Explicit high accuracy maximum resolution dispersion relation preserving schemes for computational aeroacoustics. (English) Zbl 1394.76080
Summary: A set of explicit finite difference schemes with large stencil was optimized to obtain maximum resolution characteristics for various spatial truncation orders. The effect of integral interval range of the objective function on the optimized schemes’ performance is discussed. An algorithm is developed for the automatic determination of this integral interval. Three types of objective functions in the optimization procedure are compared in detail, which show that Tam’s objective function gets the best resolution in explicit centered finite difference scheme. Actual performances of the proposed optimized schemes are demonstrated by numerical simulation of three CAA benchmark problems. The effective accuracy, strengths, and weakness of these proposed schemes are then discussed. At the end, general conclusion on how to choose optimization objective function and optimization ranges is drawn. The results provide clear understanding of the relative effective accuracy of the various truncation orders, especially the trade-off when using large stencil with a relatively high truncation order.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 76Q05 | Hydro- and aero-acoustics |
References:
| [1] | Ekaterinaris, J. A., High-order accurate, low numerical diffusion methods for aerodynamics, Progress in Aerospace Sciences, 41, 3-4, 192-300, (2005) · doi:10.1016/j.paerosci.2005.03.003 |
| [2] | Wang, Z. J.; Fidkowski, K.; Abgrall, R., High-order CFD methods: current status and perspective, International Journal for Numerical Methods in Fluids, 72, 8, 811-845, (2013) · Zbl 1455.76007 · doi:10.1002/fld.3767 |
| [3] | Liu, Y.; Vinokur, M.; Wang, Z. J., Spectral difference method for unstructured grids I: basic formulation, Journal of Computational Physics, 216, 2, 780-801, (2006) · Zbl 1097.65089 · doi:10.1016/j.jcp.2006.01.024 |
| [4] | Cockburn, B., Discontinuous Galerkin methods, ZAMM—Journal of Applied Mathematics and Mechanics, 83, 11, 731-754, (2003) · Zbl 1036.65079 · doi:10.1002/zamm.200310088 |
| [5] | Jiang, G.; Shu, C., Efficient implementation of weighted ENO schemes, DTIC Document, (1995) |
| [6] | Meitz, H. L.; Fasel, H. F., A compact-difference scheme for the Navier-Stokes equations in vorticity-velocity formulation, Journal of Computational Physics, 157, 1, 371-403, (2000) · Zbl 0960.76059 · doi:10.1006/jcph.1999.6387 |
| [7] | Bhaganagar, K.; Rempfer, D.; Lumley, J., Direct numerical simulation of spatial transition to turbulence using fourth-order vertical velocity second-order vertical vorticity formulation, Journal of Computational Physics, 180, 1, 200-228, (2002) · Zbl 1130.76348 · doi:10.1006/jcph.2002.7088 |
| [8] | Tam, C. K.; Webb, J. C.; Dong, Z., A study of the short wave components in computational acoustics, Journal of Computational Acoustics, 1, 1, 1-30, (1993) · Zbl 1360.76303 · doi:10.1142/s0218396x93000020 |
| [9] | Kim, J. W.; Morris, P. J., Computation of subsonic inviscid flow past a cone using high-order schemes, AIAA Journal, 40, 10, 1961-1968, (2002) · doi:10.2514/2.1557 |
| [10] | Lele, S. K., Compact finite difference schemes with spectral-like resolution, Journal of Computational Physics, 103, 1, 16-42, (1992) · Zbl 0759.65006 · doi:10.1016/0021-9991(92)90324-R |
| [11] | Tam, C. K.; Webb, J. C., Dispersion-relation-preserving finite difference schemes for computational acoustics, Journal of Computational Physics, 107, 2, 262-281, (1993) · Zbl 0790.76057 · doi:10.1006/jcph.1993.1142 |
