求解双曲守恒律方程的WENO型熵相容格式

程晓晗; 封建湖; 聂玉峰

doi:10.11883/1001-1455(2014)04-0501-07

摘要: 通过在单元交界面处进行高阶WENO重构，得到了一种求解双曲型守恒律方程的WENO型熵相容格式。用该格式对一维Burgers方程和Euler方程进行数值模拟，结果表明，该格式具有高精度、基本无振荡性等特点。

关键词:

Abstract: Compared with entropy stable schemes, entropy consistent schemes control entropy production more exactly and effectively eliminate phenomena such as expansion shocks and spurious oscillations. By using WENO (weighted essentially non-oscillatory) reconstruction of higher order at cell interfaces, a WENO type entropy consistent scheme for hyperbolic conservation laws is presented. The one-dimentional Burgers equation and Euler equations are used to test the proposed scheme. The numerical experiments demonstrate that the scheme is accurate and essentially non-oscillatory.

Key words:

图 1 无黏Burgers方程间断初值问题

Figure 1. Discontinuous initial value problem of Burgers equation

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图 2 一维Euler方程Sod激波管问题

Figure 2. Sod shock tube problem of 1D Euler equation

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图 3 一维Euler方程低密度流问题

Figure 3. Low density problem of 1D Euler equation

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图 4 一维Euler方程强稀疏波问题

Figure 4. Density of strong expansion problem of 1D Euler equation

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表 1 EC-WENO格式的数值精度

Table 1. Numerical accuracy of EC-WENO scheme

网格数	L₁	精度阶	L_∞	精度阶
20	1.420 0×10^-2		1.020 0×10^-2
40	4.653 5×10^-4	4.931 4	3.859 9×10^-4	4.723 9
80	1.447 5×10^-5	5.006 7	1.310 2×10^-5	4.880 7
160	4.515 7×10^-7	5.002 5	4.137 0×10^-7	4.985 1

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[1]	Roe P L. Approximate Rieman solvers, parameter vectors, and difference schemes[J]. Journal of Computational Physics, 1981, 43(2): 357-372. doi: 10.1016/0021-9991(81)90128-5
[2]	Tadmor E. The numerical viscosity of entropy stable schemes for systems of conservation laws, Ⅰ[J]. Mathematics of Computation, 1987, 49(179): 91-103. doi: 10.1090/S0025-5718-1987-0890255-3
[3]	Roe P L. Affordable, entropy-consistent, flux functions[C]//Oral Talk at Eleventh International Conference on Hyperbolic Problems: Theory, Numerics, Applications. Lyon, France, 2006.
[4]	Ismail F, Roe P L. Affordable, entropy-consistent Euler flux functions, Ⅱ: Entropy production at shocks[J]. Journal of Computational Physics, 2009, 228(15): 5410-5436. doi: 10.1016/j.jcp.2009.04.021
[5]	Tadmor E. Numerical viscosity and the entropy conditions for conservative difference schemes[J]. Mathematics of Computation, 1984, 43(168): 369-381. doi: 10.1090/S0025-5718-1984-0758189-X
[6]	Liu X D, Osher O, Chan T. Weighted essentially non-oscillatory schems[J]. Journal of Computational Physics, 1994, 115(1): 200-212. doi: 10.1006/jcph.1994.1187
[7]	Tadmor E. Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems[J]. Acta Numerica, 2003, 12: 451-512. doi: 10.1017/S0962492902000156
[8]	Fjordholm U S, Mishra S, Tadmor E. Energy preserving and energy stable schemes for the shallow water equations[R]. Hong Kong: Foundations of Computational Mathematics, 2008.
[9]	Gottlieb S, Shu C W, Tadmor E. High order time discretizations with strong stability properties[J]. SIAM Review, 2001, 43(1): 89-112. doi: 10.1137/S003614450036757X

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