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ENO reconstruction and ENO interpolation are stable. (English) Zbl 1273.65120
The authors of this article state some stability results for an essentially non-oscillatory (ENO) reconstruction and ENO interpolation procedure. They prove that both of these methods are stable in the following sense: the sign of the jumps of the reconstructed point values at cell interfaces remains the same as the sign of the jumps in the underlying cell averages across cell interfaces. To this end, suitable algorithms for a high order ENO stencil (both for the reconstruction and interpolation approach) are defined which depend on the smoothness of the data. Thus spurious oscillations are avoided and the so-called “Sign Property”, the mentioned property that the jump in the ENO reconstruction resp. interpolation “cannot have an opposite sign to the jump in the underlying cell averages”, can be proven. Furthermore, upper bounds for the relative jumps in the ENO reconstruction resp. interpolation are given. Although these results hold for arbitrary meshes and any order of reconstruction, some parts of the proofs depend on the ENO stencil procedure, hence other reconstruction procedures (as for example WENO) “will generally not satisfy a similar sign property”.

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
65D05 Numerical interpolation
35L65 Hyperbolic conservation laws
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[1] Amat, S.; Arandiga, F.; Cohen, A.; Donat, R., Tensor product multiresolution with error control, Signal Process., 82, 587-608, (2002) · Zbl 0994.94001
[2] Arandiga, F.; Cohen, A.; Donat, R.; Dyn, N., Interpolation and approximation of piecewise smooth functions, SIAM J. Numer. Anal., 43, 41-57, (2005) · Zbl 1092.65004
[3] Arandiga, F.; Cohen, A.; Donat, R.; Dyn, N.; Matei, B., Approximation of piecewise smooth functions and images by edge-adapted (ENO-EA) nonlinear multiresolution techniques, Appl. Comput. Harmon. Anal., 24, 225-250, (2008) · Zbl 1168.68592
[4] Baraniuk, R.; Claypoole, R.; Davis, G.M.; Sweldens, W., Nonlinear wavelet transforms for image coding via lifting, IEEE Trans. Image Process., 12, 1449-1459, (2003)
[5] Berzins, M., Adaptive polynomial interpolation on evenly spaced meshes, SIAM Rev., 49, 604-627, (2007) · Zbl 1133.65007
[6] Chan, T.; Zhou, H.M., ENO-wavelet transforms for piecewise smooth functions, SIAM J. Numer. Anal., 40, 1369-1404, (2002) · Zbl 1030.65147
[7] Cohen, A.; Dyn, N.; Matel, M., Quasilinear subdivison schemes with applications to ENO interpolation, Appl. Comput. Harmon. Anal., 15, 89-116, (2003) · Zbl 1046.65071
[8] Fjordholm, U.S.; Mishra, S.; Tadmor, E., Entropy stable ENO scheme, (2012) · Zbl 1284.65117
[9] U.S. Fjordholm, S. Mishra, E. Tadmor, Arbitrarily high-order essentially non-oscillatory entropy stable schemes for systems of conservation laws, SIAM J. Numer. Anal. (in press). · Zbl 1252.65150
[10] Harten, A., ENO schemes with subcell resolution, J. Comput. Phys., 83, 148-184, (1989) · Zbl 0696.65078
[11] Harten, A., Recent developments in shock-capturing schemes, Kyoto, 1990, Tokyo · Zbl 0741.65064
[12] Harten, A., Multi-resolution analysis for ENO schemes, 287-302, (1993), New York
[13] Harten, A., Adaptive multiresolution schemes for shock computations, J. Comput. Phys., 115, 319-338, (1994) · Zbl 0925.65151
[14] Harten, A., Multiresolution representation of cell-averaged data: a promotional review, No. 7, 361-391, (1998), San Diego
[15] Harten, A.; Engquist, B.; Osher, S.; Chakravarty, S.R., Some results on high-order accurate essentially non-oscillatory schemes, Appl. Numer. Math., 2, 347-377, (1986) · Zbl 0627.65101
[16] Harten, A.; Engquist, B.; Osher, S.; Chakravarty, S.R., Uniformly high order accurate essentially non-oscillatory schemes, J. Comput. Phys., 71, 231-303, (1987) · Zbl 0652.65067
[17] Jiang, G.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202-226, (1996) · Zbl 0877.65065
[18] Levy, D.; Puppo, G.; Russo, G., Central WENO schemes for hyperbolic systems of conservation laws, Math. Model. Numer. Anal., 33, 547-571, (1999) · Zbl 0938.65110
[19] B. Matei, Méthodes multirésolution non-linéaires- applications au traitement d’image. Ph.D. thesis, University Paris VI (2002). · Zbl 0994.94001
[20] Qiu, J.; Shu, C.-W., On the construction, comparison, and local characteristic decompositions for high order central WENO schemes, J. Comput. Phys., 183, 187-209, (2002) · Zbl 1018.65106
[21] Shu, C.W., Numerical experiments on the accuracy of ENO and modified ENO schemes, J. Sci. Comput., 5, 127-149, (1990) · Zbl 0732.65085
[22] C.W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. ICASE Technical report, NASA, 1997. · Zbl 0927.65111
[23] Shu, C.W.; Osher, S., Efficient implementation of essentially non-oscillatory schemes—II, J. Comput. Phys., 83, 32-78, (1989) · Zbl 0674.65061
[24] Sweby, P., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., 21, 995-1011, (1984) · Zbl 0565.65048
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