TY - JOUR
T1 - Numerical Analysis of Fully Discretized Crank–Nicolson Scheme for Fractional-in-Space Allen–Cahn Equations
AU - Hou, Tianliang
AU - TANG, Tao
AU - Yang, Jiang
N1 - Funding Information:
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.
PY - 2017/9/1
Y1 - 2017/9/1
N2 - 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 O(Nlog 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.
AB - 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 O(Nlog 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.
KW - Allen–Cahn equations
KW - Energy stability
KW - Error analysis
KW - Finite difference method
KW - Fractional derivatives
KW - Maximum principle
UR - http://www.scopus.com/inward/record.url?scp=85013852201&partnerID=8YFLogxK
U2 - 10.1007/s10915-017-0396-9
DO - 10.1007/s10915-017-0396-9
M3 - Journal article
AN - SCOPUS:85013852201
SN - 0885-7474
VL - 72
SP - 1214
EP - 1231
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
IS - 3
ER -