Workshop on fractional and nonlocal modeling

1.Sheng Chen,  Jiangsu Normal University, Jiangsu

Title: An efficient  and accurate numerical method for the fractional Laplacian equation through the Caffarelli-­Silvestre extension.

Abstract:

The fractional Laplacian equation is difficult to solve numerically due to its non-local character. An effective approach is to use the Caffarelli-Silvestre extension which transforms the d+1-dimensional nonlocal fractional Laplacian equation into a d+1-dimensional elliptic problem with only regular derivatives. However, the d+1-dimensional elliptic problem involves a singular weight in the extended direction y which renders its solution to have weak singularity at y=0 so any standard approximation method will only yield very slow convergence.

 In a seminar paper by Nochetto, Otarola & Salgado in Found.Comput. Math. Entitled  “A PDE approach to fractional diffusion in general domains: a priori error analysis”, the authors proposed an adaptive finite-element method for solving  the d+1-dimensional extended problems and derived  rigorous error estimates.  Nochetto presented this work in his plenary address at ICIAM 2015 in Beijing. However, the convergence rate in the extended direction is limited by the low-order finite-element approximation. 

  By recognizing that the particular singular weight in the y-direction is exactly the weight associated with the generalized Laguerre functions, we developed an efficient method which reduces the d+1-dimensional problem to a sequence of d-dimensional Poisson type equations.Moreover, by adding the leading terms(in the extended direction) of the solutions to the approximation space, we developed an extended spectral method for this problem which enjoys the following properties: 

(i)     It converges exponentially fast in the extended  y-direction despite the weak singularity so it is very accurate;

(ii) it only requires solving a mall number of Poisson type equation in d-dimension so it is very efficient;

(iii) it can be used with any approximation method in the original d-dimension so it is very flexible. Wepresented ample numerical results to show that this method is very effective in dealing  with fractional

Laplacian equations and outperforms the existing methods by a wide margin.

2.Buyang Li, Hong Kong Polytechnic University,  HK

Title: Correction of high-order BDF convolution quadrature for fractional evolution equations

Abstract:

We develop proper correction formulas at the starting k-1 steps to restore the desired kth-order convergence rate of the k-step BDF convolution quadrature for discretizing evolution equations involving a fractional-order derivative in time. The desired kth-order convergence rate can be achieved even if the source term is not compatible with the initial data, which is allowed to be nonsmooth. We provide complete error estimates for the subdiffusion case α ∈ (0, 1), and sketch the proof for the diffusion-wave case α ∈ (1, 2).Extensive numerical examples are provided to illustrate the effectiveness of the proposed scheme.

3.Dongling Wang,  Northwest University, Shanxi

Title: Long time stability of fractional BDFs for nonlinear F-ODEs

Abstract: 

The contractivity and dissipativity of Caputo fractional-order ordinary differential equations (F-ODEs) are recently obtained in [D. Wang and A. Xiao, Nonlinear Dynam., 80 (2015), pp. 287-294]. In this paper, we further study the corresponding numerical properties. A discrete version of fractional generalization of Leibniz rule is derived, and a new asymptotic behavior for the solution of a linear Volterra difference equation with power decay rate is provided. Those properties are used to prove that fractional backward differential formulas including Gru ̈nwald-Letnikov formula and L1 method for F-ODEs are contractive and dissipative, and can exactly preserve the contractivity rate of the continuous solution. Several numerical examples are given to validate the numerical contractivity and dissipativity, reveal the different decay rates for the initial value perturbation between F-ODEs and integer-order ODEs, and show the superiority of the structure-preserving numerical methods.