| [12] | Ekaterinaris, J. A., Implicit, high-resolution, compact schemes for gas dynamics and aeroacoustics, Journal of Computational Physics, 156, 2, 272-299, (1999) · Zbl 0957.76051 · doi:10.1006/jcph.1999.6360 |
| [13] | Zingg, D. W., Comparison of high-accuracy finite-difference methods for linear wave propagation, SIAM Journal on Scientific Computing, 22, 2, 476-502, (2000) · Zbl 0968.65061 · doi:10.1137/s1064827599350320 |
| [14] | Cunha, G.; Redonnet, S., On the effective accuracy of spectral-like optimized finite-difference schemes for computational aeroacoustics, Journal of Computational Physics, 263, 222-232, (2014) · Zbl 1349.76448 · doi:10.1016/j.jcp.2014.01.024 |
| [15] | Lockard, D. P.; Brentner, K. S.; Atkins, H. L., High-accuracy algorithms for computational aeroacoustics, AIAA Journal, 33, 2, 246-251, (1995) · Zbl 0825.76490 · doi:10.2514/3.12436 |
| [16] | Kim, J. W.; Lee, D. J., Optimized compact finite difference schemes with maximum resolution, AIAA Journal, 34, 5, 887-893, (1996) · Zbl 0900.76317 · doi:10.2514/3.13164 |
| [17] | Cheong, C.; Lee, S., Grid-optimized dispersion-relation-preserving schemes on general geometries for computational aeroacoustics, Journal of Computational Physics, 174, 1, 248-276, (2001) · Zbl 1106.76396 · doi:10.1006/jcph.2001.6904 |
| [18] | Liu, Z.; Huang, Q.; Zhao, Z.; Yuan, J., Optimized compact finite difference schemes with high accuracy and maximum resolution, International Journal of Aeroacoustics, 7, 2, 123-146, (2008) · doi:10.1260/147547208784649464 |
| [19] | Mei, Z.; Christoph, R., Computational Aeroacoustics and Its Applications, (2008) |
| [20] | Bogey, C.; Bailly, C., A family of low dispersive and low dissipative explicit schemes for flow and noise computations, Journal of Computational Physics, 194, 1, 194-214, (2004) · Zbl 1042.76044 · doi:10.1016/j.jcp.2003.09.003 |
| [21] | Venutelli, M., New optimized fourth-order compact finite difference schemes for wave propagation phenomena, Applied Numerical Mathematics, 87, 53-73, (2015) · Zbl 1394.65079 · doi:10.1016/j.apnum.2014.07.005 |
| [22] | Hu, F. Q.; Hussaini, M. Y.; Manthey, J. L., Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics, Journal of Computational Physics, 124, 1, 177-191, (1996) · Zbl 0849.76046 · doi:10.1006/jcph.1996.0052 |
| [23] | Hu, F. Q., Development of PML absorbing boundary conditions for computational aeroacoustics: a progress review, Computers & Fluids, 37, 4, 336-348, (2008) · Zbl 1237.76119 · doi:10.1016/j.compfluid.2007.02.012 |
| [24] | Edgar, N. B., A study of the accuracy and stability of high-order compact difference methods for computational aeroacoustics [Ph.D. thesis], (2001), Lawrence, Kan, USA: University of Kansas, Lawrence, Kan, USA |
| [25] | Sherer, S. E., Scattering of sound from axisymetric sources by multiple circular cylinders, The Journal of the Acoustical Society of America, 115, 2, 488-496, (2004) · doi:10.1121/1.1641790 |
| [26] | Seo, J. H.; Mittal, R., A high-order immersed boundary method for acoustic wave scattering and low-Mach number flow-induced sound in complex geometries, Journal of Computational Physics, 230, 4, 1000-1019, (2011) · Zbl 1391.76698 · doi:10.1016/j.jcp.2010.10.017 |
| [27] | Wang, J. L., The Centered Finite Difference Scheme Coefficients in This Paper, (2015), Natick, Mass, USA: MathWorks, Natick, Mass, USA |
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