A high-order compact finite difference framework for solving nonlinear multipoint boundary value problems

Abstract We present a robust numerical framework for solving general second-order multipoint boundary value problems. The methodology combines the quasilinearization method (QLM) for iterative linearization with the high-order compact finite difference (CFD) schemes for spatial discretization. The nonlinearity in the governing equation is systematically handled by transforming the problem into a sequence of linear boundary value problems. Each linear problem is then discretized using sixth-order compact schemes for both first and second-order derivatives, ensuring high-fidelity approximations. The derivation of these schemes, including the necessary non-centered boundary closures, is detailed. The formulation culminates in a sequence of matrix systems that are solved iteratively to achieve a converged solution. This framework is broadly applicable and provides high-order accuracy, making it suitable for challenging problems in science and engineering.