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

Abstract

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.

On the numerical overshoots of shock‐capturing schemes

“…Numerical simulations of compressible flows are a highly challenging topic because shock waves which cause strong density, velocity and pressure discontinuities, need to be tackled by using robust [1], accurate [2,3] and efficient [4,5] shock-capturing schemes, the development of which is still not trivial. This topic becomes especially challenging while high numerical resolution and computational robustness are expected to be provided by a unified framework of numerical schemes, since achieving high-resolution usually expects high-order polynomials for spatial approximation, and high-order polynomials tend to be oscillatory if they are crossing discontinuities (Gibbs phenomenon).…”

“…Numerical simulations of compressible flows are a highly challenging topic since shock waves, which cause strong density, velocity, and pressure discontinuities, need to be tackled by using robust, 1 accurate, 2,3 and efficient 4,5 shock-capturing schemes, the development of which, however, is still not trivial. This topic becomes even more challenging when it comes to high-order shock-capturing schemes, since for which, the satisfaction of high numerical resolution and strong computational robustness are contradictory.…”

Improving the quantification of overshooting shock-capturing oscillations

“…Therefore, the capability of capturing shocks and meanwhile resolving broad-band smooth structures is usually anticipated to simulate those complex flow problems. This has been one of the driven forces for the development of robust [1,2] and accurate [3,4,5,6] shock-capturing schemes. However, when it comes to high-order shock-capturing schemes, it is particularly challenging to achieve high-performance in both the robustness and accuracy, since the robustness is typically improved by using a diffusive damping mechanism that suppresses numerical oscillations but also frequently deteriorates the accuracy and resolution of high-order schemes.…”