Abstract
We consider numerical methods for solving the fractional-in-space Allen–Cahn equation which contains small perturbation parameters and strong nonlinearity. A standard fully discretized scheme for this equation is considered, namely, using the conventional second-order Crank–Nicolson scheme in time and the second-order central difference approach in space. For the resulting nonlinear scheme, we propose a nonlinear iteration algorithm, whose unique solvability and convergence can be proved. The nonlinear iteration can avoid inverting a dense matrix with only \(\mathcal {O}(N\log N)\) computation complexity. One major contribution of this work is to show that the numerical solutions satisfy discrete maximum principle under reasonable time step constraint. Based on the maximum stability, the nonlinear energy stability for the fully discrete scheme is established, and the corresponding error estimates are investigated. Numerical experiments are performed to verify the theoretical results.
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Acknowledgements
The research of the Tianliang Hou is supported by National Natural Science Foundation of China (No. 11526036), Scientific and Technological Developing Scheme of Jilin Province (No. 20160520108JH), and Science and Technology Research Project of Jilin Provincial Department of Education (No. 201646). The research of the Tao Tang is partially supported by Hong Kong Research Grants Council, National Science Foundation of China, and Southern University of Science and Technology.
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Hou, T., Tang, T. & Yang, J. Numerical Analysis of Fully Discretized Crank–Nicolson Scheme for Fractional-in-Space Allen–Cahn Equations. J Sci Comput 72, 1214–1231 (2017). https://doi.org/10.1007/s10915-017-0396-9
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DOI: https://doi.org/10.1007/s10915-017-0396-9
Keywords
- Fractional derivatives
- Allen–Cahn equations
- Finite difference method
- Maximum principle
- Energy stability
- Error analysis