1,721,133 research outputs found
Canonical Properties of General Linear Methods for Hamiltonian Problems
No full textThis talk is devoted to the investigation of the canonical properties of general linear methods for long time integration of Hamiltonian problems. Runge-Kutta methods exhibit some fundamental canonical properties if they are symplectic. It is known that general linear methods cannot be symplectic (see [3]), however it is possible to inherit from their nonlinear stability properties a nearly canonical behavior, known as G-symplecticity [1,2], which is
the first ingredient to obtain an accurate conservation of the invariants of an Hamiltonian problem. Due to their multivalue nature, general linear methods generates a parasitic behavior of the numerical solution which needs to be properly removed: we discuss how G-symplectic general linear methods free from parasitism can be developed. The third aspect we aim to discuss is symmetry: in particular, we explain how time reversal symmetry allows
to derive methods of a certain order by applying a reduced number of order conditions. Numerical experiments on a selection of Hamiltonian problems are discussed. 1. J. C. Butcher 2008 Numerical methods for Ordinary Differential Equations, Second Edition, Wiley. 2. J. C. Butcher, R. D’Ambrosio, Partitioned general linear methods for separable Hamiltonian problems, in preparation. 3. J. C. Butcher and L. L. Hewitt 2009 The existence of symplectic general linear methods, Numer. Algor. 51, 77-84
Recent advances in numerical modeling for differential problems
No full textIt is the purpose of this talk to analyze the behaviour of some classes of numerical methods acting as structure-preserving integrators for the numerical solution of ordinary (ODEs) and partial differential equations (PDEs), with special emphasys to Hamiltonian problems, discontinuous ODEs, reaction-diffusion problems and stochastic differential equations (SDEs). The methodology we aim to follow is unifying, i.e. we propose problem-oriented numerical solvers, able to accurately and efficiently reproduce typical properties and behaviors of the above mentioned problems. As regards ODEs, we provide a rigorous long-term error analysis in energy conservation for Hamiltonian problems by multi-value methods, obtained by means of backward error analysis arguments, and provide a study of the stability of periodic orbits in discontinuous ODEs by analyzing the behaviour of the monodromy matrix and Floquet multipliers associated the the orbit. Concerning PDEs, we present an adapted method of lines for problems with periodic or oscillatory solutions, by means of non-polynomially fitted finite differences. The coefficients of the resulting methods will depend on the unknown values of parameters related to the problem (e.g. the values of the frequencies of the oscillations): suitable techniques leading to estimates of the parameters will be discussed. Focusing on nonlinear SDEs, we analyze the conditional properties of stochastic linear multistep methods to retain dissipativity properties. Exponential mean square contractivity is analyzed, through results revealing some conditional nonlinear stability properties leading to accurate bounds for the stepsize. Numerical experiments on a selection of nonlinear problems are presented. The presented results deal with a series of joint works in collaboration with Evelyn Buckwar (Johannes Kepler University of Linz), John C. Butcher (University of Auckland), Luca Dieci and Fabio Difonzo (Georgia Institute of Technology), Ernst Hairer (University of Geneva), Beatrice Paternoster and Martina Moccaldi (University of Salerno). Part of this work has been granted by the Fulbright project "Discontinuous Dynamical Systems: An Accurate and Efficient Framework for Their Numerical Solution"
Multivalue numerical methods for partitioned differential problems: from second order ODEs to separable Hamiltonians
No full textThis talk introduces general families of multivalue methods for the numerical solution of partitioned differential problems which includes, as special cases, second order ODEs of special type and Hamiltonian problems with separable Hamiltonian. We present the family of General Nystrom methods, introduced in [2] with the initial aim to provide an unifying approach for the analysis of minimal accuracy and stability demandings. Within this family, we derive high order P-stable formulae which result to be competitive with respect to classical Runge-Kutta-Nystrom methods in numerically solving
periodic sti problems [3]. As regards problem (2), the family of partitioned general linear methods is introduced [1]. With the aim of obtaining a long-term near conservation of the Hamiltonian of (2), we present suitable notions of G-symplecticity and symmetry for these methods. A technique for the generation of numerical solutions with bounded parasitic components is also discussed. Numerical experiments on a selection of separable Hamiltonian problems are reported. The talk deals with a series of joint papers with John Butcher (University of Auckland) and Beatrice Paternoster (University of Salerno). References 1. J. C. Butcher, R. D'Ambrosio, Partitioned general linear methods for separable Hamiltonian problems, in preparation.
2. R. D'Ambrosio, E. Esposito, B. Paternoster, General linear methods for y'' = f(y(t)), Numer. Algor. 61(2), 331{349 (2012). 3. R. D'Ambrosio, B. Paternoster, P-stable General Linear Nystrom methods, submitted
On the G-symplecticity of two-step Runge-Kutta methods
No full textThis paper investigates the conservative behaviour of two-step Runge-Kutta (TSRK) methods for the numerical integration of Hamiltonian systems. In particular, the attention is focused on the existence of G-symplectic TSRK methods, according to the definition provided in Butcher's 2008 monograph
On the G-symplecticity of two-step Runge-Kutta methods
Full text linkThis paper investigates the conservative behaviour of two-step Runge-Kutta (TSRK) methods and multistep Runge-Kutta methods (MRK) for the numerical integration of Hamiltonian systems. In particular, the attention is focused on the existence of G-symplectic TSRK and MRK methods, according to the denition provided in Butcher's 2008 monograph
Stability issues in the numerical solution 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 a nonlinear system of Ito stochastic differential equation (SDE)
dX(t)=f(X(t))dt+g(X(t))dW(t), t>0.
