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