On one-dimensional arbitrary high-order WENO schemes for systems of hyperbolic conservation laws

In this paper, a one-dimensional case for an arbitrary high-order non-oscillatory finite-volume scheme is considered. This is an adaptation of the schemes presented in Dumbser and Kaser (J Comput Phys. 221:693-723, 2007). An entire polynomial inside each control volume is reconstructed in terms of hierarchical orthogonal polynomial bases over a reference element. A new numerical inter-cell flux function at the element interfaces is proposed. To validate the approach, several one-dimensional test problems with discontinuous solutions are computed. The results show an improvement on the non-oscillatory shock-capturing properties based on the new approach.