五阶FD-WENO格式与二阶Godunov格式MUSCL的数值测试与比较
|
摘要
由于求解双曲守恒律组的高阶加权实质上无振荡有限差分格式(简记为FD-WENO)是最近十年才发展起来的，它与上世纪70-80年代发展起来的著名高阶Godunov格式比较，究竟优缺点各如何，哪种格式更好，这个问题目前在国内外文献中研究尚少，尚无定评。特别是对于辐射流体力学问题的数值计算，以上两种格式中究竟哪种格式的实际效果更好，这是惯性约束聚变(ICF)数值模拟研究中迫切需要回答的问题，为此，本文作者发展了用2阶Godunov格式MUSCL求解气体动力学Euler方程的应用软件，并通过同时用五阶FD-WENO格式(下文中简记为WENO5)和二阶MUSCL格式求解各种有代表性的流体力学问题，进行了大量的数值测试，例如，通过求解线性初边值问题、一维Burgers方程初边值问题、若干Riemann问题、一维激波相互碰撞问题及二维周期漩涡问题等对这两种格式进行了测试和定量比较，通过求解Richtmyer-Meshkov(RM)不稳定性问题和Rayleigh-Taylor(RT)不稳定性问题对这两种格式进行了测试和定性比较．我们发现对于简单的Sod Riemann问题或Lax Riemann问题，两种格式都易于算出令人满意的具有较高精度和较高分辨率的数值结果，相对来说，WENO5得到的数值结果粘性稍强，MUSCL得到的结果则在局部地方有微小的非物理振荡，但计算速度更快；对于Peak Riemann问题及Woodward-Colella激波相互碰撞问题，WENO5所获数值结果在激波及稀疏波附近明显优于MUSCL，前者不仅分辨率较高，误差也较小；对于二维周期漩涡问题，采用81×81的一致网格用WENO5格式计算至时刻t=10的数值结果甚至比采用161×161的一致网格用MUSCL格式计算至时刻t=10的数值结果要好得多，但前者花费的计算时间更短；对于RM不稳定性问题和RT不稳定性问题，WENO5所得到的数值结果比MUSCL所得到的数值结果分辨率更高，图形较为符合实际情况。本文的研究结果表明，尽管对于简单的Riemann问题，二阶MUSCL具有一定优势，但对于既具有强激波又具有大面积复杂流动结构的问题，特别是对于ICF数值模拟问题，二阶格式一般难以提供具有可接受分辨率的数值结果，而高阶FD-WENO格式特别适合于求解这类问题。
A numerical study is undertaken comparing the fifth-order finite difference weighted essentially non-oscillatory scheme (FD-WEN05) to the second-order Godunov scheme MUSCL (monotonic upstream-centered scheme for conservation laws). For quantitative comparison purpose, these methods are tested on a series of problems whose true solutions are known or can be computed with high accuracy, such as linear advection problem, Sod Riemann, "peak" Riemann problem, Woodward and Colella interacting shock wave problem and the two-dimensional periodic vortex problem. For qualitative comparison purpose, these methods are tested on Rayleigh-Taylor instability and Richtmyer-Meshkov instability problems. The numerical results show that: for the Sod and Lax Riemann problems, high accuracy and high resolution results can be easily achieved by both schemes, but MUSCL is performed much faster than WEN05. For the " Peak" Riemann problem and the Woodward and Colella interacting shock wave problem, when both methods are performed on same space mesh, we find that WENO5 is better than MUSCL around the shock and the rarefaction, in more de-tail, WENO5 has not only higher resolution but also smaller errors than MUSCL. For the two-dimensional periodic vortex problem, We find that the numerical results of WENO5 when performed on the uniform mesh of 81 x 81 points are much more accurate than that of MUSCL when performed on the refined uniform mesh of 161 ×161 points, and the former takes CPU time shorter than the latter. For Rayleigh-Taylor instability and Richtmyer-Meshkov instability problems, when both methods are performed on same space mesh, we find that the numerical results of WENO5 are rich of higher resolution and the images are more realistic. In view of all the results mentioned above, we thus conclude that the second-order Godunov scheme MUSCL has advantages for many problems containing only simple shocks with almost linear smooth solutions in between, such as the solutions to most Riemann problems. However, when the solution contains both discontinuities and complex solution structures in the smooth regions, a higher order method, such as WENO5, may be more realistic and more economical in CPU time.
