Commun. Comput. Phys.,
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Volume 3.

Effectiveness of Implicit Methods for Stiff Stochastic Differential Equations

Tiejun Li 1*, Assyr Abdulle 2, Weinan E 3

1 LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China.
2 School of Mathematics and Maxwell Institute for Mathematical Sciences, King's Buildings, Edinburgh, EH9 3JZ, UK.
3 Department of Mathematics and PACM, Princeton University, Princeton, NJ 08544, USA.

Received 6 March 2007; Accepted (in revised version) 5 April 2007
Available online 27 September 2007


In this paper we study the behavior of a family of implicit numerical methods applied to stochastic differential equations with multiple time scales. We show by a combination of analytical arguments and numerical examples that implicit methods in general fail to capture the effective dynamics at the slow time scale. This is due to the fact that such implicit methods cannot correctly capture non-Dirac invariant distributions when the time step size is much larger than the relaxation time of the system.

AMS subject classifications: 65L20, 65C30, 37M25

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Key words: Implicit methods, stiff ODE, stiff SDE, invariant distribution, multiscale.

*Corresponding author.
Email: (T. Li), (A. Abdulle), (W. E)

The Global Science Journal