Piecewise-linearized methods for the solution of two-point boundary value problems in ordinary differential equations are presented. These problems are approximated by piecewise linear ones which have analytical solutions and reduced to finding the slope of the solution at the left boundary so that the boundary conditions at the right end of the interval are satisfied. This results in a rather complex system of non-linear algebraic equations which may be reduced to a single non-linear equation whose unknown is the slope of the solution at the left b o u n b of the interval and whose solution may be obtained by means of the Newton-Raphson method. This is equivalent to solving the boundary value problem as an initial value one using the piecewise-linearized technique and a shooting method. It is shown that for problems characterized by a linear operator a technique based on the superposition principle and the piecewise-linearized method may be employed. For these problems the accuracy of piecewise-linearized methods is of second order. It is also shown that for linear problems the accuracy of the piecewise-linearized method is superior to that of fourth-order-accurate techniques. For the linear singular perturbation problems considered in this paper the accuracy of global piecewise linearization is higher than that of finite difference and finite element methods. For non-linear problems the accuracy of piecewise-linearized methods is in most cases lower than that of fourth-order methods but comparable with that of second-order techniques owing to the linearization of the non-linear terms.
KEY WORDS piecewise-linearid methods; two-point boundary value problems; singular perturbations; finite differences; finite elements 1. INTRODUCTION Two-point boundary-value problems in ordinary differential equations occur in many branches o f physics; examples include the two-dimensional, incompressible, boundary layer equations, onedimensional heat transfer, etc. Numerical methods for the solution of two-point boundary value problems include finite difference, finite element and shooting methods. ' The first approximate the derivatives that appear in the differential equation by a difference quotient and yield a system of algebraic equations whose unknowns are the values of the dependent variable at a finite number of points. Finite element methods look for a weak solution in a finite-dimensional functional space to the problem.2 Finally, shooting methods solve a boundary value problem as an initial value one.