Advances in Atmospheric Sciences

, Volume 34, Issue 12, pp 1461–1471 | Cite as

A high-order spatiotemporal precision-matching Taylor–Li scheme for time-dependent problems

  • Pengfei Wang
Original Paper


Based on the Taylor series method and Li’s spatial differential method, a high-order hybrid Taylor–Li scheme is proposed. The results of a linear advection equation indicate that, using the initial values of the square-wave type, a result with third-order accuracy occurs. However, using initial values associated with the Gaussian function type, a result with very high precision appears. The study demonstrates that, when the order of the time integral is more than three, the corresponding optimal spatial difference order could be higher than six. The results indicate that the reason for why there is no improvement related to an order of spatial difference above six is the use of a time integral scheme that is not high enough. The author also proposes a recursive differential method to improve the Taylor–Li scheme’s computation speed. A more rapid and high-precision program than direct computation of the high-order space differential item is employed, and the computation speed is dramatically boosted. Based on a multiple-precision library, the ultrahigh-order Taylor–Li scheme can be used to solve the advection equation and Burgers’ equation.

Key words

Taylor–Li scheme high-order scheme Burgers’ equation 


结合Taylor级数法和Li空间微分方案的优点, 实现了高阶时间积分的Taylor-Li算法格式, 并且进行了多组数值试验.线性平流方程的试验结果表明对于高斯函数型的初值, 高阶精度算法可以取得非常好的计算效果.计算非线性无粘Burgers方程时, 高阶精度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时, 高阶算法效果非常好; 研究发现时间积分方案的阶数大于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, 这可以解释以前某些研究中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.提出了使用递归微分的方式来提高Taylor-Li算法的计算速度的技术, 所实现的快速高精度差分格式的比直接计算高阶空间微分项的方案速度显著提升, 配合多精度计算的工具库, 实现了平流方程超高阶的时间积分-空间差分格式.


Taylor-Li格式 高阶算法 Burgers方程 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author gratefully acknowledges Prof. Jianping LI for his suggestions and discussions. The work is supported by the National Natural Sciences Foundation of China (Grant Nos. 41375112 and 41530426) and the Chinese Academy of Sciences Key Technology Talent Program.