A numerical study is undertaken comparing the fifth-order finite difference weighted essentially non-oscillatory scheme (FD-WEN05) to the second-order Godunov scheme MUSCL (monotonic upstream-centered scheme for conservation laws). For quantitative comparison purpose, these methods are tested on a series of problems whose true solutions are known or can be computed with high accuracy, such as linear advection problem, Sod Riemann, "peak" Riemann problem, Woodward and Colella interacting shock wave problem and the two-dimensional periodic vortex problem. For qualitative comparison purpose, these methods are tested on Rayleigh-Taylor instability and Richtmyer-Meshkov instability problems. The numerical results show that: for the Sod and Lax Riemann problems, high accuracy and high resolution results can be easily achieved by both schemes, but MUSCL is performed much faster than WEN05. For the " Peak" Riemann problem and the Woodward and Colella interacting shock wave problem, when both methods are performed on same space mesh, we find that WENO5 is better than MUSCL around the shock and the rarefaction, in more de-tail, WENO5 has not only higher resolution but also smaller errors than MUSCL. For the two-dimensional periodic vortex problem, We find that the numerical results of WENO5 when performed on the uniform mesh of 81 x 81 points are much more accurate than that of MUSCL when performed on the refined uniform mesh of 161 ×161 points, and the former takes CPU time shorter than the latter. For Rayleigh-Taylor instability and Richtmyer-Meshkov instability problems, when both methods are performed on same space mesh, we find that the numerical results of WENO5 are rich of higher resolution and the images are more realistic. In view of all the results mentioned above, we thus conclude that the second-order Godunov scheme MUSCL has advantages for many problems containing only simple shocks with almost linear smooth solutions in between, such as the solutions to most Riemann problems. However, when the solution contains both discontinuities and complex solution structures in the smooth regions, a higher order method, such as WENO5, may be more realistic and more economical in CPU time.
引文
[1] D. Balsara and C. W. Shu, Monotonicity preserving weighted essentially nonoscillatory schemes with increasingly high order accuracy [J], J. Comput. Phys., 160(2000), 405-452.
[2] J. A. Carrillo, et al., A WENO-solver for the transients of Boltzmann-Poisson system for semiconductor devices: performance and comparisons with Monte Carlo methods [J], J. Comput. Phys., 184(2003), 498-525.
[3] B. Cockburn, G. Karniadakis and C. W. Shu, The development of discontinuous Galerkin methods, In B. Cockburn, G. Karniadakis and C. W. Shu (eds.), Discontinuous Galcrkin Methods: Theory, Computation and Applications [J], Lecture Notes in Computational Science and Engineering, Springer, 11, Part Ⅰ: Overview, 3-50, 2000.
[4] R. H. Cohen, et al., Three-dimensional simulation of a Richtmyer-Meshkov instability with a two-scale initial perturbation [J]. Physics of Fluids, 14(2002), 3692-3709.
[5] P. Colella and P. R. Woodaward, The picewise parabolic method(PPM) for gas-dynamics simulations [J], J. Comput. Phys., 54(1984), 174-201.
[6] N. Ornjaric-Zic, S. Vukovic and L. Sopta, Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equations [J], J. Comput. Phys., 200(2004), 512-548.
[7] O. Friedrichs, Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructrcd grids [J], J, Comput. Phys, 1998, 144: 194-212
[8] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong Stability-Preserving HighOrder Time Discretization Methods [J], SIAM Rev. 42(2001), 89-112.
[9] J. A. Grcenough, W. J. Ridcr, A quantitative compariosn of munerical methods for the comperssible Eulerequations: fifth-order WENO and piecewise linear Godunov [J]. J. Comput. Phys (2004), 196: 258-281.
