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Lax-Wendroff Approximate Taylor Methods with Fast and Optimized Weighted Essentially Non-oscillatory Reconstructions

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Abstract

The goal of this work is to introduce new families of shock-capturing high-order numerical methods for systems of conservation laws that combine Fast WENO (FWENO) and Optimal WENO (OWENO) reconstructions with Approximate Taylor methods for the time discretization. FWENO reconstructions are based on smoothness indicators that require a lower number of calculations than the standard ones. OWENO reconstructions are based on a definition of the nonlinear weights that allows one to unconditionally attain the optimal order of accuracy regardless of the order of critical points. Approximate Taylor methods update the numerical solutions by using a Taylor expansion in time in which, instead of using the Cauchy–Kovalevskaya procedure, the time derivatives are computed by combining spatial and temporal numerical differentiation with Taylor expansions in a recursive way. These new methods are compared between them and against methods based on standard WENO implementations and/or SSP-RK time discretization. A number of test cases are considered ranging from scalar linear 1d problems to nonlinear systems of conservation laws in 2d.

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Acknowledgements

This research has received funding from the European Union’s Horizon 2020 research and innovation program, under the Marie Sklodowska-Curie grant agreement No 642768. It has been also partially supported by the Spanish Government and FEDER through the Research project RTI2018-096064-B-C21. D. Zorío is also supported by Fondecyt Project 3170077.

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Authors and Affiliations

  1. ITESRC, Km 120, Agujita, Coahuila, Mexico

    H. Carrillo

  2. Departamento de Matemática Aplicada, Universidad de Málaga, Avda. Cervantes, 2., 29071, Málaga, Spain

    C. Parés

  3. CI2MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    D. Zorío

  4. Universidad Internacional de Valencia, Valencia, Spain

    D. Zorío

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  1. H. Carrillo
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Correspondence to H. Carrillo.

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Carrillo, H., Parés, C. & Zorío, D. Lax-Wendroff Approximate Taylor Methods with Fast and Optimized Weighted Essentially Non-oscillatory Reconstructions. J Sci Comput 86, 15 (2021). https://doi.org/10.1007/s10915-020-01380-0

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  • DOI: https://doi.org/10.1007/s10915-020-01380-0

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