Tu, Guo-Hua; Yuan, Xiang-Jiang; Lu, Li-Peng Developing shock-capturing difference methods. (English) Zbl 1231.76125 Appl. Math. Mech., Engl. Ed. 28, No. 4, 477-486 (2007). Summary: A new shock-capturing method is proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes are projected along characteristic directions. Secondly, the amplitudes of the characteristic decompositions are carefully controlled by limiters to prevent non-physical oscillations. Lastly, the schemes are converted into conservative forms, and the oscillation-free shock-capturing schemes are acquired. Two explicit upwind schemes (2nd-order and 3rd-order) and three compact upwind schemes (3rd-order, 5th-order and 7th-order) are modified by the method for hyperbolic systems and the modified schemes are checked on several one-dimensional and two-dimensional test cases. Some numerical solutions of the schemes are compared with those of a WENO scheme and a MP scheme as well as a compact-WENO scheme. The results show that the method with high order accuracy and high resolutions can capture shock waves smoothly. Cited in 2 Documents MSC: 76G25 General aerodynamics and subsonic flows 76M20 Finite difference methods applied to problems in fluid mechanics Keywords:high order scheme; shock-capturing; upwind scheme; compact scheme; high resolution; conservative scheme Software:AUSM PDFBibTeX XMLCite \textit{G.-H. Tu} et al., Appl. Math. Mech., Engl. Ed. 28, No. 4, 477--486 (2007; Zbl 1231.76125) Full Text: DOI References: [1] Harten, A., High resolution schemes for hypersonic conservation laws[J], Journal of Computational Physics, 49, 2, 367-393 (1983) · Zbl 0565.65050 [2] Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes[J], Journal of Computational Physics, 43, 2, 357-372 (1981) · Zbl 0474.65066 · doi:10.1016/0021-9991(81)90128-5 [3] Liou, M. S., A sequel to AUSM: AUSM+[J], Journal of Computational Physics, 129, 2, 364-382 (1996) · Zbl 0870.76049 · doi:10.1006/jcph.1996.0256 [4] Harten, A.; Osher, S., Uniformly high-order accurate non-oscillatory schemes I[J], SIAM Journal on Numerical Analysis, 24, 2, 279-309 (1987) · Zbl 0627.65102 · doi:10.1137/0724022 [5] Shu, C. W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes[J], Journal of Computational Physics, 77, 2, 439-471 (1988) · Zbl 0653.65072 · doi:10.1016/0021-9991(88)90177-5 [6] Jiang, G. S.; Shu, C. W., Efficient implementation of weighted ENO schemes[J], Journal of Computational Physics, 126, 1, 202-228 (1996) · Zbl 0877.65065 · doi:10.1006/jcph.1996.0130 [7] Daru, V.; Tenaud, C., High order one step monotonicity-preserving schemes for unsteady compressible flow calculations[J], Journal of Computational Physics, 193, 2, 563-594 (2004) · Zbl 1109.76338 · doi:10.1016/j.jcp.2003.08.023 [8] Tu, G.; Yuan, X.; Xia, Z., A class of compact upwind TVD difference schemes[J], Applied Mathematics and Mechanics (English Edition), 27, 6, 765-772 (2006) · Zbl 1178.76263 [9] Ravichandran, K. S., Higher order KFVS algorithms using compact upwind difference operators[J], Journal of Computational Physics, 230, 2, 161-173 (1997) · Zbl 0870.76050 · doi:10.1006/jcph.1996.5561 [10] Tu, G.; Luo, J., Improve compact schemes by limiting flux method[J], Chinese Journal of Computational Physics, 22, 4, 40-47 (2005) [11] Ren, Y. X.; Liu, M.; Zhang, H., A characteristic-wise hybrid compact-WENO schemes for solving hyperbolic conservations[J], Journal of Computational Physics, 192, 2, 365-386 (2003) · Zbl 1037.65090 · doi:10.1016/j.jcp.2003.07.006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.