[10] A. Harten and B. Engquist: et al.. Uniformly high oreer cssentially nonoscillatory schemes Ⅲ [J], J. Comput. Phys., 71(1987), 231-303.
[11] A. K. Henrick, T. D. Aslam and J. M. Powers, Mapped weighted cssentially non-oscillatory schemes: Achieving optimal order near critical points [J]. J. Comput. Phys., 207(2005), 542-567.
[12] D. J. Hill and D. I. Pulliu Hybrid tuned center-differencee-WENO method for large eddy simulations in the presence of strong shocks [J], J. Comput. Phys., 194(2004), 435-450.
[13] C. W. Hirt and B. D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries [J], J. Comput. Phys., (39)1981, 201-225.
[14] C. Hu, C.W. Shu, Weighted essentially non-oscillatory schemes on triangular meshes [J], J. Comput. Phys., 150(1999), 97-127.
[15] G. S. Jiang and D. P. Peng, Weighted ENO schemes for Hamilton+Jacobi equations [J], SIAM J. Sci. Comput., 21(2000), 2126.
[16] G. S. Jiang and C. W. Shu, Efficient implementation of weighted ENO schemes [J], J. Comput. Phys., 126(1996), 202-228.
[17] D. Levy, G. Puppo, and G. Russo, A third order central WENO scheme for 2D conservation laws [J], Appl. Numer. Math. 33(2000), 415.
[18] D. Levy, G. Puppo, and G. Russo, Compct central WENO schemes for multidimensional conservation laws [J], SIAM J. Sci. Comput. 22(2000), 656.
[19] 李德元，徐国荣，水鸿寿，何高玉，陈光南，袁国兴，二维非定常流体力学数值方法[M]，科学出版社，1998．
[20] M．J．Li，Y．Y．Y,'mg and s．Shu，High-order modifying cocfficient sdmme based on ENO sclmme[R]，总体与靶物理理论研究专题2004年度工作报告，86-92．
[21] 李寿佛，张瑗，樊华，杨水平，双曲守恒律组WENO格式的测试，总体与靶物理理论研究专题2004年度工作报告[R]，71-76．
[22] 李寿佛，叶文华，张瑗，舒适，肖爱国，高阶加权本质上无振荡有限差分格式在数值求解Rayleigh-Taylor不稳定性问题中的应用[J]，2005，待发表．
[23] 李寿佛，张瑗，刘玉珍，杨水甲，屈小妹，高阶加权本质上无振荡有限差分格式在数值求解Richtnlyer-Meshkov不稳定性问题中的应用[J]，2005，待发表．
[24] 李寿佛，叶文华，张瑗，舒适，肖爱国，流体动力学Euler方程的高效数值方法的测试、分析、比较及改进[J]，2005：待发表．
[25] 刘儒勋，舒其望，计算流体力学的若干新方法[M]，科学出版社，2003．
[26] X. D. Liu. S. Osher and T. Chan. Weighted essentially non-oscillatory schemes[J], J. Comput. Phys.. 115(1994). 200-212.
[27] P. Montarnal and C. W. Shu. Real Gas Computation Using an Energy Relaxatiou Method and tligh-Ordcr WENO Schemes [J], J. Comput. Phys., 148(1999). 59 80.
[28] S. Osher and J. A. Sethian. Fromts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations [J], J. Comput. Phys. (79)1988, 12-49.
[29] S. Pirozzoli, Conservative Hybrid Compact-WENO Schemes for ShockTurbulence Interactiou [J], J. Comput. Phys., 178(2002), 81-117.
[30] J. X. Qiu and C. W. Shu, Hermite WENO schemes and their application as limiters for Rungc-Kutta discontinuous Galerkin method: one-dimensional case [J], J. Comput. Phys. 193(2004), 115-135.
[31] J. X. Qiu and C. W. Shu, Hermite WENO schemes for Hamilton-Jacobi equations [J], J. Comput. Phys. 204(2005), 82-99.
