Andrew B. White

The numerical solution of second-order boundary value problems on nonuniform meshes

In this paper, we examine the solution of second-order, scalar boundary value problems on nonuniform meshes. We show that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order. This result illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on nonuniform meshes. A technique of reducing centered-difference approximations of first-order systems to discretizations of the underlying scalar equation is developed. We treat both vertex-centered and cell-centered difference schemes and indicate how these results apply to partial differential equations on Cartesian product grids.

Supra-convergent schemes on irregular grids

As Tikhonov and Samarskil showed for k = 2, it is not essential that k th-order compact difference schemes be centered at the arithmetic mean of the stencils points to yield second-order convergence (although it does suffice). For stable schemes and even k, the main point is seen when the k th difference quotient is set equal to the value of the k th derivative at the middle point of the stencil; the proof is particularly transparent for k = 2. For any k, in fact, there is a ( k/2J -parameter family of symmetric averages of the values of the k th derivative at the points of the stencil which, when similarly used, yield second-order convergence. The result extends to stable compact schemes for equations with lower-order terms under general boundary conditions. Although the extension of Numerovs tridiagonal scheme (approximating D2y = f with third-order truncation error) yields fourth-order con- vergence on meshes consisting of a bounded number of pieces in which the mesh size changes monotonically, it yields only third-order convergence to quintic polynomials on any three- periodic mesh with unequal adjacent mesh sizes and fixed adjacent mesh ratios. A result of some independent interest is appended (and applied): it characterizes, simply, those functions of k variables which possess the property that their average value, as one translates over one period of an arbitrary periodic sequence of arguments, is zero; i.e., those bounded functions whose average value, as one translates over arbitrary finite sequences of arguments, goes to zero as the length of the sequences increases.

Some Supraconvergent Schemes for Hyperbolic Equations on Irregular Grids

An analysis of the truncation error for finite difference schemes frequently shows an apparent loss of accuracy when a nonuniform grid is used. Some schemes exhibit the phenomenon of supraconvergence, that is, there is no loss of accuracy in the global error. We show that this is the case for smooth solutions of the color equation for an upstream conservative scheme, for two versions of the Lax-Wendroff scheme, and for a variant of the von Neumann-Richtmyer scheme for gas dynamics, if the latter three are stable.

Trailblazing with Roadrunner

The authors look at the changes occurring in computer system design, the rise of heterogeneous computing, and the effects these changes have on high-performance computing.

A calculus of difference schemes for the solution of boundary-value problems on irregular meshes

This paper derives compact-as-possible difference schemes for the solution of high-order two-point boundary-value problems on irregular meshes. This is accomplished by first writing the high-order equation as a first-order system of equations, discretizing and then algebraically reducing the discrete system to a compact-as-possible difference scheme for the original high-order equation. The reduced scheme inherits the properties of the discrete first-order system. In particular, if the first-order system represents a centered Euler scheme, then the reduced scheme will be second-order accurate despite possibly inconsistent truncation error. This phenomenon is known as supraconvergence.

On the efficient numerical solution of systems of second order boundary value problems

An explicit reduction is given of the centered Euler scheme for a first order two-dimensional system producing an accurate difference approximation for a second order scalar equation. All of the advantages of the centered Euler formula are retained except for the condition number. Results of numerical experiments are presented.

Supplement to The Numerical Solution of Second-Order Boundary Value Problems on Nonuniform Meshes

In this paper, we examine the solution of second-order, scalar boundary value problems on nonuniform meshes. We show that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order. This result illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on nonuniform meshes. A technique of reducing centered-difference approximations of first-order systems to discretizations of the underlying scalar equation is developed. We treat both vertex-centered and cell-centered difference schemes and indicate how these results apply to partial differential equations on Cartesian product grids.

Asymptotic behavior of singular values and singular functions of certain convolution operators

Abstract The singular values and singular functions of the convolution operator K · = ∫ 0 x K ( x − y ) · d y , 0≤1, are studied under the conditions that K ( u ) is mildly smooth and K (0) ≠ 0. It is shown that these singular values and functions are asymptotic to those of the operator with K ( u ) ≡ 1. A study of the kernel K ( u ) = e αu reveals that the results obtained are the best possible. Numerical and computational implications for the solution of convolution integral of the first kind, g = Kf , are briefly discussed.

Centers of Supercomputing- Supercomputing At Los Alamos National Laboratory

This paper reports on supercomputing at Los Alamos National Laboratory. Los Alamos has sought to intertwine the fields of computer science and nuclear science, while influencing the design of the computers needed to solve its scientific problems. The complexity and size of the applications prevalent at the Laboratory have dictated a continuing, ever-increasing need for computers one to two orders of magnitude faster than what is currently available. There are currently four CRAY Y-MPs and four X-MPs serving as production computers. The Central Computing Facility (CCF) and the Laboratory Data Communications Center (LDCC), a three-story building completed in 1989, house the supercomputers and associated network servers.