Two alternating direction implicit spectral methods for two-dimensional distributed-order differential equation

In this paper, two alternating direction implicit Galerkin-Legendre spectral methods for distributed-order differential equation in two-dimensional space are developed. It is proved that the schemes are unconditionally stable and convergent with the convergence orders O(Δt + σ2 + N−m) and O(Δt2 + σ2 + N−m), respectively, where Δt, σ, N, and m are the time step size, step size in distributed-order variable, polynomial degree, and regularity in the space variable of the exact solution, respectively. Moreover, the applicability and accuracy of the two schemes are demonstrated by numerical experiments to support our theoretical analysis.

[1]  Sigal Gottlieb,et al.  Spectral Methods , 2019, Numerical Methods for Diffusion Phenomena in Building Physics.

[2]  Hong Wang,et al.  A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations , 2015, J. Comput. Phys..

[3]  Anatoly A. Alikhanov,et al.  A new difference scheme for the time fractional diffusion equation , 2014, J. Comput. Phys..

[4]  Fawang Liu,et al.  Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains , 2015 .

[5]  Zhi-zhong Sun,et al.  Two unconditionally stable and convergent difference schemes with the extrapolation method for the one‐dimensional distributed‐order differential equations , 2016 .

[6]  Zhi-Zhong Sun,et al.  Some high-order difference schemes for the distributed-order differential equations , 2015, J. Comput. Phys..

[7]  Neville J. Ford,et al.  Distributed order equations as boundary value problems , 2012, Comput. Math. Appl..

[8]  Zhi-Zhong Sun,et al.  Two Alternating Direction Implicit Difference Schemes for Two-Dimensional Distributed-Order Fractional Diffusion Equations , 2016, J. Sci. Comput..

[9]  Jie Shen,et al.  Spectral Methods: Algorithms, Analysis and Applications , 2011 .

[10]  Fawang Liu,et al.  A Crank-Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation , 2014, SIAM J. Numer. Anal..

[11]  J. Faires,et al.  Numerical Methods , 2002 .

[12]  Xiaofeng Yang,et al.  Numerical approximations of Allen-Cahn and Cahn-Hilliard equations , 2010 .

[13]  Neville J. Ford,et al.  An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time , 2015 .

[14]  ShenJie.  Efficient spectral-Galerkin method I , 1994 .

[15]  Yuriko Renardy,et al.  Development and implementation of VOF-PROST for 3D viscoelastic liquid–liquid simulations , 2006 .

[16]  Yang Liu,et al.  Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem , 2015, Comput. Math. Appl..

[17]  Kai Diethelm,et al.  Numerical analysis for distributed-order differential equations , 2009 .

[18]  Jie Shen,et al.  Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials , 1994, SIAM J. Sci. Comput..

[19]  I. M. Sokolov,et al.  Fractional Fokker-Planck equation for ultraslow kinetics , 2003 .

[20]  Zhi-Zhong Sun,et al.  A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications , 2014, J. Comput. Phys..

[21]  Zhi-Zhong Sun,et al.  The Temporal Second Order Difference Schemes Based on the Interpolation Approximation for Solving the Time Multi-term and Distributed-Order Fractional Sub-diffusion Equations , 2017, J. Sci. Comput..

[22]  Anatoly N. Kochubei,et al.  Distributed order calculus and equations of ultraslow diffusion , 2008 .

[23]  Y. Sinai.  The Limiting Behavior of a One-Dimensional Random Walk in a Random Medium , 1983 .

[24]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..