On the method of modified equations. VI: Asymptotic analysis of and asymptotic successive-corrections techniques for two-point, boundary-value problems in ODE's
[For part V see ibid. 103, 241-285 (1999; Zbl 0929.65053).] The sixth paper in the series concerning the method of modified equations extends it to two-point boundary-value problems in ordinary differential equations, and the implicit centered finite difference scheme is considered as example. The proposed methods are applied to homogeneous and non-homogeneous, second-order, linear and nonlinear ordinary differential equations. Three kinds of modified equations, i.e. the equivalent, second equivalent and simply modified equations, are analyzed and it is shown that they are asymptotically equivalent when the equivalent equation is used for the boundary conditions. Furthermore, asymptotic successive-corrections techniques are applied to the three kinds of modified equations to obtain higher-order schemes. The results of the proposed techniques are compared for several second-order ordinary differential equations. The numerical order, the errors and the computational cost are also studied.
asymptotic successive-corrections techniques
implicit centered finite difference scheme
scientific article; zbMATH DE number 1464446
scientific article