Improvement and Numerical Verification of Weighting Strategy for High Precision WCNS Scheme

doi:10.16183/j.cnki.jsjtu.2022.014

Abstract

Abstract:

In order to reveal the complex flow mechanism, a series of high-order precision schemes have been proposed at home and abroad, of which, the weighted compact nonlinear scheme (WCNS) has a good shock capture ability and has been widely used in the numerical simulation of complex flows. However, it has insufficient resolution and large dissipation in the simulation of small-scale flows. In the framework of the WCNS, by using the weighting strategy of the targeted essentially non-oscillatory (TENO) scheme for reference, this paper introduces the new methods of discontinuity detection and template weighting into the construction of the WCNS scheme, and develops a WCNS7-T scheme with a 7-order accuracy. Example tests are conducted through one-dimensional shock tube problem and two-dimensional Riemann problem. By comparing with the traditional WCNS7-Z scheme, the improved performance of the new scheme is verified. The numerical experiments show that the WCNS7-T scheme can better suppress the numerical oscillation near the discontinuity, improve the resolution and shock capture ability, and further reduce the dissipation.

Key words: compressible flow, high precision scheme, shock discontinuity, nonlinear weighting

CLC Number:

V211
O355

YANG Qiang, LI Weipeng. Improvement and Numerical Verification of Weighting Strategy for High Precision WCNS Scheme[J]. Journal of Shanghai Jiao Tong University, 2023, 57(6): 719-727.

