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PDE-W-methods for parabolic problems with mixed derivatives

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Abstract

The present work considers the numerical solution of differential equations that are obtained by space discretization (method of lines) of parabolic evolution equations. Main emphasis is put on the presence of mixed derivatives in the elliptic operator. An extension of the alternating-direction-implicit (ADI) approach to this situation is presented. Our stability analysis is based on a scalar test equation that is relevant to the considered class of problems. The novel treatment of mixed derivatives is implemented in third-order W-methods. Numerical experiments and comparisons with standard methods show the efficiency of the new approach. An extension of our treatment of mixed derivatives to 3D and higher dimensional problems is outlined at the end of the article.

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Acknowledgements

Part of the work was carried out during stays of the second author at Universidad de La Laguna and of the fourth author at Université de Genève. This work has been partially supported by the Spanish Projects of Ministerio de Economia, Industria y Competitividad: MTM2013-47318-C2-2-P, MTM2016-77735-C3-3-P, and by the Fonds National Suisse, Project No. 200020_159856.

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de La Laguna, 38071, La Laguna, Spain

    S. González-Pinto, D. Hernández-Abreu & S. Pérez-Rodríguez

  2. Section de mathématiques, Université de Genève, 1211, Geneva, Switzerland

    E. Hairer

Authors
  1. S. González-Pinto
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  2. E. Hairer
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  3. D. Hernández-Abreu
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  4. S. Pérez-Rodríguez
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Correspondence to E. Hairer.

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González-Pinto, S., Hairer, E., Hernández-Abreu, D. et al. PDE-W-methods for parabolic problems with mixed derivatives. Numer Algor 78, 957–981 (2018). https://doi.org/10.1007/s11075-017-0408-8

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  • DOI: https://doi.org/10.1007/s11075-017-0408-8

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