Runge-Kutta算法与Li差分法不同阶数配合对计算精度影响研究
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摘要
为了充分发挥高阶Li空间微分方案(Li,2005)的优点,实现了时间积分为2～6阶Runge-Kutta(简称RK)格式的偏微分方程求解算法(简称RKL算法)。然后通过多组数值试验,研究了时间积分阶数对计算误差的影响。线性平流方程的试验结果表明对于方波函数型初值,2、4、5和6阶RK算法能获得和3阶精度差不多的结果,而对于高斯函数型的初值,高阶RKL算法可以取得较好的计算效果。RK为5(6)阶时,对应的Li微分阶数可达9(10)阶,总误差控制在10-7(10-8)以内。随RK阶数增加Li微分有效阶数有增加的趋势,而总误差在逐渐减小。计算非线性无粘Burgers方程时,RKL算法能否获得好的计算结果,除了受初始场形式的影响,还与计算的目标时刻有关。当目标时刻解的各阶导数连续(且未出现无穷大数值时),高阶(RK为4～6阶)算法是有效的;若出现了导数间断、或导数为无穷大,就会碰到冲击波解类型的问题,此时高阶RK算法也无法获得很高精度的数值解。此非线性的算例中,Li微分阶数仍然随RK阶数增加而增加,但增加的趋势不是线性的,具体变化关系可以通过实验结果拟合而获得。研究发现时间积分方案阶数大于3之后,对应的最优空间差分精度阶数可以比6阶提高很多,这再次证明了以前研究中6阶以上空间差分格式对结果无改进的现象,是由于没有使用足够高精度的时间积分方案引起的。相比于Taylor-Li(Wang, 2017)算法,5～6阶的RK方法编程和实现简单,计算结果的精度比3阶算法要提高很多,因此,它是一种能够对复杂方程适用的简易高阶算法方案,具有一定的实用价值。
We implement the hybrid Runge-Kutta-Li(RKL) scheme for the purpose to take full advantage of Li's high order spatial differential method(Li, 2005). A set of numerical experiments has been conducted to analyze how the computation error is affected by the order of integration scheme. The results of the linear advection equation indicate that with the square-wave type initial values, the scheme can only obtain a third-order accuracy. However, for the Gaussian function type of initial values, the scheme can obtain a better result. The fifth(sixth) order Runge-Kutta(RK) integration scheme corresponds to 9 th(10 th) order Li's difference scheme in spatial direction and the total error can be controlled within10-7(10-8). The order of Li's scheme tends to increase while the RK order increases, and the total error gradually decreases. When we compute the nonlinear Burgers' equation, whether the RKL scheme can obtain good results is not only dependent on the form of the initial field, but also related to the target computation time. When the derivative is continuous(and infinite value does not appear) at the target observation time, 4 th–6 th order RKL scheme is effective. On the contrary, if the derivative is discontinuous, or the derivative tends to infinity, the RKL scheme cannot obtain high-precision numerical solution. In this case(Burgers' with smooth initial), the order of Li's scheme still increases while the RK order increases, but the relation between them shows a nonlinear tendency(which can be specified through some fitting methods). The results indicate that when the order of time integral is more than three, the corresponding optimal spatial difference order can be higher than six. This result confirms the finding of previous studies that the order of spatial difference above six makes no improvement to the results due to the lack of high-precision time integral scheme. Compared with Taylor-Li(Wang, 2017) scheme, the 5 th–6 th order RKL scheme is easier to program and can yield more precise results than the third-order scheme. To conclude, the high order RKL scheme can be applied to some complicated types of partial differential equations and is valuable for many other similar computation cases.
We implement the hybrid Runge-Kutta-Li(RKL) scheme for the purpose to take full advantage of Li's high order spatial differential method(Li, 2005). A set of numerical experiments has been conducted to analyze how the computation error is affected by the order of integration scheme. The results of the linear advection equation indicate that with the square-wave type initial values, the scheme can only obtain a third-order accuracy. However, for the Gaussian function type of initial values, the scheme can obtain a better result. The fifth(sixth) order Runge-Kutta(RK) integration scheme corresponds to 9 th(10 th) order Li's difference scheme in spatial direction and the total error can be controlled within10-7(10-8). The order of Li's scheme tends to increase while the RK order increases, and the total error gradually decreases. When we compute the nonlinear Burgers' equation, whether the RKL scheme can obtain good results is not only dependent on the form of the initial field, but also related to the target computation time. When the derivative is continuous(and infinite value does not appear) at the target observation time, 4 th–6 th order RKL scheme is effective. On the contrary, if the derivative is discontinuous, or the derivative tends to infinity, the RKL scheme cannot obtain high-precision numerical solution. In this case(Burgers' with smooth initial), the order of Li's scheme still increases while the RK order increases, but the relation between them shows a nonlinear tendency(which can be specified through some fitting methods). The results indicate that when the order of time integral is more than three, the corresponding optimal spatial difference order can be higher than six. This result confirms the finding of previous studies that the order of spatial difference above six makes no improvement to the results due to the lack of high-precision time integral scheme. Compared with Taylor-Li(Wang, 2017) scheme, the 5 th–6 th order RKL scheme is easier to program and can yield more precise results than the third-order scheme. To conclude, the high order RKL scheme can be applied to some complicated types of partial differential equations and is valuable for many other similar computation cases.
引文
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陈显尧,宋振亚,赵伟,等.2008.气候模式系统模拟结果的不确定性分析[J].海洋科学进展,26(2):119-125.Chen Xianyao,Song Zhenya,Zhao Wei,et al.2008.Uncertainity analysis of results simulated by climate model system[J].Advances in Marine Science(in Chinese),26(2):119-125,doi:10.3969/j.issn.1671-6647.2008.02.001.
冯涛,李建平.2007.高精度迎风偏斜格式的比较与分析[J].大气科学,31(2):245-253.Feng Tao,Li Jianping.2007.A comparison and analysis of high order upwind-biased schemes[J].Chinese Journal of Atmospheric Sciences(in Chinese),31(2):245-253,doi:10.3878/j.issn.1006-9895.2007.02.06.
Hairer E,N?rsett S P,Wanner G.2000.Solving Ordinary Differential Equations I:Nonstiff Problems[M].Berlin Heidelberg:Springer,528pp.
Hopf E.1950.The partial differential equation ut+uux=μxx[J].Commun.Pure Appl.Math.,3(3):201-230,doi:10.1002/cpa.3160030302.
季仲贞,王斌.1994.一类高时间差分精度的平方守恒格式的构造及其应用检验[J].自然科学进展,4(2):149-157.Ji Zhongzhen,Wang Bin.1994.Construction and application test of a kind of high precision scheme with square-conservation[J].Progress in Natural Science(in Chinese),4(2):149-157.
季仲贞,李京,王斌.1999.紧致平方守恒格式的构造和检验[J].大气科学,23(3):323-332.Ji Zhongzhen,Li Jing,Wang Bin.1999.Construction and test of compact scheme with square-conservation[J].Chinese Journal of Atmospheric Sciences(in Chinese),23(3):323-332,doi:10.3878/j.issn.1006-9895.1999.03.08.
Lele S K.1992.Compact finite difference schemes with spectral-like resolution[J].J.Comput.Phys.,103(1):16-42,doi:10.1016/0021-9991(92)90324-R.
Li J P.2005.General explicit difference formulas for numerical differentiation[J].J.Comput.Appl.Math.,183(1):29-52,doi:10.1016/j.cam.2004.12.026.
Li J P,Zeng Q C,Chou J F.2000.Computational uncertainty principle in nonlinear ordinary differential equations(I)--Numerical results[J].Science in China(Series E),43(5):449-460.
Liao S J.2009.On the reliability of computed chaotic solutions of non-linear differential equations[J].Tellus A,61(4):550-564,doi:10.1111/j.1600-0870.2009.00402.x.
Ma Y W,Fu D X.1996.Super compact finite difference method(SCFDM)with arbitrary high accuracy[J].Comput.Fluid Dyn.J.,5(2):,259-276.
