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High-Order Flux Reconstruction Schemes with Minimal Dispersion and Dissipation

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Abstract

Modal analysis of the flux reconstruction (FR) formulation is performed to obtain the semi-discrete and fully-discrete dispersion relations, using which, the wave properties of physical as well as spurious modes are characterized. The effect of polynomial order, correction function and solution points on the dispersion, dissipation and relative energies of the modes are investigated. Using this framework, a new set of linearly stable high-order FR schemes is proposed that minimizes wave propagation errors for the range of resolvable wavenumbers. These schemes provide considerably reduced error for advection in comparison to the Discontinuous Galerkin scheme and benefit from having an explicit differential update. The corresponding resolving efficiencies compare favorably to those of standard high-order compact finite difference schemes. These theoretical expectations are verified by a comparison of proposed and existing FR schemes in advecting a scalar quantity on uniform as well as non-uniform grids.

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Acknowledgments

The authors are grateful to Prof. S. K. Lele (Professor, Department of Aeronautics and Astronautics, Stanford University) for motivating the problem and advising in the choice of relevant test cases, and to Francisco Palacios (Engineering Research Associate, Department of Aeronautics and Astronautics, Stanford University) for his suggestions regarding the manuscript. The first author would also like to thank David Williams (CFD Engineer, Flight Sciences division, Boeing Commercial Airplanes) and Manuel López (Ph.D. candidate, Department of Aeronautics and Astronautics, Stanford University) for valuable suggestions regarding the optimization problem. The first author was supported in this effort by the Thomas V. Jones Stanford Graduate Fellowship.

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  1. Department of Aeronautics and Astronautics, Stanford University, Stanford, CA , 94305, USA

    Kartikey Asthana & Antony Jameson

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  1. Kartikey Asthana
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  2. Antony Jameson
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Correspondence to Kartikey Asthana.

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Asthana, K., Jameson, A. High-Order Flux Reconstruction Schemes with Minimal Dispersion and Dissipation. J Sci Comput 62, 913–944 (2015). https://doi.org/10.1007/s10915-014-9882-5

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  • DOI: https://doi.org/10.1007/s10915-014-9882-5

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