高阶半离散中心迎风方法的研究与应用
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摘要
本文构造了一种新的四阶半离散中心迎风差分方法求解双曲守恒律及其有关问题。其主要思想是为在一个更小的、与时间步长有关的非光滑区域上对Riemann扇进行积分近似,引入了左、右局部波速。空间导数项的离散采用四阶CWENO的构造方法,使所得新方法在提高精度的同时,具有更高的分辨率。该方法同时具有一般中心差分格式的优点:不用求解Riemann问题、不用进行特征分解等,从而便于推广到复杂方程的求解。使用该方法产生的数值粘性要比交错的中心差分格式小,而且由于数值粘性与时间步长无关,从而在涉及到对流扩散方程的求解时时间步长可根据稳定性需要尽可能的小。本文通过双曲守恒律、对流扩散方程以及浅水波方程对所构造的求解方法进行了大量的数值试验,验证了该方法的高精度、高分辨率特点。
In this paper, a new fourth-order semi-discrete central-upwind scheme for hyperbolic system of conservation laws, convection-diffusion equations and shallow water equations was presented. The integration over the Riemann fans by more accurate information about one-sided local speed of wave propagation was augmented. By using the fourth-order CWENO reconstruction, the new scheme has properties of higher order accuracy and higher resolution for discontinuities. In the meantime our new scheme enjoys the main advantage of the central schemes over the upwind ones: first, no Riemann solvers are required, and second, its generalization and realization for complicated multidimensional system are considerably simpler than in the upwind case. Because the new scheme has less dissipation, which is independent of time-steps, than the staggered central scheme, it can be efficiently used with time-steps as small as the requirement of the numerical stability. A number of numerical experiments for one and two-dimensional equations were presented to illustrate the accuracy and high-resolution properties of the new central scheme. Satisfactory results were obtained.
In this paper, a new fourth-order semi-discrete central-upwind scheme for hyperbolic system of conservation laws, convection-diffusion equations and shallow water equations was presented. The integration over the Riemann fans by more accurate information about one-sided local speed of wave propagation was augmented. By using the fourth-order CWENO reconstruction, the new scheme has properties of higher order accuracy and higher resolution for discontinuities. In the meantime our new scheme enjoys the main advantage of the central schemes over the upwind ones: first, no Riemann solvers are required, and second, its generalization and realization for complicated multidimensional system are considerably simpler than in the upwind case. Because the new scheme has less dissipation, which is independent of time-steps, than the staggered central scheme, it can be efficiently used with time-steps as small as the requirement of the numerical stability. A number of numerical experiments for one and two-dimensional equations were presented to illustrate the accuracy and high-resolution properties of the new central scheme. Satisfactory results were obtained.
引文
1) S. K. Godunov, A finite difference method for the numerical computation of discontinuous solution of the fluid dynamics, Mst. Sb., (47) 1959, pp.271-290.
2) B. van Leer, Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov's method, J. Comput. Phys., 32(1979), pp.101-136.
3) A. Harten, B. Engquist, S. Osher, Uniformly high order accurate essentially non-oscillatory schemes Ⅲ, J. Comput. Phys., 71(1987), pp.231-303.
4) A. Harten, S. Osher, Uniformly high order accurate essentially non-oscillatory schemes Ⅰ, SIAM J. Sci. Comput., 24(1987), pp.279-309.
5) G.-S. Jiang, D. Levy, C.-T. Lin, S.Osher, E.admor, Non-oscillatory central schemes for multi-dimensional hyperbolic conservation laws, SIAM J. Sci. Comput., 19(1998), pp. 1892-1917.
6) X.-D.Liu, E. Tadmor, Third order non-oscillatory central scheme for hyperbolic conservation laws, Numer. Math., 79(1998), pp.397-425.
7) C.-W. Shu, Numerical experiments on the accuracy of ENO and modified ENO schemes, J. Sci. Comput., 5(1990), ppl27-149.
8) G-S. Jiang and C.-W Shu, Efficient implementation of WENO schemes, J.Comput. phys., 147(1996), pp.202-228.
9) X-D.Liu, S. Osher, and T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115(1994), pp.200-212.
