Variants for the WENO-DK reconstruction of even orders in the framework of ADER methods for very high orders of accuracy

The WENO-DK reconstruction [M. Dumbser and M. Käser, JCOMP. 221 :693-723, (2007)] is a type of WENO procedure in which, for the one-dimensional case only the leftmost, centered and rightmost stencils are involved. For even orders the central stencil contains more elements than degrees of freedom and an overdetermined system is solved by means of a least-squares approach. Here, it is numerically investigated the impact of choosing the smallest and largest central stencil around the cell of interests and proposed two variants to obtain the central polynomial where the solution of overdetermined systems is not needed. Implementations of the proposed approaches in the framework of fully discrete ADER schemes for the linear advection equation and the Euler equations of gas dynamics are reported. Comparisons with conventional WENO and conventional WENO-DK confirm that the proposed variants of WENO-DK are a suitable compromise between simplicity and accuracy in the context of ADER schemes, implemented up to the tenth order of accuracy in space and time. • Impact of central stencil is numerically investigated for WENO-DK and ADER schemes. • Two variants for central polynomial in WENO-DK, avoiding least-squares approach. • Assessment of WENO-DK variants up to tenth order of accuracy for scalar tests. • Numerical evidence shows that the WENO-DK approach cannot evade the Gibbs phenomenon. • Expected theoretical orders of accuracy are attained.