Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise

In this paper we extend the ideas of Brugnano, Iavernaro and Trigiante in their development of HBVM (s,r) methods to construct symplectic Runge-Kutta methods for all values of s and r with s>=r. However, these methods do not see the dramatic performance improvement that HBVMs can attain. Nevertheless, in the case of additive stochastic Hamiltonian problems an extension of these ideas, which requires the simulation of an independent Wiener process at each stage of a Runge-Kutta method, leads to methods that have very favourable properties. These ideas are illustrated by some simple numerical tests for the modified midpoint rule.

[1]  Henri Schurz,et al.  The Invariance of Asymptotic Laws of Linear Stochastic Systems under Discretization , 1999 .

[2]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[3]  Donato Trigiante,et al.  The lack of continuity and the role of infinite and infinitesimal in numerical methods for ODEs: The case of symplecticity , 2010, Appl. Math. Comput..

[4]  K. Burrage,et al.  Stability Criteria for Implicit Runge–Kutta Methods , 1979 .

[5]  Kevin Burrage,et al.  Parallel and sequential methods for ordinary differential equations , 1995, Numerical analysis and scientific computation.

[6]  Donato Trigiante,et al.  A note on the efficient implementation of Hamiltonian BVMs , 2010, J. Comput. Appl. Math..

[7]  Christian Soize.  Exact steady-state solution of FKP equation in higher dimension for a class of non linear Hamiltonian dissipative dynamical systems excited by Gaussian white noise , 1995 .

[8]  Kevin Burrage,et al.  Numerical Methods for Second-Order Stochastic Differential Equations , 2007, SIAM J. Sci. Comput..

[9]  L. Brugnano,et al.  A simple framework for the derivation and analysis of effective one-step methods for ODEs , 2010, Appl. Math. Comput..

[10]  L. Brugnano,et al.  Hamiltonian Boundary Value Methods ( Energy Preserving Discrete Line Integral Methods ) 1 2 , 2009 .

[11]  C. W. Gardiner,et al.  Handbook of stochastic methods - for physics, chemistry and the natural sciences, Second Edition , 1986, Springer series in synergetics.

[12]  Kevin Burrage,et al.  High order algebraically stable Runge-Kutta methods , 1978 .

[13]  M. Crouzeix.  Sur laB-stabilité des méthodes de Runge-Kutta , 1979 .

[14]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[15]  Desmond J. Higham,et al.  Numerical simulation of a linear stochastic oscillator with additive noise , 2004 .

[16]  K. Vahala.  Handbook of stochastic methods for physics, chemistry and the natural sciences , 1986, IEEE Journal of Quantum Electronics.

[17]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .