An analysis of the L1 scheme for stochastic subdiffusion problem driven by integrated space-time white noise

Xiaolei Wu,Yuyuan Yan,Yubin Yan

doi:10.1016/j.apnum.2020.05.014

Xiaolei Wu, Yuyuan Yan + Show 1 more

Open Access

https://doi.org/10.1016/j.apnum.2020.05.014

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Abstract

We consider the strong convergence of the numerical methods for solving stochastic subdiffusion problem driven by an integrated space-time white noise. The time fractional derivative is approximated by using the L1 scheme and the time fractional integral is approximated with the Lubich's first order convolution quadrature formula. We use the Euler method to approximate the noise in time and use the truncated series to approximate the noise in space. The spatial variable is discretized by using the linear finite element method. Applying the idea in Gunzburger et al. (2019) [14], we express the approximate solutions of the fully discrete scheme by the convolution of the piecewise constant function and the inverse Laplace transform of the resolvent related function. Based on such convolution expressions of the approximate solutions, we obtain the optimal convergence orders of the fully discrete scheme in spatial multi-dimensional cases by using the Laplace transform method and the corresponding resolvent estimates.

Highlights

IntroductionWe will consider the numerical methods for solving the following stochastic time-fractional partial differential equation driven by integrated noise, with α ∈ (0, 1), γ ∈ [0, 1],
In this paper, we will consider the numerical methods for solving the following stochastic time-fractional partial differential equation driven by integrated noise, with α ∈ (0, 1), γ ∈ [0, 1], C 0 Dtαu(t) + Au(t) =R0 Dt−γ dW (t), dt for 0 < t ≤ T, with u(0) = u0, (1)where A : D(A) → H is an elliptic operator, with D(A) = H01(D) ∩ H2(D) and D ⊂ Rd, d = 1, 2, 3 is some regular domain
The time fractional derivative is approximated by using the L1 scheme and the time fractional integral is approximated with the Lubich’s first order convolution quadrature formula

Summary

Introduction

We will consider the numerical methods for solving the following stochastic time-fractional partial differential equation driven by integrated noise, with α ∈ (0, 1), γ ∈ [0, 1],. Gunzburger et al [14], [15] considered the time discretization and the finite element methods for solving stochastic integral-differential equations driven by space-time white noise. We shall consider the numerical methods for solving stochastic time fractional partial differential equation driven by integrated space-time white noise. 2. A new expression of the approximate solution for stochastic time-fractional partial differential equation is developed which is based on the convolution of the piecewise constant function and the inverse Laplace transform of the resolvent related function. By c we denote a particular positive constant independent of the functions and parameters concerned

Time discretization

Space discretization

Numerical experiments

Concluding remarks