Weak Second Order Multirevolution Composition Methods for Highly Oscillatory Stochastic Differential Equations with Additive or Multiplicative Noise

We introduce a class of numerical methods for highly oscillatory systems of stochastic differential equations with general noncommutative noise. We prove global weak error bounds of order two uniformly with respect to the stiffness of the oscillations, which permits to use large time steps. The approach is based on the micro-macro framework of multi-revolution composition methods recently introduced for deterministic problems and inherits its geometric features, in particular to design integrators preserving exactly quadratic first integral. Numerical experiments, including the stochastic nonlinear Schrodinger equation with space-time multiplicative noise, illustrate the performance and versatility of the approach.

[1]  A. Debussche,et al.  Numerical simulation of focusing stochastic nonlinear Schrödinger equations , 2002 .

[2]  B. Grébert,et al.  On the energy exchange between resonant modes in nonlinear Schrödinger equations ✩ Echange d ’ énergie entre modes résonants dans une équation de Schrödinger non linéaire cubique , 2011 .

[3]  Andreas Rößler.  Second Order Runge-Kutta Methods for Itô Stochastic Differential Equations , 2009, SIAM J. Numer. Anal..

[4]  G. Mil’shtein.  Weak Approximation of Solutions of Systems of Stochastic Differential Equations , 1986 .

[5]  Jesús María Sanz-Serna,et al.  Higher-Order Averaging, Formal Series and Numerical Integration I: B-series , 2010, Found. Comput. Math..

[6]  E Weinan,et al.  The Heterognous Multiscale Methods , 2003 .

[7]  Jerrold E. Marsden,et al.  Structure preserving Stochastic Impulse Methods for stiff Langevin systems with a uniform global error of order 1 or 1/2 on position , 2010, 1006.4657.

[8]  A. Ostermann,et al.  High order splitting methods for analytic semigroups exist , 2009 .

[9]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[10]  Linda R. Petzold,et al.  Numerical solution of highly oscillatory ordinary differential equations , 1997, Acta Numerica.

[11]  Stéphane Descombes,et al.  Splitting methods with complex times for parabolic equations , 2009 .

[12]  E Weinan,et al.  The heterogeneous multiscale method* , 2012, Acta Numerica.

[13]  G. Sussman,et al.  Chaotic Evolution of the Solar System , 1992, Science.

[14]  Stig Larsson,et al.  A Trigonometric Method for the Linear Stochastic Wave Equation , 2012, SIAM J. Numer. Anal..

[15]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

[16]  Marlis Hochbruck,et al.  Error analysis of exponential integrators for oscillatory second-order differential equations , 2006 .

[17]  Michael V. Tretyakov,et al.  Numerical Methods in the Weak Sense for Stochastic Differential Equations with Small Noise , 1997 .

[18]  Assyr Abdulle,et al.  Stabilized multilevel Monte Carlo method for stiff stochastic differential equations , 2013, J. Comput. Phys..

[19]  M. V. Tretyakov,et al.  Stochastic Numerics for Mathematical Physics , 2004, Scientific Computation.

[20]  Q. Sheng.  Solving Linear Partial Differential Equations by Exponential Splitting , 1989 .

[21]  Denis Talay,et al.  Efficient numerical schemes for the approximation of expectations of functionals of the solution of a S.D.E., and applications , 1984 .

[22]  A. D. Bouard,et al.  Weak and Strong Order of Convergence of a Semidiscrete Scheme for the Stochastic Nonlinear Schrodinger Equation , 2006 .

[23]  D. Talay.  Discrétisation d'une équation différentielle stochastique et calcul approché d'espérances de fonctionnelles de la solution , 1986 .

[24]  Kristian Debrabant,et al.  Diagonally drift-implicit Runge--Kutta methods of weak order one and two for Itô SDEs and stability analysis , 2009 .

[25]  David Cohen,et al.  Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations , 2012, Numerische Mathematik.

[26]  Manuel Palacios,et al.  A new approach to the construction of multirevolution methods and their implementation , 1997 .

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

[28]  Assyr Abdulle,et al.  Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations , 2013, SIAM J. Sci. Comput..

[29]  E Weinan,et al.  Heterogeneous multiscale methods: A review , 2007 .

[30]  Ander Murua,et al.  Stroboscopic Averaging for the Nonlinear Schrödinger Equation , 2015, Found. Comput. Math..

[31]  Fernando Casas,et al.  On the necessity of negative coefficients for operator splitting schemes of order higher than two , 2005 .

[32]  Ander Murua,et al.  Multi-revolution composition methods for highly oscillatory differential equations , 2014, Numerische Mathematik.

[33]  David Cohen,et al.  On the numerical discretisation of stochastic oscillators , 2012, Math. Comput. Simul..