Mastrandrea M D,Field C B,Stocker T F,et al.2010.Guidance Note for Lead Authors of the IPCC Fifth Assessment Report on Consistent Treatment of Uncertainties[C].Jasper Ridge,CA,USA:Intergovernmental Panel on Climate Change.
Takacs L L.1985.A two-step scheme for the advection equation with minimized dissipation and dispersion errors[J].Mon.Wea.Rev.,113(6):1050-1065,doi:10.1175/1520-0493(1985)113<1050:ATSSFT>2.0.CO;2.
Tal-Ezer H.1986.Spectral methods in time for hyperbolic equations[J].SIAM J.Numer.Anal.,23(1):11-26,doi:10.1137/0723002.
Tal-Ezer H.1989.Spectral methods in time for parabolic problems[J].SIAMJ.Numer.Anal.,26(1):1-11,doi:10.1137/0726001.
Teixeira J,Reynolds C A,Judd K.2007.Time step sensitivity of nonlinear atmospheric models:Numerical convergence,truncation error growth,and ensemble design[J].J.Atmos.Sci.,64(1):175-189,doi:10.1175/JAS3824.1.
Von Neumann J,Goldstine H H.1947.Numerical inverting of matrices of high order[J].Bull.Amer.Math.Soc.,53(11):1021-1099,doi:10.1090/S0002-9904-1947-08909-6.
Wang P F.2017.A high-order spatiotemporal precision-matching Taylor-Li scheme for time-dependent problems[J].Adv.Atmos.Sci.,34(12):1461-1471,doi:10.1007/s00376-017-7018-1.
王鹏飞,王在志,黄刚.2007.舍入误差对大气环流模式模拟结果的影响[J].大气科学,31(5):815-825.Wang Pengfei,Wang Zaizhi,Huang Gang.2007.The influence of round-off error on the atmospheric general circulation model[J].Chinese Journal of Atmospheric Sciences(in Chinese),31(5):815-825,doi:10.3878/j.issn.1006-9895.2007.05.06.
Wang P F,Li J P,Li Q.2012.Computational uncertainty and the application of a high-performance multiple precision scheme to obtaining the correct reference solution of Lorenz equations[J].Numerical Algorithms,59(1):147-159,doi:10.1007/s11075-011-9481-6.
Wang P F,Liu Y,Li J P.2014.Clean numerical simulation for some chaotic systems using the parallel multiple-precision Taylor scheme[J].Chinese Sci.Bull.,59(33):4465-4472,doi:10.1007/s11434-014-0412-5.
吴声昌,刘小清.1996.KdV方程的时间谱离散方法[J].应用数学和力学,17(4):357-362.Wu Shengchang,Liu Xiaoqing.1996.Spectral method in time for KdV equations[J].Applied Mathematics and Mechanics(in Chinese),17(4):357-362.
陈显尧,宋振亚,赵伟,等.2008.气候模式系统模拟结果的不确定性分析[J].海洋科学进展,26(2):119-125.Chen Xianyao,Song Zhenya,Zhao Wei,et al.2008.Uncertainity analysis of results simulated by climate model system[J].Advances in Marine Science(in Chinese),26(2):119-125,doi:10.3969/j.issn.1671-6647.2008.02.001.
冯涛,李建平.2007.高精度迎风偏斜格式的比较与分析[J].大气科学,31(2):245-253.Feng Tao,Li Jianping.2007.A comparison and analysis of high order upwind-biased schemes[J].Chinese Journal of Atmospheric Sciences(in Chinese),31(2):245-253,doi:10.3878/j.issn.1006-9895.2007.02.06.
Hairer E,N?rsett S P,Wanner G.2000.Solving Ordinary Differential Equations I:Nonstiff Problems[M].Berlin Heidelberg:Springer,528pp.
Hopf E.1950.The partial differential equation ut+uux=μxx[J].Commun.Pure Appl.Math.,3(3):201-230,doi:10.1002/cpa.3160030302.
季仲贞,王斌.1994.一类高时间差分精度的平方守恒格式的构造及其应用检验[J].自然科学进展,4(2):149-157.Ji Zhongzhen,Wang Bin.1994.Construction and application test of a kind of high precision scheme with square-conservation[J].Progress in Natural Science(in Chinese),4(2):149-157.
