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Asymptotic properties of the space–time adaptive numerical solution of a nonlinear heat equation

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Abstract

We consider the fully adaptive space–time discretization of a class of nonlinear heat equations by Rothe’s method. Space discretization is based on adaptive polynomial collocation which relies on equidistribution of the defect of the numerical solution, and the time propagation is realized by an adaptive backward Euler scheme. From the known scaling laws, we infer theoretically the optimal grids implying error equidistribution, and verify that our adaptive procedure closely approaches these optimal grids.

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References

  1. Ascher, U., Ruuth, S., Wetton, T.R.: Implicit–explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32(3), 797–823 (1995)

    Article  MathSciNet  Google Scholar 

  2. Bebernes, J., Bricher, S.: Final time blowup profiles for semilinear parabolic equations via center manifold theory. SIAM J. Math. Anal. 23(4), 852–869 (1992)

    Article  MathSciNet  Google Scholar 

  3. Budd, C.J., Williams, J.: How to adaptively resolve evolutionary singularities in differential equations with symmetry ? J. Eng. Math. 66, 217–236 (2010)

    Article  MathSciNet  Google Scholar 

  4. Budd, C.J., Huang, W., Russell, R.D.: Moving mesh methods for problems with blow-up. SIAM J. Sci. Comput. 17, 305–327 (1996)

    Article  MathSciNet  Google Scholar 

  5. Huang, W., Russell, R .D.: Adaptive Moving Mesh Methods. Springer, New York (2010)

    MATH  Google Scholar 

  6. Kitzhofer, G., Koch, O., Pulverer, G., Simon, C., Weinmüller, E.: The new MATLAB code BVPSUITE for the solution of singular implicit boundary value problems. JNAIAM J. Numer. Anal. Ind. Appl. Math. 5, 113–134 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Pulverer, G., Söderlind, G., Weinmüller, E.: Automatic grid control in adaptive BVP solvers. Numer. Algorithms 56(1), 61–92 (2011)

    Article  MathSciNet  Google Scholar 

  8. Quarteroni, A., Saleri, F.: Approximation of Functions and Data. Springer, Berlin (2006)

    Book  Google Scholar 

  9. Söderlind, G.: Digital filters in adaptive time-stepping. ACM Trans. Math. Softw. 29, 1–26 (2003)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Chris Budd

  2. Institut für Mathematik, Universität Wien, Oskar Morgenstern-Platz 1, 1090, Vienna, Austria

    Othmar Koch

  3. Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstrasse 8–10/E101, 1040, Vienna, Austria

    Leila Taghizadeh & Ewa Weinmüller

Authors
  1. Chris Budd
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  2. Othmar Koch
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  3. Leila Taghizadeh
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  4. Ewa Weinmüller
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Correspondence to Othmar Koch.

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Budd, C., Koch, O., Taghizadeh, L. et al. Asymptotic properties of the space–time adaptive numerical solution of a nonlinear heat equation. Calcolo 55, 43 (2018). https://doi.org/10.1007/s10092-018-0286-z

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  • DOI: https://doi.org/10.1007/s10092-018-0286-z

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