ADER: Arbitrary High Order Godunov Approach

  • Authors:

  • V. A. Titarev;E. F. Toro

  • Affiliations:

  • Department of Computing and Mathematics, Manchester Metropolitan University, Manchester, United Kingdom. Titarev@mail.ru;Department of Computing and Mathematics, Manchester Metropolitan University, Manchester, United Kingdom. E.F.Toro@mmu.ac.uk, http://www.docm.mmu.ac.uk/STAFF/E.F.Toro

  • Venue:
  • Journal of Scientific Computing
  • Year:
  • 2002

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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.