Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

Abstract

To cite this version:Assyr Abdulle, Gilles Vilmart, Konstantinos Zygalakis. Weak second order explicit stabilized methods for stiff stochastic differential equations. SIAM J. Sci. Comput., Sociey for Industrial and Applied Mathematics, 2013, 35 (4) We introduce a new family of explicit integrators for stiff Itô stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer from the stepsize reduction faced by standard explicit methods. The family is based on the standard second order orthogonal Runge-Kutta Chebyshev methods (ROCK2) for deterministic problems. The convergence, mean-square and asymptotic stability properties of the methods are analyzed. Numerical experiments, including applications to nonlinear SDEs and parabolic stochastic partial differential equations are presented and confirm the theoretical results.

Brownian dynamics of confined suspensions of active microrollers

“…[41,42]. However, these schemes are too expensive to solve (1) for tens to hundreds of thousands of particles.…”

Explicit stabilized integration of stiff determinisitic or stochastic problems

“…The parameter α introduced in [5] allows to adjust the damping of the method and the value of σ α and τ α can be adjusted to obtain second order accuracy. Although a simple idea, this new way of introducing variable damping in the ROCK2 methods allows for interesting extension to stochastic problems and advection diffusion reaction problems.…”

“…where I q,r = Using the ROCK methods (3), appropriate damping and considering further two additional orthogonal polynomials P s−1 (p), P s (p) (to enhance the damping of the noise terms) weak second order methods (S-ROCK2) have been constructed in [5]. They can be seen as a stabilized version of the derivative free Talay-Milstein method.…”

Improved Stabilized Multilevel Monte Carlo Method for Stiff Stochastic Differential Equations

“…In fact, defining S SDE,a = (p, q) ∈ [−a, 0] × R | |q| ≤ √ −2p a "portion" of S exact and a * = sup {a > 0 | S SDE,a ⊂ S num }, we get for S-ROCK1 and S-ROCK2 a * ≈ c SR1 s 2 and a * ≈ c SR2 (s + 2) 2 , respectively. The parameters c SR1 and c SR2 quickly reach a value independent of the stage number that can be estimated numerically as 0.33 (S-ROCK1) and 0.42 (S-ROCK2) [3,4,1].…”

“…Here the first s − 1 represent the stabilization procedure and the last stage a finishing procedure to achieve strong order 1/2 and weak order 1 [3,4]. We will also consider the S-ROCK2 method introduced in [1]. Similar to the S-ROCK1 this scheme uses a stabilization procedure (in this case ROCK2 [5]) on the first s − 2 stages and then a finishing procedure on the last two stages to obtain a weak order of 2 and a strong order of 1/2.…”