A Uniformly Accurate Scheme for the Numerical Integration of Penalized Langevin Dynamics

Uniform strong and weak error estimates for numerical schemes applied to multiscale SDEs in a Smoluchowski–Kramers diffusion approximation regime

“…Obtaining a weak order of convergence 1{2, which is not an integer, is unusual and suprising for approximation of solutions of stochastic differential equations driven by a Wiener process and with smooth nonlinearities. Order reduction of this type when considering uniform weak error estimates for numerical schemes applied to stochastic differential equations have been obtained in the recent works [4] and [15], while it does not appear in [2]. Note that this order reduction has not been exhibited in numerical experiments so far, to the best of our knowledge.…”

“…The analysis of numerical methods for multiscale stochastic differential equations is an active research area. The recent articles [2] and [15] propose uniformly accurate methods for SDE systems which are different from (1) considered in this article. The recent article [4] has introduced a notion of asymptotic preserving schemes which applies to (1), and some uniform error estimates were proved for SDE systems in an averaging regime.…”

“…Uniform strong error estimates. In this subsection, let us consider the numerical scheme (15). One has the following result concerning the strong error Er|q ,∆t n ´q pn∆tq| 2 s for the q-component, when ∆t Ñ 0.…”

“…Theorem 3.1. Let Assumptions 1 and 2 be satisfied, and let `q ,∆t n ˘ně0 be given by the numerical scheme (15). For all T P p0, 8q, there exists CpT q P p0, 8q such that for all ∆t " T {N P p0, ∆t 0 q and n P t1, .…”

“…with the initial values q ,∆t 0 " q 0 and p ,∆t 0 " p 0 (given by Assumption 2), and we recall that the Wiener increments ∆β n are given by (8). In the scheme (15), the q-component is discretized explicitly, whereas the p-component is treated implicitly. This choice is made to ensure stability properties (in particular to be able to choose the time-step size ∆t independently of the time scale separation parameter ) in the second equation, and to ensure good behavior of q ,∆t n when vanishes, in the first equation, as will be explained below.…”