Chuying Huang

Jialin Hong, Chuying Huang, Xu Wang

We consider Hamiltonian systems driven by multi-dimensional Gaussian processes in rough path sense, which include fractional Brownian motions with Hurst parameter $H\in(1/4,1/2]$. We indicate that the phase flow preserves the symplectic structure almost surely and this property could be inherited by symplectic Runge--Kutta methods, which are implicit methods in general. If the vector fields belong to $Lip^γ$, we obtain the solvability of Runge--Kutta methods and the pathwise convergence rates. For linear and skew symmetric vector fields, we focus on the midpoint scheme to give corresponding results. Numerical experiments verify our theoretical analysis.

Jialin Hong, Chuying Huang, Xu Wang

This paper focuses on the strong convergence rate of both Runge--Kutta methods and simplified step-$N$ Euler schemes for stochastic differential equations driven by multi-dimensional fractional Brownian motions with $H\in(\frac12,1)$. Based on the continuous dependence of both stage values and numerical schemes on driving noises, order conditions of Runge--Kutta methods are proposed for the optimal strong convergence rate $2H-\frac12$. This provides an alternative way to analyze the convergence rate of explicit schemes by adding `stage values' such that the schemes are comparable with Runge--Kutta methods. Taking advantage of this technique, the optimal strong convergence rate of simplified step-N Euler scheme is obtained, which gives an answer to a conjecture in $[3]$ when $H\in(\frac12,1)$. Numerical experiments verify the theoretial convergence rate.

Jialin Hong, Chuying Huang, Zhihui Liu

For semilinear stochastic evolution equations whose coefficients are more general than the classical global Lipschitz, we present results on the strong convergence rates of numerical discretizations. The proof of them provides a new approach to strong convergence analysis of numerical discretizations for a large family of second order parabolic stochastic partial differential equations driven by space-time white noises. We apply these results to the stochastic advection-diffusion-reaction equation with a gradient term and multiplicative white noise, and show that the strong convergence rate of a fully discrete scheme constructed by spectral Galerkin approximation and explicit exponential integrator is exactly $\frac12$ in space and $\frac14$ in time. Compared with the optimal regularity of the mild solution, it indicates that the spetral Galerkin approximation is superconvergent and the convergence rate of the exponential integrator is optimal. Numerical experiments support our theoretical analysis.