Structure-preserving numerical methods for differential problems

It 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 $\lambda$-$\omega$ type by trigonometrically fitted methods}, submitted. \bibitem{sherratt94} J.A. Sherratt, {\em On the evolution of periodic plane waves in reaction-diffusion systems of $\lambda$-$\omega$ 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}