Crank-Nicolson compact difference schemes and their efficient implementations for a class of nonlocal nonlinear parabolic problems. (15th February 2023)

• Record Type: Journal Article
• Authors: Gong, Chunye; Li, Dongfang; Li, Lili; Zhao, Dan
• Is Part Of: Computers & mathematics with applications. Volume 132(2023)
• Page Start: 1
• Page End: 17
• Publication Date: 2023-02-15
• Subjects: Nonlocal initial conditions -- Compact difference scheme -- Crank-Nicolson method -- Stability -- Convergence; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2022.11.002
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: Due to the nonlocality and nonlinearity, it is usually difficult to approximate and analyze the semi-linear parabolic differential equations with nonlocal initial conditions. Several researchers have presented different numerical schemes for solving the problems. Only second-order convergence results in spatial direction are proved so far. In order to effectively solve the problem, the problem is approximated by using the fourth-order compact difference method in the spatial direction and the second-order Crank-Nicolson (CN) method in the temporal direction. The fully discrete numerical scheme gives a large system of nonlinear algebraic equations. The fixed-point iteration method with preconditioned GMRES is proposed for solving the resulting nonlinear systems. And the eigenvalues of the coefficient matrix are explored by the Hamilton-Cayley theorem and the investigation of the matrix structure. Stability and convergence of the fully discrete scheme are obtained. Numerical experiments are given to confirm the theoretical results.