Some results on uniformly high-order accurate essentially nonoscillatory schemes.

*(English)*Zbl 0627.65101The authors show how to construct for systems of conservation laws essentially nonoscillatory schemes which are uniformly high-order accurate (in the sense of global error for smooth solutions) to any finite order. The main goal of the paper is that, if the initial data \(V_ 0(x)\) are piecewise smooth, then for h sufficiently small
\[
TV(V_ h(\cdot,t+\Delta t))\leq TV(V_ h(\cdot,t))+O(h^{N+1})
\]
where N is the order of accuracy of the scheme and TV means total variation.

Reviewer: P.I.Ialamov

##### MSC:

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

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

35L65 | Hyperbolic conservation laws |

##### Keywords:

essentially nonoscillatory shock capturing methods; hyperbolic conservation laws; total variation diminishing schemes; Gibbs phenomenon; systems of conservation laws; global error##### Software:

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\textit{A. Harten} et al., Appl. Numer. Math. 2, 347--377 (1986; Zbl 0627.65101)

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##### References:

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