Abstract
This paper concerns the construction of non-oscillatory schemes of very high order of accuracy in space and time, to solve non-linear hyperbolic conservation laws. The schemes result from extending the ADER approach, which is related to the ENO/WENO methodology. Our schemes are conservative, one-step, explicit and fully discrete, requiring only the computation of the inter-cell fluxes to advance the solution by a full time step; the schemes have optimal stability condition. To compute the intercell flux in one space dimension we solve a generalised Riemann problem by reducing it to the solution a sequence of m conventional 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 of the resulting scheme. We provide numerical examples using schemes of up to fifth order of accuracy in both time and space.
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Titarev, V.A., Toro, E.F. ADER: Arbitrary High Order Godunov Approach. Journal of Scientific Computing 17, 609–618 (2002). https://doi.org/10.1023/A:1015126814947
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DOI: https://doi.org/10.1023/A:1015126814947