CTVD格式应用及相关问题研究
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摘要
本文工作分为三部分。第一部分针对磁流体力学方程组(MHD)特征异常复杂的特点,将一种无需利用雅可比矩阵对方程组进行特征解耦的格式(CTVD)推广到MHD数值模拟中。CTVD是一种具有二阶精度的TVD格式,其简单实用、无需特征解耦的特点使该格式在MHD数值模拟中具有较高的推广价值。本文采用局部的Lax-Friedrichs分裂代替CTVD格式原有的全局Lax-Friedrichs分裂,使MHD数值模拟具有更高的分辨率,在相关算例中获得了较好的结果。
第二部分是将高分辨的CTVD格式应用到多介质流体力学数值模拟中,分别采用1/(γ-1)模型的守恒方法和非守恒方法。数值实验表明,即使采用守恒方法,物质界面附近的振荡是很轻微的,加密网格后,振荡基本消失。采用无振荡的TVD格式(NOS1)来处理非守恒方程后,CTVD格式成功消除物质界面附近的振荡。
最后一部分内容是采用两种方法去克服HLLC Riemann近似解在二维欧拉方程中出现的激波不稳定现象。第一种方法是旋转方法,随后,借鉴了旋转方法的网格法线向量的速度差分解,得出了一个自适应的探测加权函数,进而得出HLLC组合方法。相关数值实验表明,在时间、健壮性方面,组合方法优于旋转方法,在对间断的分辨率方面,两种方法相当。
The paper consists of three parts. In the first part, we present CTVD scheme as a Jacobian-free solver for ideal magnetohydrodynamics(MHD) equations which have complicated characteristics. CTVD scheme, as a second order-order scheme in space and time, avoids characteristic decomposition of the Jacobian and is efficient for MHD computations. After using local Lax-Friedrichs flux splitting instead of global Lax-Friedrichs flux splitting in Yu[25], we get more accurate numerical results in MHD computations. The results of several one-dimension and two-dimension tests demonstrate that CTVD scheme is an efficiency, and has high resolution in MHD computations.
In the second part, we use high-resolution CTVD scheme to solve the Euler equations of multi-component flows. The numerical experiments of conservative CTVD scheme for the gas mixtures only produces slight oscillations near the interfaces. And the oscillations disappear when the computation is on very fine grids. We also adopt a non-oscillation TVD scheme(NOSl) to solve the non-conservative equations and the CTVD scheme eliminates the oscillations when being applied to the non-conservative equations.
In the third part, we proposed two methods to cure the numerical shock instability for HLLC Riemann solver in two-dimension Euler equations. One method uses the rotated method and the other one is the hybrid method. Taking into account the efficiency, accuracy and robustness, the hybrid method is better than rotated method for HLLC solver. A series of test problems prove this conclusion.
第二部分是将高分辨的CTVD格式应用到多介质流体力学数值模拟中,分别采用1/(γ-1)模型的守恒方法和非守恒方法。数值实验表明,即使采用守恒方法,物质界面附近的振荡是很轻微的,加密网格后,振荡基本消失。采用无振荡的TVD格式(NOS1)来处理非守恒方程后,CTVD格式成功消除物质界面附近的振荡。
最后一部分内容是采用两种方法去克服HLLC Riemann近似解在二维欧拉方程中出现的激波不稳定现象。第一种方法是旋转方法,随后,借鉴了旋转方法的网格法线向量的速度差分解,得出了一个自适应的探测加权函数,进而得出HLLC组合方法。相关数值实验表明,在时间、健壮性方面,组合方法优于旋转方法,在对间断的分辨率方面,两种方法相当。
The paper consists of three parts. In the first part, we present CTVD scheme as a Jacobian-free solver for ideal magnetohydrodynamics(MHD) equations which have complicated characteristics. CTVD scheme, as a second order-order scheme in space and time, avoids characteristic decomposition of the Jacobian and is efficient for MHD computations. After using local Lax-Friedrichs flux splitting instead of global Lax-Friedrichs flux splitting in Yu[25], we get more accurate numerical results in MHD computations. The results of several one-dimension and two-dimension tests demonstrate that CTVD scheme is an efficiency, and has high resolution in MHD computations.
In the second part, we use high-resolution CTVD scheme to solve the Euler equations of multi-component flows. The numerical experiments of conservative CTVD scheme for the gas mixtures only produces slight oscillations near the interfaces. And the oscillations disappear when the computation is on very fine grids. We also adopt a non-oscillation TVD scheme(NOSl) to solve the non-conservative equations and the CTVD scheme eliminates the oscillations when being applied to the non-conservative equations.
In the third part, we proposed two methods to cure the numerical shock instability for HLLC Riemann solver in two-dimension Euler equations. One method uses the rotated method and the other one is the hybrid method. Taking into account the efficiency, accuracy and robustness, the hybrid method is better than rotated method for HLLC solver. A series of test problems prove this conclusion.
