term and explicitly integrate the second one. This approach allows to reach stability without
increasing too much the computational cost. We will show the theoretical study of the properties of
the new method and the results of numerical experiments.
References
[1] J.A. Sherratt, On the evolution of periodic plane waves in reaction-diffusion systems of λ-ω
type, SIAM J. Appl. Math. 54, 1374-1385 (1994).
[2] J.E. Ferrell, T.Y. Tsai, Q. Yang, Modeling the cell cycle: why do certain circuits oscillate?,
Cell. 144(6), 874885 (2011).
[3] A. Atri, J. Amudson, D. Clapham, J. Sneyd, A single-pool model for intracellular calcium
oscillations and waves in Xenopus laevis Oocyte, Biophysical Journal 65, 1727-1739 (1993).
[4] N. Kopell, L.N. Howard, Plane waves solutions to reaction-diffusion equations, Studies in
Applied Mathematics 52, 291--328 (1973).
[5] R. D'Ambrosio, B. Paternoster, Numerical solution of reaction-diffusion systems of λ-ω type by
trigonometrically fitted methods, J. Comput. Appl. Math., in press
Adapted finite difference schemes advection-reaction-diffusion problems generating periodic wavefronts
No full textThe talk is focused on the numerical integration of advection-reaction-diffusion problems by finite difference schemes adapted to problem. In other terms, the numerical scheme, exploiting the a-priori knowledge of the qualitative behaviour of the solution, gains ad- vantages in terms of efficiency and accuracy with respect to classic schemes already known in literature. The adaptation is here carried out through the so-called trigonometrical fitting technique for the discretization in space, giving rise to a system of ODEs whose vector field contains both stiff and non-stiff terms. Due to this mixed nature of the vector field, an Implicit-Explicit (IMEX) method is here employed for the integration in time, based on the first order forward-backward Euler method. The coefficients of the method introduced rely on unknown parameters which have to be properly estimated: such an estimate is performed by minimizing the leading term of the local truncation error in an efficient way. The effectiveness of this problem-oriented approach is shown through a rigorous theoretical analysis and some numerical experiments.
References
[1] R. D’Ambrosio, M. Moccaldi and B. Paternoster, Adapted numerical schemes for advection-reaction-diffusion problems generating periodic wavefronts, Comp. Math. Appl. (2017).
[2] A.J. Perumpanani, J.A. Sherratt, P.K. Maini, Phase differences in reac- tion–diffusion–advection systems and applications to morphogenesis, J. Appl. Math. 55, 19-33 (1995)
Preserving structures of stochastic differential equations along numerical solutions
No full textThe aim of this talk is the analysis of the long-term behavior of stochastic linear multistep methods applied to a family of second order stochastic differential equations, modeling a stochastic damped oscillator, i.e. describing the position of a particle subject to the deterministic forcing and a random forcing whose global dynamics exhibits damped oscil- lations. In particular, the talk focuses on preserving long-term statistics related to such a dynamics; the velocity, in the stationary regime, is distributed as a Gaussian variable and uncorrelated with the position [1]. By computing the solution of a very simple ma- trix equality, we a-priori compute the long-term statistics characterizing the numerical dynamics and analyze the behaviour of a selection of methods [2].
References
[1] K. Burrage and G. Lythe, Numerical methods for second-order stochastic differential equations, SIAM J. Sci. Comput. 29 (1) (2007), 245–264.
[2] R. D’Ambrosio, M. Moccaldi and B. Paternoster, Long-term preservation of invariance laws by stochastic multistep methods, submitted
Stability issues in the discretization of stochastic differential equations
No full textThe aim of this talk is the analysis of various stability issues for numerical methods designed to solve stochastic differential equations. We first aim to consider nonlinear Ito stochastic differential equations (SDE): under suitable regularity conditions, exponential mean-square stability holds,. We aim to investigate the numerical counterpart of when trajectories are generated by stochastic linear multistep methods, in order to provide stepsize restrictions ensuring
analogous exponential mean-square stability properties also numerically [1, 4]. This is a joint
research with Evelyn Buckwar (Johannes Kepler University of Linz).
We next consider second order stochastic differential equations describing the position of a
particle subject to the deterministic forcing f(x) and a random forcing ξ(t) of amplitude ε.
The dynamics exhibits damped oscillations, with damping parameter η. We aim to analyze
long-term properties for indirect stochastic two-step methods, with special emphasis to understanding
the ability of such methods in retaining long-term invariance laws [2, 3]. This is a
joint research with Martina Moccaldi and Beatrice Paternoster (University of Salerno).
References
[1] E. Buckwar, R. D’Ambrosio, Exponential mean-square stability of linear multistep methods,
submitted.
[2] K. Burrage, G. Lythe, Numerical methods for second-order stochastic differential equations,
SIAM J. Sci. Comput. 29(1), 245–264 (2007).
[3] R. D’Ambrosio, M. Moccaldi, B. Paternoster, Numerical preservation of long-term dynamics
by stochastic two-step methods, Discr. Cont. Dyn. Sys. - B, doi: 10.3934/dcdsb.2018105
(2018).
[4] D.J. Higham, P.E. Kloeden, Numerical Methods for Nonlinear Stochastic Differential Equations
with Jumps, Numer. Math., 101, 101–119 (2005)
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