We introduce the class of skew-circulant lattice rules. These are s-dimensional lattice rules that may be generated by the rows of an s × s skewcirculant matrix. (This is a minor variant of the familiar circulant matrix.) We present briefly some of the underlying theory of these matrices and rules. We are particularly interested in finding rules of specified trigonometric degree d. We describe some of the results of computer-based searches for optimal four-dimensional skew-circulant rules. Besides determining optimal rules for δ = d + 1 ≤ 47, we have constructed an infinite sequence of rulesQ(4, δ) that has a limit rho index of 27/34 ≈ 0.79. This index is an efficiency measure, which cannot exceed 1, and is inversely proportional to the abscissa count.
a b s t r a c tWe study a numerical solution of the multi-dimensional time dependent Schrödinger equation using a split-operator technique for time stepping and a spectral approximation in the spatial coordinates. We are particularly interested in systems with near spherical symmetries. One expects these problems to be most efficiently computed in spherical coordinates as a coarse grain discretization should be sufficient in the angular directions. However, in this coordinate system the standard Fourier basis does not provide a good basis set in the radial direction. Here, we suggest an alternative basis set based on Chebyshev polynomials and a variable transformation.Furthermore, it is shown how the use of operator splitting produces a splitting error which introduces high frequency modes in the numerical solution in the case of the singular Coulomb potential. Incorporating the Coulomb potential into the radial Laplacian provides a much better splitting. Fortunately our new basis set allows this in some cases.Numerical experiments are presented which demonstrate the advantages and limitations of our technique. Details are demonstrated by 1D toy examples, while the superior efficiency is demonstrated by a 3D example.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.