[32] Y. X. Ren, M. E. Liu and H. X. Zhang, A characteristic-wise hybrid compactWENO scheme for solving hyperbolic conservatiou laws [J], J. Comput. Physics, 192(2003), 365-386.
[33] L. H. Richard, et al., Richtmyer-Meshkov instability growth: experiment, simulation and theory [J], J. Fluid Mechanics, 389(1999), 55-79.
[34] S. J. Ruuth and W. Hundsdorfer, High-order linear multistep methods with general monotonicity aud boundedness properties [J], J. Comput. Physics, 209(2005), 226-248.
[35] J. Shi, C. Hu and C. W. Shu, A technique of treating negative weights in WENO schemes [J], J. Comput. Physics, 175(2002), 108-127.
[36] J. Shi, Y. T. Zhang and C. W. Shu, Resolution of high order WENO schemes for complicated flow structures [J], J. Comput. Physics, 186(2003), 690-696.
[37] C. W. Shu, Essentially non-oscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws [R], NASA/CR-97206253, ICASE. Report No. 97-65, 1997.
[38] C. W. Shu, Essentially non-oscillatory and weighted cssentially nonoscillatory, schemes for hyperbolic conservation laws, In: B. Cockburn, C. Johnson, C. W. Shu, E. Tadmor and A. Quarteroni, (eds.), Advanced Numerical Approximation of Nonlinear Hyperbolic Equatious Lecture Notes in Mathematics [J], Springer, 1697, 325-432, 1998.
[39] C. W. Shu, High-order Finite Difference and Finite Volume WENO Schemes aud Discontinuous Galcrkin Methods for CFD, Iuternational Journal of Computational Fluid Dynamics [J], 17(2003). 2: 107-118.
[40] C. W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock capturing schemes [J]. J. Comput. Phys., 77(1988), 439-471.
[41] C. W. Shu and S. Osher, Efficient mplementation of essentially non-oscillatory shock capturing schemes Ⅱ [J], J. Comput. Phys., 83(1989), 32-78.
[42] 水鸿寿，一维流体力学差分方法[M]，北京，国防工业出版社．
[43] G. Strang, On the construction and comparison of difference schemes [J], SIAM J. Numer. Anal. 5, 506(1968).
[44] V. A. Titarev and E. F. Toro, Finite-volume WENO schemes for threedimensional conservation laws [J], J. Comput. Physics, 201(2004), 238-260.
[45] M. Torrilhon and D. S. Balsara, High order WENO schemes: investigations on non-uniform convergence for MHD Riemann problems [J], J. Comput. Physics, 201(2004), 586-600.
[46] B. Van Leer, Stabilization of difference schemes for the equations of inviscid compressible flow by artificial diffusion [J], J. Comput. Phys., 3(1969), 473-485.
[47] B. Van Leer, Towards the ultimate conservative difference scheme, Ⅱ, Monotonicity and conservation combined in a second order scheme [J], J. Comput. Phys., 14(1974), 361-370.
[48] B. Van Leer, Towards the ultimate conservative difference scheme, Ⅲ, Upstream-centered and conservation finite difference schemes for ideal compressible flow [J], J. Comput. Phys., 23(1977), 263-275.
[49] B. Van Leer, Towards the ultimate conservative difference scheme, Ⅳ, A new approach to numerical convection [J], J. Comput. Phys., 23(1977), 276-299.
[50] B. Van Leer, Towards the ultimate conservative difference scheme, Ⅴ, A second order sequel to Godunov's method [J], J. Comput. Plays., 32(1979), 101-136.
[51] S. Vukovic and L. Sopta, ENO and WENO Schemes with the Exact Conservation Property for One-Dimensional Shallow Water Equations [J], J. Comput. Phys., 179(2002), 593-621.
[52] Z. J. Wang and R. F. Chen, Optimized Weighted Essentially Nonoscillatory Schemes for Linear Waves with Discontinuity [J], J. Comput. Phys., 174(2001), 381-404.
