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**Convex ENO high order multi-dimensional schemes without field by field decomposition or staggered grids.**
*(English)*
Zbl 0941.65082

The main purpose of the paper is the presentation of a new central essentially nonoscillatory (ENO) scheme, which is of third order and does not require field-by-field decomposition. First, the standard ENO schemes developed by C.-W. Shu and S. Osher [J. Comput. Phys. 83, No. 1, 32-78 (1989; Zbl 0674.65061)] are revisited by using point values instead of cell averages, which avoids staggering, as well as component-wise limiting. Next the authors introduce a new type of decision process called “convex ENO”, which yields the desired scheme. They insist on the fact that their flux does degenerate to first order at discontinuities. Furthermore, their scheme is easily extended to multi-dimensions. Numerical results are reported in detail for 10 significant test problems in one or two dimensions. They are excellent. It is also pointed out that the component-wise calculation is twice as fast as the field-by-field decomposition version of the scheme in each dimension.

Reviewer: S.Benzoni (Lyon)

### 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 |

35L65 | Hyperbolic conservation laws |

### Keywords:

central schemes; essentially nonoscillatory scheme; component-wise calculation; high resolution methods; multi-dimensions; numerical results; ENO scheme### Citations:

Zbl 0674.65061
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\textit{X.-D. Liu} and \textit{S. Osher}, J. Comput. Phys. 142, No. 2, 304--330 (1998; Zbl 0941.65082)

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