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**Towards very high order Godunov schemes.**
*(English)*
Zbl 0989.65094

Toro, E. F. (ed.), Godunov methods. Theory and applications. International conference, Oxford, GB, October 1999. New York, NY: Kluwer Academic/ Plenum Publishers. 907-940 (2001).

Summary: We present an approach, called ADER, for constructing non-oscillatory advection schemes of very high order of accuracy in space and time; the schemes are explicit, one step and have optimal stability condition for one and multiple space dimensions. The approach relies on essentially non-oscillatory reconstructions of the data and the solution of a generalized Riemann problem via solutions of derivative Riemann problems. The schemes may thus be viewed as Godunov methods of very high order of accuracy. We present the ADER formulation for the linear advection equation with constant coefficients, in one and multiple space dimensions. Some preliminary ideas for extending the approach to nonlinear problems are also discussed. Numerical results for one- and two-dimensional problems using schemes of upto 10th-order accuracy are presented.

For the entire collection see [Zbl 0978.00036].

For the entire collection see [Zbl 0978.00036].

### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35L45 | Initial value problems for first-order hyperbolic systems |

35L65 | Hyperbolic conservation laws |