On Spline Basis Selection for Solving Differential Equations

The suitability of B-splines as a basis for piecewise polynomial solution representation for solving differential equations is challenged. Two alternative local solution representations are considered in the context of collocating ordinary differential equations: "Hermite-type" and "monomial". Both are much easier and shorter to implement and somewhat more efficient than B-splines.

A new condition number estimate for the B-splines and Hermite-type representations is presented. One choice of the Hermite-type representation is experimentally determined to produce roundoff errors at most as large as those for B-splines. The monomial representation is shown to have a much smaller condition number than the other ones, and correspondingly produces smaller roundoff errors, especially for extremely nonuniform meshes. The operation counts for the two local representations considered are about the same, the Hermite-type representation being slightly cheaper. It is concluded that both representations are preferable, and the monomial representation is particularly recommended.