Аннотация:In this paper, we carry out the extension of the ADER approach to multidimensional non-linear systems of conservation laws. We implement non-linear schemes of up to fourth order of accuracy in both time and space. Numericalresults for the compressible Euler equations illustrate the very high order of accuracy and non-oscillatory propertiesof the new schemes. Compared to the state-of-art finite-volume WENO schemes the ADER schemes are faster, moreaccurate, need less computer memory and have no theoretical accuracy barrier