Figures/Tables 9

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References 30

[1]	HARTEN A. High resolution schemes for hyperbolic conservation laws[J]. Journal of Computational Physics, 1997, 135(2): 260-278. doi: 10.1006/jcph.1997.5713 URL
[2]	LIOU M S. A sequel to AUSM: AUSM⁺[J]. Journal of Computational Physics, 1996, 129(2): 364-382. doi: 10.1006/jcph.1996.0256 URL
[3]	HARTEN A, ENGQUIST B, OSHER S, et al. Uniformly high order accurate essentially non-oscillatory schemes, III[J]. Journal of Computational Physics, 1987, 71(1): 231-303. doi: 10.1016/0021-9991(87)90031-3 URL
[4]	SHU C W, OSHER S. Efficient implementation of essentially non-oscillatory shock-capturing schemes[J]. Journal of Computational Physics, 1989, 77(2): 439-471. doi: 10.1016/0021-9991(88)90177-5 URL
[5]	JIANG G S, SHU C W. Efficient implementation of weighted ENO schemes[J]. Journal of Computational Physics, 1996, 126(1): 202-228. doi: 10.1006/jcph.1996.0130 URL
[6]	DENG X, ZHANG H. Developing high-order weighted compact nonlinear schemes[J]. Journal of Computational Physics, 2000, 165(1): 22-44. doi: 10.1006/jcph.2000.6594 URL
[7]	FU L, HU X Y, ADAMS N A. A new class of adaptive high-order targeted ENO schemes for hyperbolic conservation laws[J]. Journal of Computational Physics, 2018, 374: 724-751. doi: 10.1016/j.jcp.2018.07.043 URL
[8]	YE C C, WAN Z H, SUN D J. An alternative formulation of targeted ENO scheme for hyperbolic conservation laws[J]. Computers & Fluids, 2022, 238: 105368. doi: 10.1016/j.compfluid.2022.105368 URL
[9]	BORGES R, CARMONA M, COSTA B, et al. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws[J]. Journal of Computational Physics, 2008, 227(6): 3191-3211. doi: 10.1016/j.jcp.2007.11.038 URL
[10]	MARTIN M P, TAYLOR E M, WU M, et al. A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence[J]. Journal of Computational Physics, 2006, 220(1): 270-289.
[11]	HU X Y, WANG Q, ADAMS N A. An adaptive central-upwind weighted essentially non-oscillatory scheme[J]. Journal of Computational Physics, 2010, 229(23): 8952-8965. doi: 10.1016/j.jcp.2010.08.019 URL
[12]	BALSARA D S, SHU C W. Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy[J]. Journal of Computational Physics, 2000, 160(2): 405-452. doi: 10.1006/jcph.2000.6443 URL
[13]	GEROLYMOS G A D. SÉNÉCHAL , VALLET I. Very-high-order WENO schemes[J]. Journal of Computational Physics, 2009, 228(23): 8481-8524. doi: 10.1016/j.jcp.2009.07.039 URL
[14]	PIROZZOLI S. On the spectral properties of shock-capturing schemes[J]. Journal of Computational Physics, 2006, 219(2): 489-497. doi: 10.1016/j.jcp.2006.07.009 URL
[15]	JOHNSEN E, LARSSON J, BHAGATWALA A V, et al. Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves[J]. Journal of Computational Physics, 2010, 229(4): 1213-1237. doi: 10.1016/j.jcp.2009.10.028 URL
[16]	DENG X, MAEKAWA H. Compact high-order accurate nonlinear schemes[J]. Journal of Computational Physics, 1997, 130(1): 77-91. doi: 10.1006/jcph.1996.5553 URL
[17]	PENG J, LIU S, LI S, et al. An efficient targeted ENO scheme with local adaptive dissipation for compressible flow simulation[J]. Journal of Computational Physics, 2021, 425: 109902. doi: 10.1016/j.jcp.2020.109902 URL
[18]	ZHANG H, ZHANG F, XU C. Towards optimal high-order compact schemes for simulating compressible flows[J]. Applied Mathematics and Computation, 2019, 355: 221-237. doi: 10.1016/j.amc.2019.03.001 URL
[19]	马燕凯, 刘化勇, 燕振国, 等. 基于 HWCNS 格式的紧致插值方法研究[J]. 计算力学学报, 2015, 32(3): 388-393.
	MA Yankai, LIU Huayong, YAN Zhenguo, et al. Research on compact interpolation method based on HWCNS scheme[J]. Chinese Journal of Computational Mechanics, 2015, 32(3): 388-393.
[20]	ROE P L. Approximate Riemann solvers, parameter vectors, and difference schemes[J]. Journal of Computational Physics, 1997, 135(2): 250-258. doi: 10.1006/jcph.1997.5705 URL
[21]	LIOU M S, STEFFEN C J. A new flux splitting scheme[J]. Journal of Computational Physics, 1993, 107(1): 23-39. doi: 10.1006/jcph.1993.1122 URL
[22]	VAN LEER B. Flux-vector splitting for the Euler equations[J]. Lecture Notes in Physics, 1982, 170(1): 507-512.
[23]	HIEJIMA T. A high-order weighted compact nonlinear scheme for compressible flows[J]. Computers & Fluids, 2022, 232: 105199. doi: 10.1016/j.compfluid.2021.105199 URL
[24]	EF T, SPRUCE M, SPEARES W. Restoration of the contact surface in the HLL-Riemann solver[J]. Shock Waves, 1994, 4(1): 25-34. doi: 10.1007/BF01414629 URL
[25]	GOTTLIEB S, SHU C W. Total variation diminishing Runge-Kutta schemes[J]. Mathematics of Computation, 1998, 67(221): 73-85. doi: 10.1090/mcom/1998-67-221 URL
[26]	WONGA M L, LELEA S K. High-order localized dissipation weighted compact nonlinear scheme for shock-and interface-capturing in compressible flows[J]. Journal of Computational Physics, 2017, 339: 179-209. doi: 10.1016/j.jcp.2017.03.008 URL
[27]	LAX P D, LIU X D. Solution of two-dimensional Riemann problems of gas dynamics by positive schemes[J]. Siam Journal on Scientific Computing, 1998, 19(2): 319-340. doi: 10.1137/S1064827595291819 URL
[28]	ZHANG H, ZHANG F, LIU J, et al. A simple extended compact nonlinear scheme with adaptive dissipation control[J]. Communications in Nonlinear Science and Numerical Simulation, 2020, 84: 105191. doi: 10.1016/j.cnsns.2020.105191 URL
[29]	SHI J, ZHANG Y T, SHU C W. Resolution of high order WENO schemes for complicated flow structures[J]. Journal of Computational Physics, 2003, 186(2): 690-696. doi: 10.1016/S0021-9991(03)00094-9 URL
[30]	ZHAO G, SUN M, XIE S, et al. Numerical dissipation control in an adaptive WCNS with a new smoothness indicator[J]. Applied Mathematics and Computation, 2018, 330: 239-253. doi: 10.1016/j.amc.2018.01.019 URL

N_x	WCNS7-Z				WCNS7-T
N_x	E₁	P₁	E₂	P₂	E₁	P₁	E₂	P₂
10²	4.34×10^-3	0.00	3.12×10^-3	0.00	2.17×10^-3	0.00	1.47×10^-3	0.00
20²	5.74×10^-5	6.24	3.91×10^-5	6.32	1.86×10^-5	6.87	1.02×10^-5	7.17
40²	5.41×10^-7	6.73	3.46×10^-7	6.82	1.33×10^-7	7.13	8.08×10^-8	6.98
80²	4.17×10^-9	7.02	2.51×10^-9	7.11	8.15×10^-10	7.35	5.46×10^-10	7.21
160²	3.34×10^-11	6.96	1.86×10^-11	7.08	5.98×10^-12	7.09	3.90×10^-12	7.13
320²	2.19×10^-13	7.25	1.43×10^-13	7.02	4.04×10^-14	7.21	3.01×10^-14	7.02