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A Note on Local Refinement for Direction Splitting Methods

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Numerical Methods and Applications (NMA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8962))

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Abstract

In this note we propose a grid refinement procedure for direction splitting schemes for parabolic problems that can be easily extended to the incompressible Navier-Stokes equations. The procedure is developed to be used in conjunction with a direction splitting time discretization. Therefore, the structure of the resulting linear systems is tridiagonal for all internal unknowns, and only the Schur complement matrix for the unknowns at the interface of refinement has a four diagonal structure. Then the linear system in each direction can be solved either by a kind of domain decomposition iteration or by a direct solver, after an explicit computation of the Schur complement. The numerical results on a manufactured solution demonstrate that this grid refinement procedure does not alter the spatial accuracy of the finite difference approximation and seems to be unconditionally stable.

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Authors and Affiliations

  1. Department of Flows and Materials Simulation, Fraunhofer Institute for Industrial Mathematics, Fraunhofer-Platz 1, 67663, Kaiserslautern, Germany

    T. Gornak & O. Iliev

  2. Germany and Technical University of Kaiserslautern, Kaiserslautern, Germany

    T. Gornak & O. Iliev

  3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    P. Minev

Authors
  1. T. Gornak
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  2. O. Iliev
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  3. P. Minev
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Corresponding author

Correspondence to P. Minev .

Editor information

Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Stefka Fidanova

  3. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Lirkov

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Gornak, T., Iliev, O., Minev, P. (2015). A Note on Local Refinement for Direction Splitting Methods. In: Dimov, I., Fidanova, S., Lirkov, I. (eds) Numerical Methods and Applications. NMA 2014. Lecture Notes in Computer Science(), vol 8962. Springer, Cham. https://doi.org/10.1007/978-3-319-15585-2_1

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  • DOI: https://doi.org/10.1007/978-3-319-15585-2_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-15584-5

  • Online ISBN: 978-3-319-15585-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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