[53] Y. L. Xing and C. W. Shu, High order finite difference WENO schemes with the exact conservation property for the shallow water equations [J], J. Comput. Phys., 208(2005), 206-227.
[54] Z. F. Xu and C. W. Shu, Anti-diffusive flux corrections for high order finite difference WENO schemes [J], J. Comput. Phys., 205(2005). 458-485.
[55] Y．Y．Yang，et al．，High order schemes based on upwind schemes with modifled coefficients[C]，总体与靶物理理论研究专题2004年度工作报告，978-982．J．Comput．Appl．Math．in press．
[56] 杨水平，李寿佛，屈小妹，张瑗，刘玉珍，五阶WENO格式和二阶Godunov格式的数值测试与比较[J]，2005，待发表．
[57] 叶文华，张维岩，陈光南，数值模拟激光烧蚀瑞利泰勒不稳定性[J]，内部通信，1997．
[58] M. P. Zhang, et al., A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model [J], J. Comput. Phys., 191(2003), 639-659.
[59] T. Zhou, Y. Li and C. W. Shu, Numerical comparison of WENO finite volume and Runge-Kutta discontinuous Galerkin methods [J], Journal of Scientific Computing, 16(2001), 145-171.
[2] J. A. Carrillo, et al., A WENO-solver for the transients of Boltzmann-Poisson system for semiconductor devices: performance and comparisons with Monte Carlo methods [J], J. Comput. Phys., 184(2003), 498-525.
[3] B. Cockburn, G. Karniadakis and C. W. Shu, The development of discontinuous Galerkin methods, In B. Cockburn, G. Karniadakis and C. W. Shu (eds.), Discontinuous Galcrkin Methods: Theory, Computation and Applications [J], Lecture Notes in Computational Science and Engineering, Springer, 11, Part Ⅰ: Overview, 3-50, 2000.
[4] R. H. Cohen, et al., Three-dimensional simulation of a Richtmyer-Meshkov instability with a two-scale initial perturbation [J]. Physics of Fluids, 14(2002), 3692-3709.
[5] P. Colella and P. R. Woodaward, The picewise parabolic method(PPM) for gas-dynamics simulations [J], J. Comput. Phys., 54(1984), 174-201.
[6] N. Ornjaric-Zic, S. Vukovic and L. Sopta, Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equations [J], J. Comput. Phys., 200(2004), 512-548.
[7] O. Friedrichs, Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructrcd grids [J], J, Comput. Phys, 1998, 144: 194-212
[8] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong Stability-Preserving HighOrder Time Discretization Methods [J], SIAM Rev. 42(2001), 89-112.
[9] J. A. Grcenough, W. J. Ridcr, A quantitative compariosn of munerical methods for the comperssible Eulerequations: fifth-order WENO and piecewise linear Godunov [J]. J. Comput. Phys (2004), 196: 258-281.
[10] A. Harten and B. Engquist: et al.. Uniformly high oreer cssentially nonoscillatory schemes Ⅲ [J], J. Comput. Phys., 71(1987), 231-303.
[11] A. K. Henrick, T. D. Aslam and J. M. Powers, Mapped weighted cssentially non-oscillatory schemes: Achieving optimal order near critical points [J]. J. Comput. Phys., 207(2005), 542-567.
[12] D. J. Hill and D. I. Pulliu Hybrid tuned center-differencee-WENO method for large eddy simulations in the presence of strong shocks [J], J. Comput. Phys., 194(2004), 435-450.
[13] C. W. Hirt and B. D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries [J], J. Comput. Phys., (39)1981, 201-225.
[14] C. Hu, C.W. Shu, Weighted essentially non-oscillatory schemes on triangular meshes [J], J. Comput. Phys., 150(1999), 97-127.
[15] G. S. Jiang and D. P. Peng, Weighted ENO schemes for Hamilton+Jacobi equations [J], SIAM J. Sci. Comput., 21(2000), 2126.
[16] G. S. Jiang and C. W. Shu, Efficient implementation of weighted ENO schemes [J], J. Comput. Phys., 126(1996), 202-228.