引文
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[2]刘儒勋,舒其望,计算流体力学的若干新方法。北京:科学出版社,2003。
[3]刘儒勋,王志峰,数值模拟方法和运动界面追踪。合肥:中国科学技术大学出版社,2001。
[4]任玉新,陈海昕,计算流体力学基础。北京:清华大学出版社,2006。
[5]水鸿寿,一维流体力学差分方法。北京:国防工业出版社,1998。
[6]于恒,无振荡选取法构造高分辨激波捕捉格式,博士学位论文,上海:复旦大学,2000。
[7]楼剑锋,于恒,非定常爆轰问题的高分辨CTVD格式数值模拟,中国工程物理研究院硕士学位论文,北京,2004
[8]李大潜,秦铁虎,物理学与偏微分方程。北京:高等教育出版社,2005。
[9]傅德薰,马延文,计算流体力学。北京:高等教育出版社,2002。
[10]陆金甫,偏微分方程数值解法。北京:清华大学出版社,2003。
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[2]刘儒勋,舒其望,计算流体力学的若干新方法。北京:科学出版社,2003。
[3]刘儒勋,王志峰,数值模拟方法和运动界面追踪。合肥:中国科学技术大学出版社,2001。
[4]任玉新,陈海昕,计算流体力学基础。北京:清华大学出版社,2006。
[5]水鸿寿,一维流体力学差分方法。北京:国防工业出版社,1998。
[6]于恒,无振荡选取法构造高分辨激波捕捉格式,博士学位论文,上海:复旦大学,2000。
[7]楼剑锋,于恒,非定常爆轰问题的高分辨CTVD格式数值模拟,中国工程物理研究院硕士学位论文,北京,2004
[8]李大潜,秦铁虎,物理学与偏微分方程。北京:高等教育出版社,2005。
[9]傅德薰,马延文,计算流体力学。北京:高等教育出版社,2002。
[10]陆金甫,偏微分方程数值解法。北京:清华大学出版社,2003。
[11]胡文瑞,宇宙磁流体力学。北京:科学出版社,1987。
[12]贝特曼,磁流体力学不稳定性。北京:原子能出版社,1982。
[13]Pieter Wesseling,Principles of computiational fluid dynamics.Springer-Verlag Berlin Heidelberg,2001.
[14]J.von Neumann,R.D.Richtmyer,A method for the numerical calculations of hydrodynamical shocks.Jounal of Applied Physics,1950,21:232.
[15] A. Harten, G. Zwas, Self-adjusting hybrid schemes for shock computations. Journal of Computational Physics, 1973, 11:38-69.
[16] J. P. Boris, D. L. Book, Flux-corrected transport. I. SHASTA, A fluid transport accuracy analysis of finite-difference methods. Journal of Computational Physics, 1974, 14:159-179.
[17] P. Colella, P. R. Woodward, The piecewise parabolic method (PPM) for gas-dynamical simulations. Journal of Computational Physics, 1984, 54:174-201.
[18] A. Harten, High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, 1983, 49:357-393.
[19] A. Harten, S. Osher, Uniformly high-order accurate non-oscillatory schemes I. SIAM Journal of Numerical Analysis, 1987, 24:279-309.
[20] A. Harten, B. Engquist, S. Osher and R. Chakravarthy, Uniformly high-order accurate non-oscillatory schemes III. Journal of Computational Physics, 1987, 71:231-303.
[21] C. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics, 1988, 77:439-471.
[22] C. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes II. Journal of Computational Physics, 1989, 82,362-397.
[23] X. Liu, S. Osher, Weighted essentially non-oscillatory schemes. Journal of Computational Physics, 1994, 115:319-338.
[24] E. F. Toro, Riemann solver and Numerical Methods for Fluid Dynamics (2nd edn). Springer,Berlin 1999.
[25] H. Yu, Y. Liu, A Second-Order Accurate, Component-Wise TVD Scheme for Nonlinear, Hyperbolic Conservation Laws. Journal of Computational Physics, 2001, 173:1-16.
[26]S.K.Godunov,A finite difference method for computation of discontinuous solutions of the equations of fluid dynamics.Mathematicheskii Sbornik,1959,47:271-306.
[27]C.Shu,Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws,ICASR Report No.97-65,NASA/CR-97-206253(1997).
[28]P.K.Sweby,High resolution schemes using flux limiters for hyperbolic conservation laws.SIAM Journal of Numerical Analysis,1984,21:995-1011.
[29]J.Adams,P.Swarztrauber,R.Sweet,Fishpack.Available from http://www.scd.ucar.edu/css/software/fishpack/.
[30]S.A.Orazag and C.M.Tang,Small-scale structure of two-dimensional magnetohydrodynamic turbulence.J.Fluid.Mech,1979,90:129-143.
[31]J.Balbasa,E.Tadmor,C.C.Wu,Non-oscillatory central schemes for oneand twodimensional MHD equations:Ⅰ.Journal of Computational Physics,2004,201:261-285.
[32]J.U.Brackbill and D.C.Barnes,Note:The effect of nonzero ▽.B on the numerical solution of the magnetohydrodynamic equations.Journal of Computational Physics,1980,35:426-461.
[33]M.Brio,C.C.Wu,An upwind differencing scheme for the equations of magnetohydrodynamics.Journal of Computational Physics,1987 75(2):400-422.
[34]P.Cargo and C.Gallice,Roe matrices for ideal MHD and systematic construction of Roe matrices for systems of conservation laws.Journal of Computational Physics,1997,136:446-466.
[35]A.Harten,Self aduusting grid methods for one-dimensional hyperbolic conservation laws.Journal of Computational Physics,1983,49:357-393.
[36] P. L. Roe. Approximate riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 1981, 43:357-372.
[37] G. -S. Jiang, C.C. Wu, A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics. Journal of Computational Physics. 1999, 150:1892-1917.
[38] R. B. Dahlburg and J. M. Picone, Evolution of the Orszag - Tang vortex system in a compressible medium. I. Initial average subsonic flow. Phys. Fluids B, 1989, 1:2153-2171.
[39] W. Dai, P.R. Woodward, A simple finite difference scheme for multidimensional magnetohydrodynamical equations. Journal of Computational Physics, 1998, 142:331-369.
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