## Preprint 2018-43

## Raimund Bürger, David Zorío:

### Hybrid essentially non-oscillatory schemes for hiperbolic conservation laws

### Abstract:

A novel family of high-order shock-capturing schemes is dened by combining highorder and low-order (rst-order) reconstructions of numerical fluxes through a single non-linear, scale-independent and non-dimensional weight. This weight is designed to attain the desired order of accuracy near smooth data and to reduce the scheme to first order near discontinuities, and involves a tuning parameter that introduces a degree of tolerance towards high gradients. The resulting hybrid essentially non-oscillatory (HENO) reconstruction is easy to implement on unstructured meshes and is computationally cheap, especially on uniform meshes, in which the number of operations grows linearly with respect to the order. Numerical experiments for solving hyperbolic conservation laws with discontinuous solutions in one and two space dimensions illustrate that if the tuning parameter of HENO reconstructions is chosen properly, then a scheme of this type attains similar or even better results than weighted essentially non-oscillatory (WENO) schemes of the same formal order of accuracy, but does so at a lower computational cost.