Optimal explicit stabilized integrator of weak order one for stiff and ergodic stochastic differential equations

TL;DR

This paper introduces a new explicit stabilized integrator of weak order one for stiff and ergodic SDEs, which improves stability and accuracy in sampling invariant measures, outperforming existing methods.

Contribution

The paper presents a novel explicit stabilized scheme that matches deterministic stability properties and extends mean-square stability for stiff SDEs, with enhanced convergence for ergodic sampling.

Findings

Achieves quadratic stability domain growth with internal stages.

Attains order two convergence in invariant measure sampling.

Outperforms existing methods in stability and efficiency.

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].