High-order orthogonal spline collocation methods for two-point boundary value problems with interfaces

摘要

Orthogonal spline collocation methods (OSC) are used to solve two-point boundary value problems (BVPs) with interfaces. We first consider the one-dimensional Helmholtz equation with piecewise wave numbers solved using the standard OSC approach. For the solution of self-adjoint two-point BVPs with interfaces, we employ OSC with monomial bases of degree r, where r =3,4. In each case, the results of numerical experiments involving numerous examples from the literature exhibit optimal accuracy in the L~∞ and L~2 norms of order r + 1, and order r accuracy in the H~1 norm. Moreover, superconvergence of order 2r - 2 in the nodal error in the OSC approximation and also in its derivative when r = 4 is observed. Each OSC approach gives rise to almost block diagonal linear systems which are solved using standard software.