Preface to the Republication of “Uniformly High Order Essentially Non-oscillatory Schemes, III,” by Harten, Engquist, Osher, and Chakravarthy

✍ Scribed by Chi-Wang Shu

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 129 KB
• Volume: 131
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1996.5630

✦ Synopsis

The classic paper by Harten, Engquist, Osher, and Chak-By carefully designing such limiters, the TVD (total variation diminishing) property could be achieved for nonlinear ravarthy on ENO schemes [1] has had a tremendous influence on research in numerical solutions of hyperbolic con-scalar one-dimensional problems. One disadvantage of this approach is that accuracy necessarily degenerated to first servation laws since its publication.

The original and beautiful idea of this paper is a uni-order near smooth extrema.

The ENO idea proposed in [1] is the first successful formly high-order interpolation recipe with an adaptive stencil, termed ENO (Essentially Non-Oscillatory) recon-attempt to obtain a uniformly high-order accurate, yet essentially non-oscillatory, interpolation (i.e., the magni-struction. It is well known that the wider the stencil, the higher the order of accuracy of the interpolation, provided tude of the oscillations decay as O(h r ), where r is the order of accuracy) for piecewise smooth functions. The generic the function being interpolated is smooth inside the stencil. Traditional finite difference methods are based on fixed solution for hyperbolic conservation laws is in the class of piecewise smooth functions. The reconstruction in [1] is a stencil interpolations. For example, to obtain an interpolation for cell i to third-order accuracy, the information of natural extension of an earlier second-order version of Harten and Osher [2]. In [1], Harten, Engquist, Osher, the three cells i Ϫ 1, i, and i ϩ 1 can be used. This works well for globally smooth problems. The resulting scheme and Chakravarthy investigated different ways of measuring local smoothness to determine the local stencil, and devel-is linear for linear PDEs; hence stability can be analyzed by Fourier transforms. However, fixed stencil interpolation oped a hierarchy that begins with one or two cells, then adds one cell at a time to the stencil from the two candi-of second-or higher-order accuracy is necessarily oscillatory near a discontinuity. Such oscillations (called the dates on the left and right, based on the size of the two relevant Newton divided differences. This seems to be the Gibbs phenomenon in spectral methods) do not decay in magnitude when the mesh is refined. It is a nuisance to most robust way for a wide range of grid sizes, h, both before and inside the asymptotic regime. say the least for practical calculations, and often leads to numerical instabilities in nonlinear problems containing

As one can see from the numerical examples in [1] and in later papers, ENO schemes are indeed uniformly high-discontinuities.

Before 1987, there were two common ways to eliminate order accurate and resolve shocks with sharp and monotone (to the eye) transitions. ENO schemes are especially or reduce such spurious oscillations near discontinuities. One way was to add an artificial viscosity. This could be suitable for problems containing both shocks and complicated smooth flow structures, such as occur in shock inter-tuned so that it was large enough near the discontinuity to suppress, or at least to reduce, the oscillations, but was actions with turbulent flow.

This paper of Harten, Engquist, Osher, and Chakra-small elsewhere to maintain high-order accuracy. One disadvantage of this approach is that fine tuning of the param-varthy [1] has been cited 144 times, according to the ISI database. The original authors and many other researchers eter controlling the size of the artificial viscosity is problem dependent. Another way was to apply limiters to eliminate have followed the pioneer work of [1], improving the methodology and expanding the area of its applications. ENO the oscillations. In effect, one reduced the order of accuracy of the interpolation near the discontinuity (e.g., using a schemes based on point values and TVD Runge-Kutta time discretizations, which can save computational costs linear rather than a quadratic interpolant near the shock).

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