Adaptive algorithms based on exact difference schemes for nonlinear BVPs on the half-axis

The boundary value problem (BVP) for a scalar nonlinear second order differential equation on the half-axis is considered. A constructive method is proposed to derive from the three-point exact difference scheme (EDS) a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely chosen natural number. The n-TDS has the order of accuracy n ¯ = 2 [ ( n + 1 ) / 2 ] , i.e., the global error is of the form O ( | h | n ¯ ) , where | h | is the maximum step size and [ ⋅ ] denotes the entire part of the expression in brackets. This n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which confirm the efficiency and reliability of our algorithm.