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A PRACTICAL SPLITTING METHOD FOR STIFF SDES WITH APPLICATIONS TO PROBLEMS WITH SMALL NOISE
 

Summary: A PRACTICAL SPLITTING METHOD FOR STIFF SDES WITH
APPLICATIONS TO PROBLEMS WITH SMALL NOISE
HECTOR D. CENICEROS AND GEORGE O. MOHLER
Abstract. We present an easy to implement drift splitting numerical method for the approxi-
mation of stiff, nonlinear stochastic differential equations (SDEs). The method is an adaptation of
the SBDF multi-step method for deterministic differential equations and allows for a semi-implicit
discretization of the drift term to remove high order stability constraints associated with explicit
methods. For problems with small noise, of amplitude , we prove that the method converges
strongly with order O(t2 + t + 2t1/2) and thus exhibits second order accuracy when the time
step is chosen to be on the order of or larger. We document the performance of the scheme with
numerical examples and also present as an application a discretization of the stochastic Cahn-Hilliard
equation which removes the high order stability constraints for explicit methods.
Key words. stochastic differential equations, mean-square convergence, weak convergence,
multi-step methods, IMEX methods, Cahn-Hilliard equation, conservative phase field models, Langevin
equations.
AMS subject classifications. 60H35, 65L06
1. Introduction. Stochastic Partial Differential Equations (SPDEs) are an im-
portant and essential modeling tool in a wide range of fields from nonlinear filtering
to continuum physics [1]. Often the SPDEs employed in modeling a physical process
involve nonlinear and high order derivative terms and have an additional random force

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics