Optimal Explicit Stabilized Integrator of Weak Order 1 for Stiff and Ergodic Stochastic Differential Equations

Abstract

A new explicit stabilized scheme of weak order one for stiff and ergodic stochastic differential equations (SDEs) is introduced. In the absence of noise, the new method coincides with the classical deterministic stabilized scheme (or Chebyshev method) for diffusion dominated advection-diffusion problems and it inherits its optimal stability domain size that grows quadratically with the number of internal stages of the method. For mean-square stable stiff stochastic problems, the scheme has an optimal extended mean-square stability domain that grows at the same quadratic rate as the deterministic stability domain size in contrast to known existing methods for stiff SDEs [A. Abdulle and T. Li. Commun. Math. Sci., 6(4), 2008, A. Abdulle, G. Vilmart, and K. C. Zygalakis, SIAM J. Sci. Comput., 35(4), 2013]. Combined with postprocessing techniques, the new methods achieve a convergence rate of order two for sampling the invariant measure of a class of ergodic SDEs, achieving a stabilized version of the non-Markovian scheme introduced in [B. Leimkuhler, C. Matthews, and M. V. Tretyakov, Proc. R. Soc. A, 470, 2014].

A fully adaptive explicit stabilized integrator for advection-diffusion-reaction problems

“…In this paper we use the idea of stabilization by combining first and second kind Chebyshev polynomials introduced in [4] to derive explicit stabilized methods for advection-diffusion problems with, possibly, costly non-stiff reaction terms, ∂ t u(x, t) = ∇ • (D∇u(x, t)) − ∇ • (vu(x, t)) + r(u(x, t)), (x, t) ∈ Ω × [0, T ], with initial and boundary conditions, where Ω ∈ R d , D is the matrix of diffusion coefficients, and v is the velocity vector. The function r represents non-stiff, but possibly costly, reaction terms.…”

“…Explicit stabilized Runge-Kutta-Chebyshev methods were originally introduced in the context of purely diffusive or diffusion dominated advection-diffusion problems (very small Peclet number) as a compromise between costly implicit methods and restrictive usual explicit schemes [1,2,3,6,17]. Due to their versatility, they were extended to many other types of problems such as advection-diffusion-reaction equations [7,18,19,20], stochastic differential equations (SDEs) [4,5,8], and optimal control problems [10].…”

Explicit Stabilized Integrators for Stiff Optimal Control Problems

“…the survey [2]. It was extended to the stochastic context first in [4,5] and recently in [3] for the design of explicit stabilized integrators with optimally large stability domains in the context of mean-square stable stiff and ergodic problems.…”

Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs

“…The aim of this paper is to provide a unified algebraic framework based on aromatic trees and B-series, with a set of trees independent of the dimension d of the problem, for the systematic study of the order conditions for the invariant measure of a class of numerical integrators that includes Runge-Kutta type schemes for problems of the form (1.1). We show that the new framework permits to recover some schemes and simplify the calculations in [3] and for postprocessed integrators in [47,10,1]. Analogously to [41] (we study here the additive noise case), we consider in this paper Runge-Kutta methods of the form 2…”