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On a Nonlinear Three-Point Subdivision Scheme Reproducing Piecewise Constant Functions

Author

Listed:
  • Sofiane Zouaoui

    (Ecole Supérieure des Sciences Appliquées-Alger, Algiers 16001, Algeria
    Laboratory of EDPNL & HM of ENS Kouba, Kouba 16308, Algeria)

  • Sergio Amat

    (Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain)

  • Sonia Busquier

    (Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain)

  • Juan Ruiz

    (Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain)

Abstract

In this article, a nonlinear binary three-point non-interpolatory subdivision scheme is presented. It is based on a nonlinear perturbation of the three-point subdivision scheme: A new three point approximating C 2 subdivision scheme, where the convergence and the stability of this linear subdivision scheme are analyzed. It is possible to prove that this scheme does not present Gibbs oscillations in the limit functions obtained. The numerical experiments show that the linear scheme is stable even in the presence of jump discontinuities. Even though, close to jump discontinuities, the accuracy is loosed. This order reduction is equivalent to the introduction of some diffusion. Diffusion is a good property for subdivision schemes when the discontinuities are numerical, i.e., they appear when discretizing a continuous function close to high gradients. On the other hand, if the initial control points come from the discretization of a piecewise continuous function, it can be interesting that the subdivision scheme produces a piecewise continuous limit function. For instance, in the approximation of conservation laws, real discontinuities appear as shocks in the solution. The nonlinear modification introduced in this work allows to attain this objective. As far as we know, this is the first subdivision scheme that appears in the literature with these properties.

Suggested Citation

  • Sofiane Zouaoui & Sergio Amat & Sonia Busquier & Juan Ruiz, 2022. "On a Nonlinear Three-Point Subdivision Scheme Reproducing Piecewise Constant Functions," Mathematics, MDPI, vol. 10(15), pages 1-20, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2790-:d:881590
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    References listed on IDEAS

    as
    1. Tan, Jieqing & Sun, Jiaze & Tong, Guangyue, 2016. "A non-stationary binary three-point approximating subdivision scheme," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 37-43.
    2. Amat, S. & Ruiz, J. & Trillo, J.C., 2012. "Adaptive interpolation of images using a new nonlinear cell-average scheme," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(9), pages 1586-1596.
    3. Siddiqi, Shahid S. & Salam, Wardat us & Rehan, Kashif, 2015. "Binary 3-point and 4-point non-stationary subdivision schemes using hyperbolic function," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 120-129.
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