 

4.Xiaobo Yin,  Central China Normal University, Hubei

报告题目：可积核函数非局部问题的数值解法

摘要：

对于可积核函数的非局部积分方程，通常解在边界处会有间断，此时采用连续有限元方法去离散原方程是不合适的。若采用间断有限元方法，要尽量减少自由度的个数，对于一维问题，可选取边界上的两个点处间断而其它点连续的有限元空间，这样做自由度少且解的精度高；对于二维问题，对于三角形网格很难选择自由度少的间断有限元空间，在这个报告中我们介绍了一种新的方法来求解可积核函数的非局部积分方程，这种方法得到的离散问题自由度少，并且收敛阶是最优的，数值算例也验证了这一点。

 

5.Huifang Yuan, Hong Kong Baptist University, HK

Title: A Hermite spectral collocation method for fractional PDEs in unbounded domain

Abstract:

This work is concerned with spectral collocation methods for fractional PDEs in unbounded domains. The method consists of expanding the solution with proper global basis functions and imposing collocation conditions on the Gauss-Hermite points. In this work, two Hermite-type functions are employed to serve as basis functions. Our main task is to find corresponding differentiation matrices which are computed recursively. Several numerical examples are carried out to demonstrate the effectiveness of the proposed methods.

6.Jiwei Zhang, Beijing Computational Science Research Center,

Title: Nonlocal Wave Propagation in Unbounded Multi-scale Mediums

Abstract:

This paper focuses on the simulation of nonlocal wave propagations in unbounded multi-scale mediums. To this end, we consider two folds: (a) the design of artificial/absorbing boundary conditions; and (b) the construction of an asymptotically compatible (AC) scheme for nonlocal operator with general kernel. The design of ABCs facilitates us to reformulate the problem on unbounded domains into a problem on bounded domains, which is useful to reduce the computational cost. The construction of AC scheme facilitates us that the simulations of nonlocal wave propagations in multi-scale mediums are valid.  With the proposed ABCs and AC scheme, wave propagation behaviors in the ``local” and nonlocal mediums are investigated through three numerical examples. Furthermore, accuracy of our approach is also validated. This is a joint work with Qiang Du, Houde Han and Chunxiong Zheng.

 

7.Xuan Zhao, Southeast University, Jiangsu

Title: Numerical methods for the space fractional problem

Abstract:

In this talk, Finite difference method, Finite element method and Spectral method for the space fractional problem involving Riesz fractional derivative and fractional Laplacian for multi-dimension will be introduce respectively. The main discussions are focused on the sigularity of the problem and the computational cost for the numerical methods.

 

8.Chunxiong Zheng, Tsinghua University, Beijing

Title: Stability and error analysis for a second-order fast approximation of the 1D Schrodinger equation under absorbing boundary conditions

Abstract:

A second-order Crank-Nicolson finite difference method, integrating a fast approximation of the discrete absorbing boundary conditions derived through convolution quadrature, is proposed for solving the one-dimensional Schrodinger equation under absorbing boundary conditions. The fast approximation is based on dyadic decomposition and Gaussian quadrature approximation of the con-volution coefficients in the discrete absorbing boundary conditions. It approximates the discrete convolution in time by a system of O(log2(N)) ordinary differential equations at each time step. Stability and error estimate are presented for the numerical solutions given by the proposed fast algorithm. Numerical experiments are provided, which agree with the theoretical results and show the performance of the proposed method. This is a joint work with Buyang Li and Jiwei Zhang.

9.Zhi Zhou, Hong Kong Polytechnic University,HK

 Title: Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint

Abstract: 

In this work, we present numerical analysis for a distributed optimal control problem, with box constraint on the control, governed by a subdiffusion equation which involves a fractional derivative of order (0,1)in time. The fully discrete scheme is obtained by applying the conforming linear Galerkin finite element method in space, L1 scheme/backward Euler convolution quadrature in time, and the control variable by a variational type discretization.With a space mesh size h and time stepsize τ, we prove the following order of convergence for the numerical solutions of the optimal control problem: O(τ^{min(1/2+α−ǫ,1)} +h²) in the discrete L²(0, T; L²(Ω)) norm and O(τ^α−ǫ +ℓ_h²h²) in the discrete L^∞(0,T;L²(Ω)) norm, with an arbitrarily small positive number ǫ and a logarithmic factor ℓ_h = ln(2+1/h). Numerical experiments are provided to support the theoretical results.