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An adaptive numerical method to handle blow-up in a parabolic system

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Abstract

We study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval, coupled at the boundary in a nonlinear way. In certain cases the system develops a blow-up singularity in finite time. Fixed mesh methods are not well suited to approximate the problem near the singularity. As an alternative to reproduce the behaviour of the continuous solution, we present an adaptive in space procedure. The scheme recovers the conditions for blow-up and non-simultaneous blow-up. It also gives the correct non-simultaneous blow-up rate and set. Moreover, the numerical simultaneous blow-up rates coincide with the continuous ones in the cases when the latter are known. Finally, we present numerical experiments that illustrate the behaviour of the adaptive method.

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Authors and Affiliations

  1. Departamento de Matemáticas, U. Autónoma de Madrid, 28049, Madrid, Spain

    Cristina Brändle & Fernando Quirós

  2. IMAFF, CSIC, c/ Serrano 123, 28006, Madrid, Spain

    Julio D. Rossi

  3. Departamento de Matemática, FCEyN, U. de Buenos Aires, Ciudad Universitaria Pab. I, (1428), Buenos Aires, Argentina

    Julio D. Rossi

Authors
  1. Cristina Brändle
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  2. Fernando Quirós
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  3. Julio D. Rossi
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Correspondence to Cristina Brändle.

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Brändle, C., Quirós, F. & D. Rossi, J. An adaptive numerical method to handle blow-up in a parabolic system. Numer. Math. 102, 39–59 (2005). https://doi.org/10.1007/s00211-005-0638-x

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  • DOI: https://doi.org/10.1007/s00211-005-0638-x

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