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

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.


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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[1] Adams, N. A.; Shariff, K., A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems, J. Comput. Phys., 127, 27-51 (1996) · Zbl 0859.76041
[2] Balsara, D. S.; Shu, C.-W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160, 405-452 (2000) · Zbl 0961.65078
[3] Boersma, B. J., A staggered compact finite difference formulation for the compressible Navier-Stokes equations, J. Comput. Phys., 208, 675-690 (2005) · Zbl 1329.76222
[4] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods in Fluid Dynamics (1987), Springer-Verlag: Springer-Verlag New York · Zbl 0636.76009
[5] Carpenter, M. H.; Gottlieb, D.; Abarbanel, S., The stability of numerical boundary treatments for compact high-order finite-difference schemes, J. Comput. Phys., 108, 272-295 (1993) · Zbl 0791.76052
[6] Cockburn, B.; Shu, C.-W., Nonlinearly stable compact schemes for shock calculations, SIAM J. Numer. Anal., 31, 607-627 (1994) · Zbl 0805.65085
[7] G.S. Constantinescu, S.K. Lele, Large eddy simulation of a near sonic turbulent jet and its radiated noise, AIAA Paper, 2001, 2001-0376.; G.S. Constantinescu, S.K. Lele, Large eddy simulation of a near sonic turbulent jet and its radiated noise, AIAA Paper, 2001, 2001-0376.
[8] Deng, X.; Maekawa, H., Compact high-order accurate nonlinear schemes, J. Comput. Phys., 130, 77-91 (1997) · Zbl 0870.65075
[9] Deng, X.; Zhang, H., Developing high-order weighted compact nonlinear schemes, J. Comput. Phys., 165, 22-44 (2000) · Zbl 0988.76060
[10] Fu, D.; Ma, Y.; Hong, L., Upwind compact schemes and applications, (Proceedings of the Fifth Symp. on Comput. Fluid Dyn., vol. 1 (1993), Japan Soc. of Comput. Fluid Dyn.)
[11] Gottlieb, D.; Orszag, S. A., Numerical Analysis of Spectral Methods (1977), SIAM: SIAM Philadelphia · Zbl 0412.65058
[12] Grasso, F.; Pirozzoli, S., Shock wave-thermal inhomogeneity interactions: analysis and numerical simulations of sound generation, Phys. Fluids, 12, 205-219 (2000) · Zbl 1149.76390
[13] Grasso, F.; Pirozzoli, S., Shock-wave-vortex interactions: shock and vortex deformations, and sound production, Theor. Comput. Fluid Dyn., 13, 421-456 (2000) · Zbl 0972.76051
[14] Grasso, F.; Pirozzoli, S., Simulations and analysis of the coupling process of compressible vortex pairs: free evolution and shock induced coupling, Phys. Fluids, 13, 1343-1366 (2001) · Zbl 1184.76194
[15] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes, III, J. Comput. Phys., 71, 231-303 (1987) · Zbl 0652.65067
[16] Henrick, A. K.; Aslam, T. D.; Powers, J. M., Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points, J. Comput. Phys., 207, 542-567 (2005) · Zbl 1072.65114
[17] Hirsh, R. S., High order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J. Comput. Phys., 19, 90-109 (1975) · Zbl 0326.76024
[18] Inoue, O.; Hattori, Y., Sound generation by shock-vortex interactions, J. Fluid Mech., 380, 81-116 (1999) · Zbl 0953.76080
[19] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202-228 (1996) · Zbl 0877.65065
[20] Jiang, G.-S.; Wu, C.-C., A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics, J. Comput. Phys., 150, 561-594 (1999) · Zbl 0937.76051
[21] Jiang, L.; Shan, H.; Liu, C. Q., Weighted compact scheme, Int. J. Comput. Fluid Dyn., 15, 147-155 (2001) · Zbl 1044.76046
[22] Kim, J.; Moin, P.; Moser, R., Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., 177, 133-166 (1987) · Zbl 0616.76071
[23] Kreiss, H.-O.; Orszag, S. A.; Israeli, M., Numerical simulation of viscous incompressible flow, Annu. Rev. Fluid Mech., 6, 281-318 (1974) · Zbl 0295.76016
[24] Lax, P. D., Commun. Pure Appl. Math., 7, 159 (1954) · Zbl 0055.19404
