Rate of convergence of Local Linearization schemes for random differential equations

Abstract

Recently, two Local Linearization (LL) schemes for the numerical integration of random differential equation have been proposed, which differ with respect to the algorithm that is used for the numerical implementation of the Local Linear discretization. However, in contrast with the Local Linear discretization, the order of convergence of the LL schemes have not been studied so far. In this paper, a general theorem about this matter is presented and, on that base, additional results are derived for each particular scheme.

Locally Linearized Runge Kutta method of Dormand and Prince

“…In this case, because of the flexibility in the numerical implementation of the LLRK methods mentioned in the introduction, the local linearization of the embedded Runge Kutta formulas of Dormand and Prince can be easily formulated in terms of the Krylov-type methods for exponential matrices. In effect, this can be done just by replacing the Padé formula in (10) and (12) by the Krylov-Padé formula as performed in [5,16,17] for the local linearizations schemes for ordinary, random and stochastic differential equations. In this way, the Locally Linearized formulas of Dormand and Prince could be applied to high dimensional ODEs with a reasonable computational cost [23].…”

“…where H is the 12-dimensional Hilbert matrix (with conditioned number 1.69 × 10 16 ), x i (t 0 ) = 1, i = 1 . .…”