Methods and Applications of Analysis

Volume 26 (2019)

Number 3

Special Issue in Honor of Roland Glowinski (Part 2 of 2)

Guest Editors: Xiaoping Wang (Hong Kong University of Science and Technology) and Xiaoming Yuan (The University of Hong Kong)

An alternating direction explicit method for time evolution equations with applications to fractional differential equations

Pages: 249 – 268



Hao Liu (School of Mathematics, Georgia Institute of Technology, Atlanta, Ga., U.S.A.)

Shingyu Leung (Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong)


We derive and analyze the alternating direction explicit (ADE) method for time evolution equations with the time-dependent Dirichlet boundary condition and with the zero Neumann boundary condition. The original ADE method is an additive operator splitting (AOS) method, which has been developed for treating a wide range of linear and nonlinear time evolution equations with the zero Dirichlet boundary condition. For linear equations, it has been shown to achieve the second order accuracy in time yet is unconditionally stable for an arbitrary time step size. For the boundary conditions considered in this work, we carefully construct the updating formula at grid points near the boundary of the computational domain and show that these formulas maintain the desired accuracy and the property of unconditional stability. We also construct numerical methods based on the ADE scheme for two classes of fractional differential equations. We will give numerical examples to demonstrate the simplicity and the computational efficiency of the method.


numerical methods, partial differential equations, fractional derivatives, operator splitting methods

2010 Mathematics Subject Classification

26A33, 65M06, 65M12

Dedicated to Professor Roland Glowinski on the occasion of his 80th birthday

The work of Leung was supported in part by the Hong Kong RGC grants 16303114 and 16309316.

Received 22 December 2017

Accepted 19 September 2019

Published 2 April 2020