Under suitable regularity conditions, exponential mean-square stability holds, i.e. any two solutions and of the SDE with and satisfy
with . We aim to investigate its numerical counterpart when trajectories are generated by stochastic linear multistep methods, in order to provide stepsize restrictions ensuring analogous exponential mean-square stability properties also numerically.
We next move to the following second order stochastic differential equation
describing the position of a particle subject to the deterministic forcing and a random forcing of amplitude . The dynamics exhibits damped oscillations, with damping parameter . We aim to analyze asymptotic mean-square stability properties for partitioned Runge-Kutta methods and multistep linear methods, thought as applied to the system equivalent to
This is a joint work with E. Buckwar (Univ. of Linz), M. Moccaldi and B. Paternoster (Univ. of Salerno).
[1] 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)
Structure-preserving numerical methods for differential problems
No full textIt is the purpose of this talk to analyze the behaviour of multi-value numerical methods acting as structure-preserving integrators for the numerical solution of ordinary and partial differential equations (PDEs), with special emphasys to Hamiltonian problems, reaction-diffusion problems and stochastic differential equations (SDEs).
The methodology we aim to follow is unifying, i.e. we propose problem-oriented numerical solvers, able to accurately and efficiently reproduce typical properties and behaviors of the above mentioned problems. Thus, according to this clear perspective, the methods we consider are adapted to the problem, in contrast with general purpose solvers that, on the contrary, do not take into account specific features of the problems. A descriptions of the issues regarding each of the above mentioned problems now follows, with the aim to describe how this unifying structure-preserving approach is adapted in concrete to each of the mentioned operators.
As regards Hamiltonian problems, we analyze the nearly conservative behavior of multi-value numerical methods. Such methods, even though they cannot be symplectic, may act as structure-preserving integrators if the fulfill the properties of G-symplecticity \cite{bh2}, symmetry and bounded parasitic components which come into the numerical solution due to the multi-value nature of the solver. In particular, we provide a rigorous long-term error analysis regarding energy conservation for Hamiltonian problems \cite{bh1}, obtained by means of backward error analysis arguments, leading to sharp estimates for the parasitic solution components and for the error in the Hamiltonian. We also discuss the way these features characterize partitioned multi-value methods, specific for solving separable Hamiltonian problems, by pointing out how they can lead to overall explicit numerical schemes, also in comparison with existing symplectic partitioned Runge-Kutta methods \cite{but}.
As regards PDEs, we present novel finite difference schemes for problems with periodic or oscillatory solutions of interest in the mathematical modeling of oscillatory biological systems \cite{sherratt94,sherratt09}. We mainly focus our attention on problem-oriented numerical schemes as in \cite{ref2,ref1}, based on adapted finite difference formulae arising from a twofold level of adaptation to the problem: along space, by approximating the spatial derivatives appearing in the operator by means of finite differences based on non-polynomial fitting techniques; along time, by integrating the semi-discretized problems via special purpose numerical solvers. The coefficients of the resulting methods will depend on the unknown values of parameters related to the problem (e.g. the values of the frequencies of the oscillations): suitable techniques leading to estimates of the parameters will be discussed. Moreover, further issues on the possibility to improve the efficiency of the solvers by assessing adapted IMEX numerical schemes will also be briefly described.
As regards SDEs, the perspective is that of analyzing the potential of stochastic linear multistep methods to act as structure-preserving integrators, with special emphasys to numerically retaining dissipativity properties possessed by the problem \cite{bd}. Exponential mean square contractivity is analyzed, through results revealing some conditional nonlinear stability properties leading to accurate bounds for the stepsize. Numerical experiments on a selection of nonlinear problems are presented.
The presented results deal with a series of joint works in collaboration with Evelyn Buckwar (Johannes Kepler University of Linz), John C. Butcher (University of Auckland), Ernst Hairer (University of Geneva) and Beatrice Paternoster (University of Salerno).
\begin{thebibliography}{4}
\bibitem{bd} E. Buckwar, R. D'Ambrosio, {\em Mean square contractivity of stochastic linear multistep methods}, in preparation.
\bibitem{but} J.C. Butcher, R. D'Ambrosio, {\em Partitioned general linear methods for separable Hamiltonian problems}, in preparation.
\bibitem{bh1} R. D'Ambrosio, E. Hairer, {\em Long-term stability of multi-value methods for ordinary differential equations}, J. Sci. Comput. 60(3), 627--640 (2014).
\bibitem{bh2} R. D'Ambrosio, E. Hairer, C.J. Zbinden, {\em G-symplecticity implies conjugate-symplecticity of the underlying one-step method}, BIT 53, 867-872 (2013).
\bibitem{ref2} R. D'Ambrosio, B. Paternoster, {\em Numerical solution of a diffusion problem by exponentially fitted finite difference methods}, Springer Plus 3, 425--431 (2014).
\bibitem{ref1} R. D'Ambrosio, B. Paternoster, {\em Numerical solution of reaction-diffusion systems of - type by trigonometrically fitted methods}, submitted.
\bibitem{sherratt94} J.A. Sherratt, {\em On the evolution of periodic plane waves in reaction-diffusion systems of - type}, SIAM J. Appl. Math. 54(5), 1374--1385 (1994).
\bibitem{sherratt09} M.J. Smith, J.D.M. Rademacher, J.A. Sherratt, {\em Absolute stability of wavetrains can explain spatiotemporal dynamics in reaction-diffusion systems of lambda-omega type}, SIAM J. Appl. Dyn. Systems 8, 1136--1159 (2009).
\end{thebibliography
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