一种求解二维辐射流体力学方程组的显隐式格式

doi:10.19596/j.cnki.1001-246x.8312

摘要/Abstract

摘要：

构造一种求解二维辐射流体力学方程组的有限体积方法。相较于Euler方程组，辐射流体力学方程组的数值格式设计更为困难。不仅辐射压力与辐射能量的强非线性增加了数值计算的难度，而且求解强激波问题也是一大难点。与此同时，物质量以声速传播，辐射量以光速传播也增加了该系统求解的难度。为了克服这些难点，我们使用MUSCL-Hancock方法求解模型的双曲部分，利用TR/BDF2的变换格式求解其扩散部分。给出了灰非平衡扩散模型关于时间的渐近分析。最后，通过数值实验展现该格式的表现。

关键词: 辐射流体力学, Godunov-type格式, MUSCL Hancock, 二维空间维度

Abstract:

We study a finite volume solver for radiation hydrodynamical models in two space dimensions. For nonlinearity of radiation pressure and radiation energy, it is not a trivial task to develop a numerical scheme for radiation hydrodynamics compared to Euler equations. The current strategy is that the convective part is solved by MUSCL-Hancock schemes, while the diffusive part is handled through an alternative form of TR/BDF2 (trapezoidal rule with second order backward difference formula) method. An asymptotic analysis for the grey nonequilibrium diffusion model in time is presented. Numerical experiments are given to show performance of the IMEX scheme.

Key words: radiation hydrodynamics, Godunov-type scheme, MUSCL Hancock, two space dimensions

中图分类号:

O241.82

房尧立, 王一. 一种求解二维辐射流体力学方程组的显隐式格式[J]. 计算物理, 2021, 38(4): 401-417.

Yaoli FANG, Yi WANG. An IMEX Method for 2-D Radiation Hydrodynamics[J]. Chinese Journal of Computational Physics, 2021, 38(4): 401-417.

图/表 13

Table 1 The L2 error in density

N	L²-error	L²-order
20	0.003 157 7
40	0.000 602 5	2.38
80	0.000 134 1	2.16
160	0.000 032 2	2.05
320	0.000 007 1	2.17

Fig.1 A shock-tube problem: The solution of density, fluid temperature and radiation temperature of our scheme with 200 mesh cells at t=0.2 against the reference solution

Fig.2 A shock-tube problem: Comparison results of Example 2 for K=0.000 1, 0.001, 0.01 at t=0.2

Fig.3 A shock-tube problem: The solution of density, fluid temperature and radiation temperature of our scheme with 200 mesh cells at t=0.04 against the reference solution

Fig.4 A shock-tube problem: The solution of density, fluid temperature and radiation temperature of our scheme with 200 mesh cells at t=0.018 against the reference solution

Fig.5 A steady shock problem: The solution of density, fluid temperature and radiation temperature for u0=1.05 on 100 mesh cells

Fig.6 A steady shock problem: The solution of density, fluid temperature and radiation temperature for u0=1.2 on 100 mesh cells

Fig.7 A steady shock problem: The solution of density, fluid temperature and radiation temperature for u0=1.4 on 100 mesh cells

Fig.8 A steady shock problem: The solution of density, fluid temperature and radiation temperature for u0=3.0 on 100 mesh cells

Fig.9 Solutions of density at t=0.6 with 200 ×100 uniform cells and 30 equally contour lines, where K=0

Fig.10 Solutions of temperature at t=0.6 with 200 ×100 uniform cells and 30 equally contour lines, where K=0

Fig.11 Solutions of density at t=0.6 with 200 ×100 uniform cells and 30 equally contour lines, where K=0.001

Fig.12 Solutions of temperature at t=0.6 with 200 ×100 uniform cells and 30 equally contour lines, where K=0.001

参考文献 21

1	BEN-ARTZI M, LI J Q, WARNECKE G. A direct Eulerian GRP scheme for compressible fluid flows[J]. J Comput Phys, 2006, 218 (1): 19- 43. DOI
2	GODUNOV S K. A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[J]. Math Sb(N S), 1959, 47 (3): 271- 306.
3	LEVEQUE R J. Numerical methods for conservation laws[M]. Basel: Birkhäuser, 1992.
4	TORO E F. Riemann solvers and numerical methods for fluid dynamics: A practical introduction[M]. 3rd ed. Springer, 2009.
5	VAN L B. Towards the ultimate conservative difference scheme V: A second-order sequel to Godunov's method[J]. J Comput Phys, 1979, 32 (1): 101- 136. DOI
6	SUN C, LI X, SHEN Z J. A Godunov method with staggered Lagrangian discretization applicable to isentropic flows[J]. Chinese J Comput Phys, 2020, 37 (5): 529- 538.
7	CHEN H Q. Implicit gridless method for Euler equations[J]. Chinese J Comput Phys, 2003, 20 (1): 9- 13.
8	YAN G W, HU S X, SHI W P. An Euler solver based on lattice Boltzmann Godunov method with bi-distribution functions[J]. Chinese J Comput Phys, 1998, 15 (2): 193- 198.
9	DAI W, WOODWARD P R. Numerical simulations for radiation hydrodynamics I: Diffusion limit[J]. J Comput Phys, 1998, 142 (1): 182- 207. DOI
10	TANG H Z, WU H M. Kinetic flux vector splitting for radiation hydiation hydrodynamical equations[J]. Comput Fluids, 2000, 29 (8): 917- 933. DOI
11	KUANG Y Y, TANG H Z. Second-order direct Eulerian GRP schemes for radiation hydrohynamical equations[J]. Comput Fluids, 2019, 179 (1): 163- 177.
12	CHENG J, SHU C W, SONG P. High order conservative Lagrangian schemes for one-dimensional radiation hydrodynamics equations in the equilibrium-diffusion limit[J]. J Comput Phys, 2020, 421 (15): 109724.
13	QAMAR S, ASHRAF W. Application of central schemes for solving radition hydrodynamical models[J]. Comput Phys Comm, 2013, 184 (5): 1349- 1363. DOI
14	FANG Y L. An implicit-explicit scheme for the radiation hydrodynamics[J]. The Adva in App Math and Mech, 2021, 13 (2): 404- 425. DOI
15	BOLDING S, HANSEL J, EDWARDS J D, et al. Second-order discretization in space and time for radiation-hydrodynamics[J]. J Comput Phys, 2017, 338 (1): 511- 526.
16	LOWRIE R B, RAUENZAHN R M. Radiative shock solutions in the equilibrium diffusion limit[J]. Shock Wave, 2007, 16 (6): 445- 453. DOI
17	LOWRIE R B, EDWARDS J D. Radiative shock solutions with grey nonequilibrium diffusion[J]. Shock Wave, 2008, 18 (2): 129- 143. DOI
18	LARSEN E W, MOREL J E. Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes Ⅱ[J]. J Comput Phys, 1989, 83 (1): 212- 236. DOI
19	LARSEN E W, MOREL J E. Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes[J]. J Comput Phys, 1987, 69 (2): 283- 324. DOI
20	KNOLL D A, LOWRIE R B, MOREL J E. Numerical analysis of time integrarion errors for nonequilibrium radiation diffusion[J]. J Comput Phys, 2007, 226 (2): 1332- 1347. DOI
21	DENSMORE J D, LARSEN E W. Asymptotic equilibrium diffusion analysis of time-dependent Monte Carlo methods for grey radiative transfer[J]. J Comput Phys, 2004, 199 (1): 175- 204. DOI

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