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Efficient Implementation of Adaptive Order Reconstructions

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Abstract

Including polynomials with small degree and stencil when designing very high order reconstructions is surely beneficial for their non oscillatory properties, but may bring loss of accuracy on smooth data unless special care is exerted. In this paper we address this issue with a new Central \(\mathsf {WENOZ}\) (\(\mathsf {CWENOZ}\)) approach, in which the reconstruction polynomial is computed from a single set of non linear weights, but the linear weights of the polynomials with very low degree (compared to the final desired accuracy) are infinitesimal with respect to the grid size. After proving general results that guide the choice of the \(\mathsf {CWENOZ}\) parameters, we study a concrete example of a reconstruction that blends polynomials of degree six, four and two, mimicking already published Adaptive Order \(\mathsf {WENO}\) reconstructions (Arbogast et al. in SIAM J Numer Anal 56(3):1818-1947, 2018),(Balsara et al. in J Comput Phys 326:780-804, 2016). The novel reconstruction yields similar accuracy and oscillations with respect to the previous ones, but saves up to 20% computational time since it does not rely on a hierarchic approach and thus does not compute multiple sets of nonlinear weights in each cell.

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  1. For this test, claw1dArena was compiled with the GNU Compiler and -O3 optimization level, profiling data were collected with the callgrind utility of the valgrind suite and analyzed with kcachegrind. The data reported refer to the linear advection of the Jiang–Shu profile and to the Lax shock tube with characteristic projection.

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Authors and Affiliations

  1. Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio, 11, Como, Italy

    M. Semplice

  2. Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, Aachen, Germany

    G. Visconti

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  1. M. Semplice
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  2. G. Visconti
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This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC-2023 Internet of Production—390621612 and by INDAM GNCS-2019 grant “Approssimazione numerica di problemi di natura iperbolica ed applicazioni”.

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Semplice, M., Visconti, G. Efficient Implementation of Adaptive Order Reconstructions. J Sci Comput 83, 6 (2020). https://doi.org/10.1007/s10915-020-01156-6

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