Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. (English) Zbl 1392.76048

Summary: In this paper the weighted ENO (essentially non-oscillatory) scheme developed for the one-dimensional case by Liu, Osher, and Chan is applied to the case of unstructured triangular grids in two space dimensions. Ideas from Jiang and Shu, especially their new way of smoothness measuring, are considered. As a starting point for the unstructured case we use an ENO scheme like the one introduced by Abgrall. Beside the application of the weighted ENO ideas the whole reconstruction algorithm is analyzed and described in detail. Here we also concentrate on technical problems and their solution. Finally, some applications are given to demonstrate the accuracy and robustness of the resulting new method. The whole reconstruction algorithm described here can be applied to any kind of data on triangular unstructured grids, although it is used in the framework of compressible flow computation in this paper only.


76M25 Other numerical methods (fluid mechanics) (MSC2010)
Full Text: DOI


[1] Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. Comput. Phys., 114, 45, (1994) · Zbl 0822.65062
[2] Test Cases for Inviscid Flow Field Methods
[3] Friedrich, O., A new method for generating inner points of triangulations in two dimensions, Comput. Methods Appl. Mech. Engrg., 104, 77, (1993) · Zbl 0771.76056
[4] Harten, A.; Chakravarthy, S. R., Multi-Dimensional ENO Schemes for General Geometries, (1991)
[5] Hietel, D.; Meister, A.; Sonar, Th., On the comparision of four different implementations of an implicit third-order ENO scheme of box type for the computation of unsteady compressible flow, Numer. Algorithms, 13, 77, (1996) · Zbl 0865.76052
[6] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202, (1996) · Zbl 0877.65065
[7] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115, 200, (1994) · Zbl 0811.65076
[8] Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic conservation laws, Math. Comp., 38, 339, (1982) · Zbl 0483.65055
[9] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439, (1988) · Zbl 0653.65072
[10] Sonar, Th., On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations: polynomial recovery, accuracy and stencil selection, Comput. Methods Appl. Mech. Engrg., 140, 157, (1997) · Zbl 0898.76086
[11] Woodward, P.; Colella, Ph., The numerical simulation of two-dimensional fluid flows with strong shocks, J. Comput. Phys., 54, 115, (1984) · Zbl 0573.76057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.