Two-grid quasilinearization approach to ODEs with applications to model problems in physics and mechanics

In this paper we propose a two-grid quasilinearization method for solving high order nonlinear differential equations. In the first step, the nonlinear boundary value problem is discretized on a coarse grid of size H. In the second step, the nonlinear problem is linearized around an interpolant of the computed solution (which serves as an initial guess of the quasilinearization process) at the first step. Thus, the linear problem is solved on a fine mesh of size h  , h≪Hh≪H. On this base we develop two-grid iteration algorithms, that achieve optimal accuracy as long as the mesh size satisfies h=O(Hr2)h=O(H2r), r=1,2,…r=1,2,… , where r is the rth Newton's iteration for the linearized differential problem. Numerical experiments show that a large class of NODEs, including the Fisher–Kolmogorov, Blasius and Emden–Fowler equations solving with two-grid algorithm will not be much more difficult than solving the corresponding linearized equations and at the same time with significant economy of the computations.