[25] Lee, S. S.; Lele, S. K.; Moin, P., Interaction of isotropic turbulence with shock waves: effect of shock strength, J. Fluid Mech., 340, 225 (1997) · Zbl 0899.76194
[26] Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103, 16-42 (1992) · Zbl 0759.65006
[27] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115, 200-212 (1994) · Zbl 0811.65076
[28] Mahesh, K., A family of high order finite difference schemes with good spectral resolution, J. Comput. Phys., 145, 332-358 (1998) · Zbl 0926.76081
[29] Mahesh, K.; Lele, S. K.; Moin, P., The influence of entropy fluctuations on the interaction of turbulence with a shock wave, J. Fluid Mech., 334, 353-379 (1997) · Zbl 0899.76193
[30] Moin, P.; Squires, K.; Cabot, W.; Lee, S., A dynamic subgrid-scale model for compressible turbulence and scalar transport, Phys. Fluids A, 3, 2746 (1991) · Zbl 0753.76074
[31] Nagarajan, S.; Lele, S. K.; Ferziger, J. H., A robust high-order compact method for large eddy simulation, J. Comput. Phys., 191, 392-419 (2003) · Zbl 1051.76030
[32] Pirozzoli, S., Conservative hybrid compact-WENO schemes for shock-turbulence interaction, J. Comput. Phys., 178, 81-117 (2002) · Zbl 1045.76029
[33] Ponziani, D.; Pirozzoli, S.; Grasso, F., Development of optimized weighted-ENO schemes for multiscale compressible flows, Int. J. Numer. Methods Fluids, 42, 953-977 (2003) · Zbl 1055.76039
[34] Saad, M. A., Compressible Fluid Flow (1993), Prentice-Hall · Zbl 0782.76001
[35] Sebastian, K.; Shu, C.-W., Multi domain WENO finite difference method with interpolation at sub-domain interfaces, J. Sci. Comput., 19, 405-438 (2003) · Zbl 1081.76577
[36] Cockburn, B.; Johnson, C.; Shu, C.-W.; Tadmor, E., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, (Quarteroni, A., Lecture Notes in Mathematics, vol. 1697 (1998), Springer), 325-432 · Zbl 0927.65111
[37] C.-W. Shu, High order weighted essentially non-oscillatory schemes for convection dominated problems, SIAM Rev. (in press).; C.-W. Shu, High order weighted essentially non-oscillatory schemes for convection dominated problems, SIAM Rev. (in press). · Zbl 1160.65330
[38] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 77, 439-471 (1988) · Zbl 0653.65072
[39] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. Comput. Phys., 83, 32-78 (1989) · Zbl 0674.65061
[40] Shukla, R. K.; Zhong, X., Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation, J. Comput. Phys., 204, 404-429 (2005) · Zbl 1067.65088
[41] Sod, G. A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27, 1-31 (1978) · Zbl 0387.76063
[42] Tam, C. K.W.; Webb, J. C., Dispersion-relation-preserving finite difference schemes for computational acoustics, J. Comput. Phys., 107, 262-281 (1993) · Zbl 0790.76057
[43] Wang, Z. J.; Chen, R. F., Optimized weighted essentially non-oscillatory schemes for linear waves with discontinuity, J. Comput. Phys., 174, 381-404 (2001) · Zbl 1106.76412
[44] Zhang, S.; Shu, C.-W., A new smoothness indicator for WENO schemes and its effect on the convergence to steady state solutions, J. Sci. Comput., 31, 273-305 (2007) · Zbl 1151.76542
[45] Zhang, S.; Zhang, Y.-T.; Shu, C.-W., Multistage interaction of a shock wave and a strong vortex, Phys. Fluid, 17, 116101 (2005) · Zbl 1188.76183
[46] Zhang, S.; Zhang, Y.-T.; Shu, C.-W., Interaction of an oblique shock wave with a pair of parallel vortices: shock dynamics and mechanism of sound generation, Phys. Fluids, 18, 126101 (2006) · Zbl 1146.76581
[47] Zhang, Y.-T.; Shi, J.; Shu, C.-W.; Zhou, Y., Numerical viscosity and resolution of high-order weighted essentially nonoscillatory schemes for compressible flows with high Reynolds numbers, Phys. Rev. E, 68, 046709 (2003)
[48] Zhong, X., High-order finite-difference schemes for numerical simulation of hypersonic boundary-layer transition, J. Comput. Phys., 144, 662-709 (1998) · Zbl 0935.76066
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