SPECTRAL PROPERTIES OF DERIVATIVE OPERATORS IN THE BASIS-SPLINE COLLOCATION METHOD

We discuss the basis-spline collocation method for the lattice solutio n of boundary-value differential equations, drawing particular attenti on to the difference between lattice and continuous collocation method s. Spectral properties of the basis-spline lattice representation of t he first and second spatial derivatives are studied for the case of pe riodic boundary conditions with homogeneous lattice spacing and compar ed to spectra obtained using traditional finite-difference schemes. Ba sis-spline representations are shown to give excellent resolution on s mall-length scales and to satisfy the chain rule with good fidelity fo r the lattice-derivative operators using high-order splines. Applicati on to the one-dimensional Dirac equation shows that very high-order sp line representations of the Hamiltonian on odd lattices avoid the noto rious spectral-doubling problem.