On the method of modified equations. VI: Asymptotic analysis of and asymptotic successive-corrections techniques for two-point, boundary-value problems in ODE's

Abstract

The modified equation technique is extended to two-point, boundary-value problems, and a second-order accurate, implicit, centered, finite difference scheme for non-homogeneous, second-order, ordinary differential equations with linear boundary conditions is analyzed. The first, second and third modified equations, or equivalent, second equivalent and (simply) modified equations, respectively, for this scheme and its boundary conditions are presented. It is shown that the three kinds of modified equations are asymptotically equivalent when the equivalent equation is used for the boundary conditions, since an asymptotic analysis of these equations with the grid size as small parameter yields exactly the same results. For a linear problem, multiple scales and summed-up asymptotic techniques are used and the resulting uniform asymptotic expansions are shown to be equivalent to the solution of the original finite difference scheme. Asymptotic successive-corrections techniques are also applied to the three kinds of modified equations to obtain higher-order schemes. Higher-order boundary conditions are easily treated in the asymptotic successive-corrections technique, although these boundary conditions must be obtained by using the equivalent equation in order to obtain a correct estimate of the global error near the domain boundaries. The methods introduced in this paper are applied to homogeneous and non-homogeneous, second-order, linear and non-linear, ordinary differential equations, and yield very accurate results.