A fourth-order compact finite difference method for nonlinear higher-order multi-point boundary value problems

A fourth-order compact finite difference method is proposed for a class of nonlinear 2n2nth-order multi-point boundary value problems. The multi-point boundary condition under consideration includes various commonly discussed boundary conditions, such as the three- or four-point boundary condition, (n+2)(n+2)-point boundary condition and 2(n−m)2(n−m)-point boundary condition. The existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. The convergence and the fourth-order accuracy of the method are proved. An efficient monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems. Various sufficient conditions for the construction of upper and lower solutions are obtained. Some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach.