Higher order approximations using interpolation applied to collocation solutions of two-point boundary value problems

摘要

When solving a system of first order differential equations involving boundary conditions using collocation at k Gauss points that are the zeros of Pk(x), the kth degree Legendre polynomial, to produce a continuous solution over the problem interval, it is well known that the global error at the mesh points is O(h2k) and at other points the global error is O(hk+1). In addition, it is known that the error in the derivative of the solution at non-mesh points is O(hk). This paper shows that the error in the solution at the zeros of Pk′(x) in each subinterval is O(hk+2) and that the error in the derivative of the solution at the zeros of Pk(x) is O(hk+1). This higher order accuracy can be used in various interpolation schemes (Hermite–Birkhoff interpolation in the case that includes derivative data) to produce global errors O(hk+2) in the solution without any additional collocation and without any additional function evaluations. Only the evaluation of the collocation solution or its derivative at certain special points is necessary. Hence, this increased accuracy is almost free compared to other methods which give global errors O(h2k) but at considerable extra expense. Some numerical results illustrating the higher accuracy at these special points are given at the end of the paper along with comparisons, in the form of diagrams, of results produced from the package COLSYS and the results produced from the interpolation schemes (which use COLSYS as a starting point).