Strong convergence order for the scheme of fractional diffusion equation driven by fractional Gaussion noise

Abstract

Fractional Gaussian noise models the time series with long-range dependence; when the Hurst index H > 1/2, it has positive correlation reflecting a persistent autocorrelation structure. This paper studies the numerical method for solving stochastic fractional diffusion equation driven by fractional Gaussian noise. Using the operator theoretical approach, we present the regularity estimate of the mild solution and the fully discrete scheme with finite element approximation in space and backward Euler convolution quadrature in time. The O(τ H−ρα ) convergence rate in time and O(h min(2,2−2ρ, H α ) ) in space are obtained, showing the relationship between the regularity of noise and convergence rates, where ρ is a parameter to measure the regularity of noise and α ∈ (0, 1). Finally, numerical experiments are performed to support the theoretical results.

Numerical Approximation for Stochastic Nonlinear Fractional Diffusion Equation Driven by Rough Noise

“…As we all know, fractional noises exist widely in the natural world, such as flows in porous media, the rough Hamiltonian systems, water flows in hydrology and so on [6,11]. In recent years, there have been many discussions about numerically solving stochastic partial differential equations driven by fractional Brownian motion with Hurst parameter H ∈ ( 1 2 , 1) (it can be called as "smoother noise") [4,25,29,30]. But for the case H ∈ (0, 1 2 ) (called as "rough noise"), the existing discussions seem to be few.…”

Numerical analysis for stochastic time-space fractional diffusion equation driven by fractional Gaussion noise

“…Extensive numerical schemes for the deterministic fractional diffusion equation (1.3) have been proposed in [2, 5,28]. Also, there have been many works for numerically solving stochastic partial differential equations (PDEs) involving Laplace and spectral fractional Laplacian; one can refer to [7,16,17,20,21,23,29,33].…”