  1. Barrio, R., 2005: Performance of the Taylor series method for ODEs/DAEs. Applied Mathematics and Computation, 163, 525–545, doi: 10.1016/j.amc.2004.02.015.CrossRefGoogle Scholar
  2. Barrio, R., M. Rodríguez, A. Abad, and F. Blesa, 2011: Breaking the limits: The Taylor series method. Applied Mathematics and Computation, 217, 7940–7954, doi: 10.1016/j.amc.2011.02.080.CrossRefGoogle Scholar
  3. Feng, T., and J. P. Li, 2007: A comparison and analysis of high order upwind-biased schemes. Chinese Journal of Atmospheric Sciences, 31, 245–253, doi: 10.3878/j.issn.1006-9895.2007.02.06. (in Chinese with English abstract)Google Scholar
  4. H´enon, M., and C. Heiles, 1964: The applicability of the third integral of motion: Some numerical experiments. Astronomical Journal, 69, 73–79, doi: 10.1086/109234.CrossRefGoogle Scholar
  5. Hopf, E., 1950: The partial differential equation ut + uux = μxx. Commun. Pure Appl. Math., 3, 201–230, doi: 10.1002/cpa.3160030302.CrossRefGoogle Scholar
  6. Ji, Z. P., J. Li, and B. Wang, 1999: Construction and test of compact scheme with square-conservation. Chinese Journal of Atmospheric Sciences, 23, 323–332, doi: 10.3878/j.issn.1006-9895.1999.03.08. (in Chinese with English abstract)Google Scholar
  7. Lele, S. K., 1992: Compact finite difference schemes with spectrallike resolution. J. Computat. Phys., 103, 16–42, doi: 10.1016/0021-9991(92)90324-R.CrossRefGoogle Scholar
  8. Li, J. P., 2005: General explicit difference formulas for numerical differentiation. Journal of Computational and Applied Mathematics, 183, 29–52, doi: 10.1016/ Scholar
  9. Li, J. P., Q. C. Zeng, and J. F. Chou, 2000: Computational uncertainty principle in nonlinear ordinary differential equations (I)-Numerical results. Science in China Series E, 43, 449–460.CrossRefGoogle Scholar
  10. Liao, S. J., 2009: On the reliability of computed chaotic solutions of non-linear differential equations. Tellus A, 61, 550–564, Scholar
  11. Lorenz, E. N., 2006: Computational periodicity as observed in a simple system. Tellus A, 58, 549–557, Scholar
  12. Ma, Y., and D. Fu, 1996: Super compact finite difference method (SCFDM) with arbitrary high accuracy. Computational Fluid Dynamics, 5, 259–276.Google Scholar
  13. Mastrandrea, M. D., and Coauthors, 2010: Guidance note for lead authors of the IPCC fifth assessment report on consistent treatment of uncertainties. Intergovermental Panel on Climate Change (IPCC). [Available online from]Google Scholar
  14. Moore, R. E., 1966: Interval Analysis. Prentice-Hall, Englewood Cliffs, NY, USA.Google Scholar
  15. Moore, R. E., 1979: Methods and Applications of Interval Analysis. SIAM, Philadelphia, USA, 190 pp.CrossRefGoogle Scholar
  16. Neamtan, S. M., 1946: The motion of harmonic waves in the atmosphere. J. Meteor., 3, 53–56, doi: 10.1175/1520-0469(1946)003<0053:TMOHWI>2.0.CO;2.CrossRefGoogle Scholar
  17. Song, Z., F. Qiao, X. Lei, and C. Wang, 2012: Influence of parallel computational uncertainty on simulations of the Coupled General Climate Model. Geoscientific Model Development, 5, 313–319, doi: 10.5194/gmd-5-313-2012.CrossRefGoogle Scholar
  18. Sun, G. Q., 2016: Mathematical modeling of population dynamics with Allee effect. Nonlinear Dynamics, 85, 1–12, doi: 10.1007/s11071-016-2671-y.CrossRefGoogle Scholar
  19. Sun, G. Q., Z. Y. Wu, Z. Wang, and Z. Jin, 2016: Influence of isolation degree of spatial patterns on persistence of populations. Nonlinear Dynamics, 83, 811–819, doi: 10.1007/s11071-015-2369-6.CrossRefGoogle Scholar
  20. Takacs, L. L., 1985: A two-step scheme for the advection equation with minimized dissipation and dispersion errors. Mon. Wea. Rev., 113, 1050–1065, doi: 10.1175/1520-0493(1985)113<1050:ATSSFT>2.0.CO;2.CrossRefGoogle Scholar
  21. Tal-Ezer, H., 1986: Spectral methods in time for hyperbolic equations. SIAM Journal on Numerical Analysis, 23, 11–26, doi: 10.1137/0723002.CrossRefGoogle Scholar
  22. Tal-Ezer, H., 1989: Spectral methods in time for parabolic problems. SIAM Journal on Numerical Analysis, 26, 1–11, doi: 10.1137/0726001.CrossRefGoogle Scholar
  23. Teixeira, J., C. A. Reynolds, and K. Judd, 2007: Time step sensitivity of nonlinear atmospheric models: Numerical convergence, truncation error growth, and ensemble design. J. Atmos. Sci., 64, 175–189, doi: 10.1175/JAS3824.1.CrossRefGoogle Scholar
  24. von Neumann, J., and H. H. Goldstine, 1947: Numerical inverting of matrices of high order. Bulletin of the American Mathematical Society, 53, 1021–1099, doi: 10.1090/S0002-9904-1947-08909-6.CrossRefGoogle Scholar
  25. Wang, P. F., 2016: Forward period analysis method of the periodic hamiltonian system. PLoS One, 11, e0163303, doi: 10.1371/journal.pone.0163303.CrossRefGoogle Scholar
  26. Wang, P. F., Z. Z. Wang, and G. Huang, 2007: The influence of round-off error on the atmospheric general circulation model. Chinese Journal of Atmospheric Sciences, 31, 815–825, doi: 10.3878/j.issn.1006-9895.2007.05.06. (in Chinese with English abstract)Google Scholar
  27. Wang, P. F., J. P. Li, and Q. Li, 2012: Computational uncertainty and the application of a high-performance multiple precision scheme to obtaining the correct reference solution of Lorenz equations. Numerical Algorithms, 59, 147–159, doi: 10.1007/s11075-011-9481-6.CrossRefGoogle Scholar
  28. Wang, P. F., Y. Liu, and J. P. Li, 2014: Clean numerical simulation for some chaotic systems using the parallel multiple-precision Taylor scheme. Chinese Science Bulletin, 59, 4465–4472, doi: 10.1007/s11434-014-0412-5.CrossRefGoogle Scholar

Copyright information

© Chinese National Committee for International Association of Meteorology and Atmospheric Sciences, Institute of Atmospheric Physics, Science Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Center for Monsoon System Research, Institute of Atmospheric PhysicsChinese Academy of SciencesBeijingChina
  2. 2.State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric PhysicsChinese Academy of SciencesBeijingChina

Personalised recommendations