Аннотация:This paper concerns the construction of non-oscillatory schemes of very highorder of accuracy in space and time, to solve non-linear hyperbolic conservationlaws. The schemes result from extending the ADER approach, which is relatedto the ENO/WENO methodology. Our schemes are conservative, one-step,explicit and fully discrete, requiring only the computation of the inter-cell fluxesto advance the solution by a full time step; the schemes have optimal stabilitycondition. To compute the intercell flux in one space dimension we solve ageneralised Riemann problem by reducing it to the solution a sequence of mconventional Riemann problems for the kth spatial derivatives of the solution,with k=0, 1,..., m − 1, where m is arbitrary and is the order of the accuracy ofthe resulting scheme. We provide numerical examples using schemes of up tofifth order of accuracy in both time and space.