季仲贞,李京,王斌.1999.紧致平方守恒格式的构造和检验[J].大气科学,23(3):323-332.Ji Zhongzhen,Li Jing,Wang Bin.1999.Construction and test of compact scheme with square-conservation[J].Chinese Journal of Atmospheric Sciences(in Chinese),23(3):323-332,doi:10.3878/j.issn.1006-9895.1999.03.08.
Lele S K.1992.Compact finite difference schemes with spectral-like resolution[J].J.Comput.Phys.,103(1):16-42,doi:10.1016/0021-9991(92)90324-R.
Li J P.2005.General explicit difference formulas for numerical differentiation[J].J.Comput.Appl.Math.,183(1):29-52,doi:10.1016/j.cam.2004.12.026.
Li J P,Zeng Q C,Chou J F.2000.Computational uncertainty principle in nonlinear ordinary differential equations(I)--Numerical results[J].Science in China(Series E),43(5):449-460.
Liao S J.2009.On the reliability of computed chaotic solutions of non-linear differential equations[J].Tellus A,61(4):550-564,doi:10.1111/j.1600-0870.2009.00402.x.
Ma Y W,Fu D X.1996.Super compact finite difference method(SCFDM)with arbitrary high accuracy[J].Comput.Fluid Dyn.J.,5(2):,259-276.
Mastrandrea M D,Field C B,Stocker T F,et al.2010.Guidance Note for Lead Authors of the IPCC Fifth Assessment Report on Consistent Treatment of Uncertainties[C].Jasper Ridge,CA,USA:Intergovernmental Panel on Climate Change.
Takacs L L.1985.A two-step scheme for the advection equation with minimized dissipation and dispersion errors[J].Mon.Wea.Rev.,113(6):1050-1065,doi:10.1175/1520-0493(1985)113<1050:ATSSFT>2.0.CO;2.
Tal-Ezer H.1986.Spectral methods in time for hyperbolic equations[J].SIAM J.Numer.Anal.,23(1):11-26,doi:10.1137/0723002.
Tal-Ezer H.1989.Spectral methods in time for parabolic problems[J].SIAMJ.Numer.Anal.,26(1):1-11,doi:10.1137/0726001.
Teixeira J,Reynolds C A,Judd K.2007.Time step sensitivity of nonlinear atmospheric models:Numerical convergence,truncation error growth,and ensemble design[J].J.Atmos.Sci.,64(1):175-189,doi:10.1175/JAS3824.1.
Von Neumann J,Goldstine H H.1947.Numerical inverting of matrices of high order[J].Bull.Amer.Math.Soc.,53(11):1021-1099,doi:10.1090/S0002-9904-1947-08909-6.
Wang P F.2017.A high-order spatiotemporal precision-matching Taylor-Li scheme for time-dependent problems[J].Adv.Atmos.Sci.,34(12):1461-1471,doi:10.1007/s00376-017-7018-1.
王鹏飞,王在志,黄刚.2007.舍入误差对大气环流模式模拟结果的影响[J].大气科学,31(5):815-825.Wang Pengfei,Wang Zaizhi,Huang Gang.2007.The influence of round-off error on the atmospheric general circulation model[J].Chinese Journal of Atmospheric Sciences(in Chinese),31(5):815-825,doi:10.3878/j.issn.1006-9895.2007.05.06.
Wang P F,Li J P,Li Q.2012.Computational uncertainty and the application of a high-performance multiple precision scheme to obtaining the correct reference solution of Lorenz equations[J].Numerical Algorithms,59(1):147-159,doi:10.1007/s11075-011-9481-6.
Wang P F,Liu Y,Li J P.2014.Clean numerical simulation for some chaotic systems using the parallel multiple-precision Taylor scheme[J].Chinese Sci.Bull.,59(33):4465-4472,doi:10.1007/s11434-014-0412-5.
吴声昌,刘小清.1996.KdV方程的时间谱离散方法[J].应用数学和力学,17(4):357-362.Wu Shengchang,Liu Xiaoqing.1996.Spectral method in time for KdV equations[J].Applied Mathematics and Mechanics(in Chinese),17(4):357-362.