Numerical approximation of nonlinear BVPs by means of BVMs

Abstract

Boundary Value Methods (BVMs) would seem to be suitable candidates for the solution of nonlinear Boundary Value Problems (BVPs). They have been successfully used for solving linear BVPs together with a mesh selection strategy based on the conditioning of the linear systems. Our aim is to extend this approach so as to use them for the numerical approximation of nonlinear problems. For this reason, we consider the quasi-linearization technique that is an application of the Newton method to the nonlinear differential equation. Consequently, each iteration requires the solution of a linear BVP. In order to guarantee the convergence to the solution of the continuous nonlinear problem, it is necessary to determine how accurately the linear BVPs must be solved. For this goal, suitable stopping criteria on the residual and on the error for each linear BVP are given. Numerical experiments on stiff problems give rather satisfactory results, showing that the experimental code, called TOM, that uses a class of BVMs and the quasi-linearization technique, may be competitive with well known solvers for BVPs.