A Crank--Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation | Zendy

Fanhai Zeng | Zendy; Fawang Liu | Zendy; Changpin Li | Zendy; Kevin Burrage | Zendy; Ian Turner | Zendy; Vo Anh | Zendy

Open Access

A Crank--Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation

Author(s) -

Fanhai Zeng,

Fawang Liu,

Changpin Li,

Kevin Burrage,

Ian Turner,

Vo Anh

Publication year - 2014

Publication title -

siam journal on numerical analysis

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 2.78

H-Index - 134

eISSN - 1095-7170

pISSN - 0036-1429

DOI - 10.1137/130934192

Subject(s) - mathematics , crank–nicolson method , discretization , galerkin method , alternating direction implicit method , nonlinear system , convergence (economics) , spectral method , mathematical analysis , fractional calculus , legendre polynomials , space (punctuation) , stability (learning theory) , finite difference method , linguistics , philosophy , physics , quantum mechanics , machine learning , computer science , economics , economic growth

In this paper, a new alternating direction implicit Galerkin--Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank--Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order 2 in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh--Nagumo model. Numerical results are provided to verify the theoretical analysis

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