Arbitrary high‐order extended essentially non‐oscillatory schemes for hyperbolic conservation laws

Abstract

AbstractAchieving high numerical resolution in smooth regions and robustness near discontinuities within a unified framework is the major concern while developing numerical schemes solving hyperbolic conservation laws, for which the essentially non‐oscillatory (ENO) type scheme is a favorable solution. Therefore, an arbitrary‐high‐order ENO‐type framework is designed in this article. With using a typical five‐point smoothness measurement as the shock‐detector, the present schemes are able to detect discontinuities before spatial reconstructions, and thus more spatial information can be exploited to construct incremental‐width stencils without crossing discontinuities, ensuring ENO property and high‐order accuracy at the same time. The present shock‐detection procedure is specifically examined for justifying its performance of resolving high‐frequency waves, and a standard metric for discontinuous solutions is also applied for measuring the shock‐capturing error of the present schemes, especially regarding the amplitude error in post‐shock regions. In general, the present schemes provide high‐resolution, and more importantly, the schemes are more efficient compared with the typical WENO schemes since only a five‐point smoothness measurement is applied for arbitrary‐high‐order schemes. Numerical results of canonical test cases also provide evidences of the overall performance of the present schemes.