An explicit Euler scheme for generalized [formula omitted]-dimensional second-order differential equations with initial value conditions driven by additive Gaussian white noises

Abstract

The current paper is devoted to the numerical integration of generalized n-dimensional second-order differential equations with initial value conditions driven by additive Gaussian white noises. We submitted the Euler integrator for the integral form of this type of differential equation. Convergence analysis shows that the proposed scheme order of strong superconvergence is 1.0 when diffusion terms, in general form gl2(t,ρ), l=1,2, satisfies gl2(t,t)=0. While, the strong convergence order of the scheme is 1/2 if gl2(t,t)≠0. Especially, for convolution diffusion terms gl2(t,ρ)=gl2(t−ρ) our scheme has strong superconvergence order 1.0 when gl2(0)=0. And scheme strongly convergent with order 1/2 if gl2(0)≠0. Finally, four problems are considered to exhibit the efficiency and compatibility of the numerical integrator.