10) C.-W. Shu, and S. Osher., Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. Comput. Phys., 83(1989), pp.23-78.
11) P.D. Lax, Weak solutions of nonlinear hyperbolic equations and their mumerical computation, Comm. Pure Appl. Math., 7(1954), pp.159-193.
12) K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math., 7(1954), pp.345-392.
13) H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys., 87(1990), pp.408-463.
14) X.-D. Liu, E. Tadmor, Third order non-oscillatory central scheme for hyperbolic conservation laws. Numerische Mathematik, 79(1998), pp.397-425.
15) G-S. Jiang, C.-W. Shu, Non-oscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput., 19(1998), pp.892-917.
16) D. Levy, G. Puppo, and G. Russo, A third-order central WENO scheme for 2D conservation laws, Appl. Numer. Math., 33(2000), pp.407-414.
17) D. Levy, G. Puppo and G Russo, Central WENO schemes for hyperbolic systems of conservation laws, Math. Model. Numer. Anal., 33(1999), pp.547-571.
18) F. Bianco, G. Puppo, and G. Russo, High order central schemes for hyperbolic systems of conservation laws. SIAM J. Sci. Comput., 21(1999), pp294-322.
19) D. Levy, G. Puppo and G. Russo, Compact central WENO schemes for multidimensional conservation laws, SIAM J. Sci. Comput., 22(2000), pp.656-672.
20) D. Levy, G. Puppo and G. Russo, A fourth-order central WENO schemes for
multidimensional hyperbolic systems for conservation laws, SIAM J. Sci. Comput, 24(2002), pp.480-506.
21) D. Levy, G. Puppo and G. Russo, On the behavior of the total variation in CWENO methods for hyperbolic conservation laws, Appl. Numer. Math., 33(2000), pp.656-672.
22) A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160(2000), pp.241-282.
23) A. Kurganov and E. Tadmor, A third-order semi-discrete central scheme for conservation laws and convection-diffusion equations, SIAM J. Sci. Comput., 22(2000), pp.1461-1488.
24) A. Kurganov, G. Petrova, A third-order semi-discrete genuinely multidimensional central schemes for hyperbolic conservation laws and related problems, Numerische Mathematik, 88(2001), pp.683-729.
25) A. Kurganov, S. Nolle and G. Petrova, Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., 23(2001), pp.707-740.
26) M. Seaid, T. Darmstadt, Non-oscillatory relaxation methods for the shallow water equations in one and two space dimensions, J. Int. Num. Meth. Fluids, 170(2004), pp.217-239.
27) S. Roberts and C. Zoppou, Robust and efficient solution of the 2D shallow water equation with domains containing dry beds, ANZIAM J.42(E), (2000), pp.C1260-C1282.
28) S. Jin, A steady-state capturing method for hyperbolic system with geometrical source terms, Math .Model. Num. Anal., 35(2001), pp.631-646.
29) S. Jin and X. Wen, An efficient method for computing hyperbolic systems with geometrical source terms having concentrations, J.Comp. Math., 22(2004), pp.230-249.
30) S.Jin and X. Wen, Two interface type numerical methods for computing hyperbolic systems with geometrical source terms having concentration, http://www.math.wisc.edu/~jin/research.html.
31) 陈正.《广义特征坐标系在浅水波方程中的应用》[硕士学位论文],北京:清华大学工程力学系,2003.
32) T. Gallouet, J.-M Herard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography, Computers & Fluids, 32(2003), pp. 479-513.
33) S. Osher, J. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton Jacobi formulations, J. Comput. Phys., 79(1988), pp. 12-40.
34) S. Bryson, D. Levy, High- order central WENO schemes for Multi-Dimensional Hamilton Jacobi equations, SIAM J. Num .Anal., 41(2003), pp.1339-1369.
35) S. Bryson, D. Levy, High-order semi-discrete schemes for multi-dimensional Hamilton Jacobi equations, J. Comput. Phys., 189(2003), pp. 63-87.
36) R. Kupferman, Simulation of viscoelastic fluids: Couette-Tayler flow, J. Comput.
Phys.,147(1998),pp.22-59.
