Cool WENO schemes

TL;DR

This paper develops and compares high-order WENO schemes for hyperbolic balance laws, introducing new measures of spurious artifacts and proposing CWENOZ schemes that are less prone to oscillations while maintaining accuracy.

Contribution

It introduces the concept of distorsive errors and temperature to evaluate WENO schemes, and proposes CWENOZ schemes that are less oscillatory and more accurate.

Findings

CWENOZ schemes are cooler than existing WENO schemes.

CWENOZ schemes maintain good non-oscillatory properties.

The new measures effectively evaluate scheme artifacts.

Abstract

This work is dedicated to the development and comparison of WENO-type reconstructions for hyperbolic systems of balance laws. We are particularly interested in high order shock capturing non-oscillatory schemes with uniform accuracy within each cell and low spurious effects. We need therefore to develop a tool to measure the artifacts introduced by a numerical scheme. To this end, we study the deformation of a single Fourier mode and introduce the notion of distorsive errors, which measure the amplitude of the spurious modes created by a discrete derivative operator. Further we refine this notion with the idea of temperature, in which the amplitude of the spurious modes is weighted with its distance in frequency space from the exact mode. Following this approach linear schemes have zero temperature, but to prevent oscillations it is necessary to introduce nonlinearities in the scheme, thus increasing their temperature. However it is important to heat the linear scheme just enough to prevent spurious oscillations. With several tests we show that the newly introduced CWENOZ schemes are cooler than other existing WENO-type operators, while maintaining good non-oscillatory properties.

Full text

See pages 1-last of coolWENO.pdf