Cool WENO schemes

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. As a tool to measure the artifacts introduced by a numerical scheme, 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. In this analysis, linear schemes have zero temperature. Of course, in order to prevent oscillations it is necessary to introduce nonlinearities in the scheme, thus increasing their temperature. However it is important to warm up 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.