Discontinuity, Nonlinearity, and Complexity
Parallel Computation of Reliable Chaotic Solutions of Saltzman’s Equations by Means of the Clean Numerical Simulation
Discontinuity, Nonlinearity, and Complexity 2(4) (2013) 345355  DOI:10.5890/DNC.2013.11.004
Peng Yang$^{2}$, Zhiliang Lin$^{2}$, Shijun Liao$^{1}$,$^{2}$,$^{3}$,$^{4}$
$^{1}$ State Key Laboratory of Ocean Engineering
$^{2}$ School of Naval Architecture, Ocean and Civil Engineering
$^{3}$ Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
$^{4}$ Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University (KAU), Jeddah, Saudi Arabia
Download Full Text PDF
Abstract
The method of the socalled “Clean Numerical Simulation” (CNS) is applied to gain reliable chaotic solutions of Saltzman’s dynamic system, a simplified model for convection flows of fluid. Based on the highorder Taylor series method with data in multiple precision library and a validation check of global reliability of result, the CNS provides us a practical way to gain reliable, accurate enough solutions of chaotic dynamic systems in a finite but long enough time interval. The parallel computation is used to greatly increase the computational efficiency. The numerical noises of the CNS can be controlled to be so small that even the influence of the microlevel inherent uncertainty of initial conditions can be investigated in details. It is found that the microlevel inherent physical uncertainty (i.e. the unavoidable statistical fluctuation of temperature and velocity of fluid) of initial conditions of chaotic Saltzman’s system transfers into macroscopic randomness. This suggests that chaos might be a bridge between microlevel inherent physical uncertainty and macroscopic randomness. The current work illustrates that the above conclusion holds not only for Lorenz equation with three ODEs but also for Saltzman’s equation with upto nine ODEs, and thus has general meanings.
Acknowledgments
Thanks to Dr. P.F. Wang for his kindly assistance on multiple precision library and parallel computation. This work is partly supported by the State Key Lab of Ocean Engineering (Approval No. GKZD0100566) and the National Natural Science Foundation of China (Approval No. 11272209).
References

[1]  Lorenz, E.N. (1963), Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20, 130141. 

[2]  Lorenz, E.N. (1989), Computational chaos: a prelude to computational instability, Physica D, 35, 299317. 

[3]  Lorenz, E.N. (2006), Computational periodicity as observed in a simple system, TellusA, 58, 549557. 

[4]  Yao, L.S. and Hughes, D. (2008), Comment on 'Computational periodicity as observed in a simple system' By Edward N. Lorenz (2006), TellusA, 60, 803805. 

[5]  Barrio, R. (2005), Performance of the Taylor series method for ODEs/DAEs, Applied Mathematics and Computation , 163, 525545. 

[6]  Barrio, R., Rodriguez, M., Abad, A., and Blesa, F. (2011), Breaking the limits: the Taylor series method, Applied Mathematics and Computation, 217, 79407954. 

[7]  Corliss, G. and Chang, Y.F. (1982), Solving ordinary differential equations using Taylor series, Association for Computing Machinery Transactions on Mathematical Software , 8, 114144. 

[8]  Deprit, A. and Zahar, R. (1966), Numerical integration of an orbit and its concomitant variations, Z. Angew. Math. Phys., 17, 425430. 

[9]  Nedialkov, N.S. and Pryce, J.D. (2005), Solving differentialalgebraic equations by Taylor series (I): computing Taylot coefficients, BIT, 45, 561591. 

[10]  Liao, S.J. (2009), On the reliability of computed chaotic solution of nonlinear differential equations, TellusA, 61, 550564. 

[11]  Liao, S.J. (2012), Chaos: A bridge from microscopic uncertainty to macroscopic randomness, Communications in Nonlinear Science and Numerical Simulation, 17, 25642569. 

[12]  Liao, S.J. (2013), On the numerical simulation of propagation of microlevel uncertainty for chaotic dynamic systems, Chaos, Solitons and Fractals, 47, 112. 

[13]  Wang, P.F., Li, J.P., and Li, Q. (2012), Computational uncertainty and the application of a highperformance multiple precision scheme to obtaining the correct reference solution of lorenz equations, Numerical Algorithms, 59, 147159. 

[14]  Sarra, S.A. and Meador, C. (2011), On the numerical simulation ofchaotic dynamic systems using extend precision floating point arithmetic and very high order numerical methods, Nonlinear Analysis  Modelling and Control, 16, 340352. 

[15]  Saltzman, B. (1962), Finite amplitude free convection as an initial value problem(1), J. Atmos. Sci, 19, 329341. 

[16]  Nikolaevskaya, E., Khimich, A., and Chistyakova, T. (2012), Programming with Multiple Precision, Springer. 