The study of building blocks for essentially non-oscillatory (ENO) schemes.

摘要

ENO schemes have proved to be very useful in the computation of discontinuous solutions of hyperbolic conservation laws. The problem of designing these schemes basically reduces to a problem of interpolation. Currently, ENO schemes make use of polynomial interpolation (pol-ENO) in Newton form. However, polynomial interpolations are not always optimal and trigonometric based ENO schemes (trig-ENO) appear to be attractive. Addition of one point at a time is preferred by the ENO stencil idea and this causes a difficulty since it corresponds to a transition from an incomplete to a complete trigonometric basis, or vice versa. Thus, I developed a new Newton-type trigonometric interpolation which is valid for both even and odd number of interpolation points. The crucial step in its development is the determination of a convenient trigonometric basis and its leading coefficients. The procedure involves three different types of trigonometric divided differences (dd's). They are all shown to possess the necessary symmetries that allow the use of the ENO stencil idea. Recurrence relations for their calculation have been obtained. A pointwise error relation is also given and the local truncation error was shown to be of {dollar}{lcub}cal O{rcub}(Delta xsp{lcub}n{rcub}),{dollar} as in the case of pol-ENO, where n is the number of points in the stencil.; An important distinction between the trig-ENO and the pol-ENO approaches is the role played by the new term (or "correction") added to the Newton series. In the trigonometric case, whereas a comparison of errors is still equivalent to a comparison of dd's, it is no longer equivalent to that of corrections.; This introduces two different possibilities of selecting the trig-ENO stencil--correction and error based. The error based stencil is more theoretically appealing. Numerical experiments verify its superiority, especially in the presence of discontinuities, rapid oscillations, and any type of waves. Another possible scheme is one which uses polynomial based stencils and trigonometric interpolations (trig/pol-ENO). Numerical experiments showed that the two trigonometric schemes were superior to pol-ENO when interpolating rapidly oscillating functions and functions with wave like features and as good as the pol-ENO scheme in capturing discontinuities. Moreover, these results remained true when the schemes were used to solve hyperbolic conservation laws.; Other building blocks, such as rational functions and exponential functions coupled with polynomials, are also considered and mentioned briefly.