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A new fifth-order trigonometric WENO scheme for hyperbolic conservation laws and highly oscillatory problems. (English) Zbl 1488.65297

Summary: In this paper, we propose trigonometric polynomial reconstructions based on one five-point stencil and two two-point stencils, instead of algebraic polynomial reconstructions defined on three three-point stencils [G.-S. Jiang and C.-W. Shu, J. Comput. Phys. 126, No. 1, 202–228 (1996; Zbl 0877.65065); C.-W. Shu, SIAM Rev. 51, No. 1, 82–126 (2009; Zbl 1160.65330)], as a building block for designing a fifth-order trigonometric weighted essentially non-oscillatory (TWENO) scheme to solve hyperbolic conservation laws and highly oscillatory problems. The main objective of the paper is to extremely reduce the difficulty in computing the linear weights, could get less absolute truncation errors in smooth regions, and keep sharp shock transitions in nonsmooth regions. Extensive benchmark numerical tests including some highly oscillatory problems are provided to verify the good performance of the new scheme.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35B06 Symmetries, invariants, etc. in context of PDEs
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