37) R. Kupferman, A numerical study of the axisymmetric Couette-Tayler problem using a fast high-resolution second-order central scheme, SLAM J. Sci. Comput. , 20(1998), pp.858-877.
38) R. Kupferman and E. Tamdor, A fast high-resolution second-order central scheme for incompressible flows, Proc. Natl, Acad.Sci. USA,94(1997), pp .4848-4852.
39) E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. Spinger-Verlag,1997.
40) M. Zennaro, Natural Continuous Extensions of Runge-Kutta Methods, Math. Comp., 46(1986), pp.119-133.
41) S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high order time discretization methods, SIAM Review, 43(2001), pp.89-112.
42) L. Pareschi, G. Puppo and G. Russo, C.-W. Shu, Central Runge-Kutta Schemes for Conservation Laws, http://www.hyke.org/preprint/2003/02/028.pdf.
43) J.J. Stokes, Water waves: the mathematical theory with applications, John Wiley & Sons, 2001.
44) J. Lighthill, Waves in fluids, Cambridge University Press, 1978.
45) F.M. Henderson, Open channel flow, MacMillan Publishing Co., 1966.
46) M.B. Abbott, Computational hydraulics, Ashgate Publishing Company, 1992.
47) T. Weiyan, Shallow water hydraulics, Elsevier, Amsterdam, Elsevier Oceanography Series, 55, 1992.
48) E.F. Toro, Shock-capturing methods for free-surface shallow flows, John Wiley & Sons, 2001.
2) B. van Leer, Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov's method, J. Comput. Phys., 32(1979), pp.101-136.
3) A. Harten, B. Engquist, S. Osher, Uniformly high order accurate essentially non-oscillatory schemes Ⅲ, J. Comput. Phys., 71(1987), pp.231-303.
4) A. Harten, S. Osher, Uniformly high order accurate essentially non-oscillatory schemes Ⅰ, SIAM J. Sci. Comput., 24(1987), pp.279-309.
5) G.-S. Jiang, D. Levy, C.-T. Lin, S.Osher, E.admor, Non-oscillatory central schemes for multi-dimensional hyperbolic conservation laws, SIAM J. Sci. Comput., 19(1998), pp. 1892-1917.
6) X.-D.Liu, E. Tadmor, Third order non-oscillatory central scheme for hyperbolic conservation laws, Numer. Math., 79(1998), pp.397-425.
7) C.-W. Shu, Numerical experiments on the accuracy of ENO and modified ENO schemes, J. Sci. Comput., 5(1990), ppl27-149.
8) G-S. Jiang and C.-W Shu, Efficient implementation of WENO schemes, J.Comput. phys., 147(1996), pp.202-228.
9) X-D.Liu, S. Osher, and T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115(1994), pp.200-212.
10) C.-W. Shu, and S. Osher., Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. Comput. Phys., 83(1989), pp.23-78.
11) P.D. Lax, Weak solutions of nonlinear hyperbolic equations and their mumerical computation, Comm. Pure Appl. Math., 7(1954), pp.159-193.
12) K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math., 7(1954), pp.345-392.
13) H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys., 87(1990), pp.408-463.
14) X.-D. Liu, E. Tadmor, Third order non-oscillatory central scheme for hyperbolic conservation laws. Numerische Mathematik, 79(1998), pp.397-425.
15) G-S. Jiang, C.-W. Shu, Non-oscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput., 19(1998), pp.892-917.
16) D. Levy, G. Puppo, and G. Russo, A third-order central WENO scheme for 2D conservation laws, Appl. Numer. Math., 33(2000), pp.407-414.
17) D. Levy, G. Puppo and G Russo, Central WENO schemes for hyperbolic systems of conservation laws, Math. Model. Numer. Anal., 33(1999), pp.547-571.
18) F. Bianco, G. Puppo, and G. Russo, High order central schemes for hyperbolic systems of conservation laws. SIAM J. Sci. Comput., 21(1999), pp294-322.
19) D. Levy, G. Puppo and G. Russo, Compact central WENO schemes for multidimensional conservation laws, SIAM J. Sci. Comput., 22(2000), pp.656-672.
