Abstract
A third-order WENO reconstruction has been recently proposed (Fjordholm and Ray, J Sci Comput, 68(1):42–63, 2016, [5]) in the context of finite difference schemes for conservation laws and tested for scalar conservation laws. The method, which is called SP-WENO, satisfies the sign property required for constructing high-order finite difference schemes for conservation laws that are provably entropy stable. In the present work, we extend the reconstruction procedure to systems of conservation laws in multiple space dimensions, with a focus on the compressible Euler equations. SP-WENO in its original form can lead to large overshoots near discontinuities when tested with the Euler equations. We show that SP-WENO can be modified to control oscillations near discontinuities, without compromising on the accuracy for smooth solutions.
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Acknowledgements
This research work benefited from the support of the AIRBUS Group Corporate Foundation Chair in Mathematics of Complex Systems, established in TIFR/ICTS, Bangalore.
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Ray, D. (2018). A Third-Order Entropy Stable Scheme for the Compressible Euler Equations. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems II. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 237. Springer, Cham. https://doi.org/10.1007/978-3-319-91548-7_38
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