[17] D. Levy, G. Puppo, and G. Russo, A third order central WENO scheme for 2D conservation laws [J], Appl. Numer. Math. 33(2000), 415.
[18] D. Levy, G. Puppo, and G. Russo, Compct central WENO schemes for multidimensional conservation laws [J], SIAM J. Sci. Comput. 22(2000), 656.
[19] 李德元，徐国荣，水鸿寿，何高玉，陈光南，袁国兴，二维非定常流体力学数值方法[M]，科学出版社，1998．
[20] M．J．Li，Y．Y．Y,'mg and s．Shu，High-order modifying cocfficient sdmme based on ENO sclmme[R]，总体与靶物理理论研究专题2004年度工作报告，86-92．
[21] 李寿佛，张瑗，樊华，杨水平，双曲守恒律组WENO格式的测试，总体与靶物理理论研究专题2004年度工作报告[R]，71-76．
[22] 李寿佛，叶文华，张瑗，舒适，肖爱国，高阶加权本质上无振荡有限差分格式在数值求解Rayleigh-Taylor不稳定性问题中的应用[J]，2005，待发表．
[23] 李寿佛，张瑗，刘玉珍，杨水甲，屈小妹，高阶加权本质上无振荡有限差分格式在数值求解Richtnlyer-Meshkov不稳定性问题中的应用[J]，2005，待发表．
[24] 李寿佛，叶文华，张瑗，舒适，肖爱国，流体动力学Euler方程的高效数值方法的测试、分析、比较及改进[J]，2005：待发表．
[25] 刘儒勋，舒其望，计算流体力学的若干新方法[M]，科学出版社，2003．
[26] X. D. Liu. S. Osher and T. Chan. Weighted essentially non-oscillatory schemes[J], J. Comput. Phys.. 115(1994). 200-212.
[27] P. Montarnal and C. W. Shu. Real Gas Computation Using an Energy Relaxatiou Method and tligh-Ordcr WENO Schemes [J], J. Comput. Phys., 148(1999). 59 80.
[28] S. Osher and J. A. Sethian. Fromts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations [J], J. Comput. Phys. (79)1988, 12-49.
[29] S. Pirozzoli, Conservative Hybrid Compact-WENO Schemes for ShockTurbulence Interactiou [J], J. Comput. Phys., 178(2002), 81-117.
[30] J. X. Qiu and C. W. Shu, Hermite WENO schemes and their application as limiters for Rungc-Kutta discontinuous Galerkin method: one-dimensional case [J], J. Comput. Phys. 193(2004), 115-135.
[31] J. X. Qiu and C. W. Shu, Hermite WENO schemes for Hamilton-Jacobi equations [J], J. Comput. Phys. 204(2005), 82-99.
[32] Y. X. Ren, M. E. Liu and H. X. Zhang, A characteristic-wise hybrid compactWENO scheme for solving hyperbolic conservatiou laws [J], J. Comput. Physics, 192(2003), 365-386.
[33] L. H. Richard, et al., Richtmyer-Meshkov instability growth: experiment, simulation and theory [J], J. Fluid Mechanics, 389(1999), 55-79.
[34] S. J. Ruuth and W. Hundsdorfer, High-order linear multistep methods with general monotonicity aud boundedness properties [J], J. Comput. Physics, 209(2005), 226-248.
[35] J. Shi, C. Hu and C. W. Shu, A technique of treating negative weights in WENO schemes [J], J. Comput. Physics, 175(2002), 108-127.
[36] J. Shi, Y. T. Zhang and C. W. Shu, Resolution of high order WENO schemes for complicated flow structures [J], J. Comput. Physics, 186(2003), 690-696.
[37] C. W. Shu, Essentially non-oscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws [R], NASA/CR-97206253, ICASE. Report No. 97-65, 1997.