20) D. Levy, G. Puppo and G. Russo, A fourth-order central WENO schemes for
multidimensional hyperbolic systems for conservation laws, SIAM J. Sci. Comput, 24(2002), pp.480-506.
21) D. Levy, G. Puppo and G. Russo, On the behavior of the total variation in CWENO methods for hyperbolic conservation laws, Appl. Numer. Math., 33(2000), pp.656-672.
22) A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160(2000), pp.241-282.
23) A. Kurganov and E. Tadmor, A third-order semi-discrete central scheme for conservation laws and convection-diffusion equations, SIAM J. Sci. Comput., 22(2000), pp.1461-1488.
24) A. Kurganov, G. Petrova, A third-order semi-discrete genuinely multidimensional central schemes for hyperbolic conservation laws and related problems, Numerische Mathematik, 88(2001), pp.683-729.
25) A. Kurganov, S. Nolle and G. Petrova, Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., 23(2001), pp.707-740.
26) M. Seaid, T. Darmstadt, Non-oscillatory relaxation methods for the shallow water equations in one and two space dimensions, J. Int. Num. Meth. Fluids, 170(2004), pp.217-239.
27) S. Roberts and C. Zoppou, Robust and efficient solution of the 2D shallow water equation with domains containing dry beds, ANZIAM J.42(E), (2000), pp.C1260-C1282.
28) S. Jin, A steady-state capturing method for hyperbolic system with geometrical source terms, Math .Model. Num. Anal., 35(2001), pp.631-646.
29) S. Jin and X. Wen, An efficient method for computing hyperbolic systems with geometrical source terms having concentrations, J.Comp. Math., 22(2004), pp.230-249.
30) S.Jin and X. Wen, Two interface type numerical methods for computing hyperbolic systems with geometrical source terms having concentration, http://www.math.wisc.edu/~jin/research.html.
31) 陈正.《广义特征坐标系在浅水波方程中的应用》[硕士学位论文],北京:清华大学工程力学系,2003.
32) T. Gallouet, J.-M Herard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography, Computers & Fluids, 32(2003), pp. 479-513.
33) S. Osher, J. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton Jacobi formulations, J. Comput. Phys., 79(1988), pp. 12-40.
34) S. Bryson, D. Levy, High- order central WENO schemes for Multi-Dimensional Hamilton Jacobi equations, SIAM J. Num .Anal., 41(2003), pp.1339-1369.
35) S. Bryson, D. Levy, High-order semi-discrete schemes for multi-dimensional Hamilton Jacobi equations, J. Comput. Phys., 189(2003), pp. 63-87.
36) R. Kupferman, Simulation of viscoelastic fluids: Couette-Tayler flow, J. Comput.
Phys.,147(1998),pp.22-59.
37) R. Kupferman, A numerical study of the axisymmetric Couette-Tayler problem using a fast high-resolution second-order central scheme, SLAM J. Sci. Comput. , 20(1998), pp.858-877.
38) R. Kupferman and E. Tamdor, A fast high-resolution second-order central scheme for incompressible flows, Proc. Natl, Acad.Sci. USA,94(1997), pp .4848-4852.
39) E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. Spinger-Verlag,1997.
40) M. Zennaro, Natural Continuous Extensions of Runge-Kutta Methods, Math. Comp., 46(1986), pp.119-133.
41) S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high order time discretization methods, SIAM Review, 43(2001), pp.89-112.
42) L. Pareschi, G. Puppo and G. Russo, C.-W. Shu, Central Runge-Kutta Schemes for Conservation Laws, http://www.hyke.org/preprint/2003/02/028.pdf.
43) J.J. Stokes, Water waves: the mathematical theory with applications, John Wiley & Sons, 2001.
44) J. Lighthill, Waves in fluids, Cambridge University Press, 1978.
45) F.M. Henderson, Open channel flow, MacMillan Publishing Co., 1966.
46) M.B. Abbott, Computational hydraulics, Ashgate Publishing Company, 1992.
47) T. Weiyan, Shallow water hydraulics, Elsevier, Amsterdam, Elsevier Oceanography Series, 55, 1992.
48) E.F. Toro, Shock-capturing methods for free-surface shallow flows, John Wiley & Sons, 2001.