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Zhang, Shuhai; Jiang, Shufen; Shu, Chi-Wang

Development of nonlinear weighted compact schemes with increasingly higher order accuracy. (English) Zbl 1152.65094

J. Comput. Phys. 227, No. 15, 7294-7321 (2008).
Summary: We design a class of high order accurate nonlinear weighted compact schemes that are higher order extensions of the nonlinear weighted compact schemes proposed by X. Deng and H. Zhang [J. Comput. Phys. 165, No. 1, 22–44 (2000; Zbl 0988.76060)] and the weighted essentially non-oscillatory (WENO) schemes of G.-S. Jiang and C.-W. Shu, [ibid. 126, No. 1, 202–228 (1996; Zbl 0877.65065)] and D. S. Balsara and C.-W. Shu [ibid. 160, No. 2, 405–452 (2000; Zbl 0961.65078)]. These nonlinear weighted compact schemes are proposed based on the cell-centered compact scheme of S. K. Lele [ibid. 103, No. 1, 16–42 (1992; Zbl 0759.65006)].
Instead of performing the nonlinear interpolation on the conservative variables as done by Deng and Zhang [loc. cit.], we propose to directly interpolate the flux on its stencil. Using the Lax-Friedrichs flux splitting and characteristic-wise projection, the resulted interpolation formulae are similar to those of the regular WENO schemes. Hence, the detailed analysis and even many pieces of the code can be directly copied from those of the regular WENO schemes. Through systematic test and comparison with the regular WENO schemes, we observe that the nonlinear weighted compact schemes have the same ability to capture strong discontinuities, while the resolution of short waves is improved and numerical dissipation is reduced.

Cited in 5 Reviews
Cited in 65 Documents

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws

Keywords:

compact scheme; weighted interpolation; WENO scheme; smoothness indicator; conservation laws; finite difference method; numerical examples

Citations:

Zbl 0988.76060; Zbl 0877.65065; Zbl 0961.65078; Zbl 0759.65006
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\textit{S. Zhang} et al., J. Comput. Phys. 227, No. 15, 7294--7321 (2008; Zbl 1152.65094)
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