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

Achieving 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.