[38] C. W. Shu, Essentially non-oscillatory and weighted cssentially nonoscillatory, schemes for hyperbolic conservation laws, In: B. Cockburn, C. Johnson, C. W. Shu, E. Tadmor and A. Quarteroni, (eds.), Advanced Numerical Approximation of Nonlinear Hyperbolic Equatious Lecture Notes in Mathematics [J], Springer, 1697, 325-432, 1998.
[39] C. W. Shu, High-order Finite Difference and Finite Volume WENO Schemes aud Discontinuous Galcrkin Methods for CFD, Iuternational Journal of Computational Fluid Dynamics [J], 17(2003). 2: 107-118.
[40] C. W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock capturing schemes [J]. J. Comput. Phys., 77(1988), 439-471.
[41] C. W. Shu and S. Osher, Efficient mplementation of essentially non-oscillatory shock capturing schemes Ⅱ [J], J. Comput. Phys., 83(1989), 32-78.
[42] 水鸿寿，一维流体力学差分方法[M]，北京，国防工业出版社．
[43] G. Strang, On the construction and comparison of difference schemes [J], SIAM J. Numer. Anal. 5, 506(1968).
[44] V. A. Titarev and E. F. Toro, Finite-volume WENO schemes for threedimensional conservation laws [J], J. Comput. Physics, 201(2004), 238-260.
[45] M. Torrilhon and D. S. Balsara, High order WENO schemes: investigations on non-uniform convergence for MHD Riemann problems [J], J. Comput. Physics, 201(2004), 586-600.
[46] B. Van Leer, Stabilization of difference schemes for the equations of inviscid compressible flow by artificial diffusion [J], J. Comput. Phys., 3(1969), 473-485.
[47] B. Van Leer, Towards the ultimate conservative difference scheme, Ⅱ, Monotonicity and conservation combined in a second order scheme [J], J. Comput. Phys., 14(1974), 361-370.
[48] B. Van Leer, Towards the ultimate conservative difference scheme, Ⅲ, Upstream-centered and conservation finite difference schemes for ideal compressible flow [J], J. Comput. Phys., 23(1977), 263-275.
[49] B. Van Leer, Towards the ultimate conservative difference scheme, Ⅳ, A new approach to numerical convection [J], J. Comput. Phys., 23(1977), 276-299.
[50] B. Van Leer, Towards the ultimate conservative difference scheme, Ⅴ, A second order sequel to Godunov's method [J], J. Comput. Plays., 32(1979), 101-136.
[51] S. Vukovic and L. Sopta, ENO and WENO Schemes with the Exact Conservation Property for One-Dimensional Shallow Water Equations [J], J. Comput. Phys., 179(2002), 593-621.
[52] Z. J. Wang and R. F. Chen, Optimized Weighted Essentially Nonoscillatory Schemes for Linear Waves with Discontinuity [J], J. Comput. Phys., 174(2001), 381-404.
[53] Y. L. Xing and C. W. Shu, High order finite difference WENO schemes with the exact conservation property for the shallow water equations [J], J. Comput. Phys., 208(2005), 206-227.
[54] Z. F. Xu and C. W. Shu, Anti-diffusive flux corrections for high order finite difference WENO schemes [J], J. Comput. Phys., 205(2005). 458-485.
[55] Y．Y．Yang，et al．，High order schemes based on upwind schemes with modifled coefficients[C]，总体与靶物理理论研究专题2004年度工作报告，978-982．J．Comput．Appl．Math．in press．
[56] 杨水平，李寿佛，屈小妹，张瑗，刘玉珍，五阶WENO格式和二阶Godunov格式的数值测试与比较[J]，2005，待发表．
[57] 叶文华，张维岩，陈光南，数值模拟激光烧蚀瑞利泰勒不稳定性[J]，内部通信，1997．
[58] M. P. Zhang, et al., A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model [J], J. Comput. Phys., 191(2003), 639-659.
[59] T. Zhou, Y. Li and C. W. Shu, Numerical comparison of WENO finite volume and Runge-Kutta discontinuous Galerkin methods [J], Journal of Scientific Computing, 16(2001), 145-171.