To gain a reproducible/reliable numerical simulation of chaotic system, Liao [1] proposed a brand-new numerical strategy, namely the "Clean Numerical Simulation" (CNS) [2-4], to control the background numerical noise, say, truncation and round-off errors, during a temporal interval [0, Tc], where Tc is the so-called "critical predictable time" and this temporal interval should be long enough for calculating statistics. In the frame of the CNS [1-10], the temporal and spatial truncation errors can be decreased to a required small level via using the Taylor expansion method with a high enough order in the temporal dimension and adopting a fine enough discretization method in the spatial dimension (such as the high-order spatial Fourier expansion), respectively. Significantly, all of the physical and numerical variables/parameters should be represented by means of the multiple precision with a large enough number of significant digits, and thus the round-off error can also be decreased to a required small level. In this way, different from some other general numerical algorithms, the CNS is able to give the reproducible/reliable numerical simulation of a chaotic dynamical system within a finite but long enough temporal interval.
Up to now, the CNS has been applied to many chaotic dynamical systems successfully with the corresponding computer-generated simulations being reproducible and of course reliable.
In 2014, Liao & Wang [4] obtained a reproducible/convergent numerical simulation of the Lorenz system in quite a long temporal interval [0, 10000] (Lorenz unit time) by means of a parallel algorithm of the CNS using the 3500th-order Taylor expansion with the constant time step ∆t = 0.01 and 4180-digit multiple precision for all physical and numerical variables/parameters. It took 220.9 hours (i.e. about 9 days and 5 hours) using 1200 CPUs of the National Supercomputer TH-1A at Tianjin, China.
For the code, please refer to the website:
"Lorenz.c" is the main program written in C language using the MPFR library and MPI parallel technique, which needs two header files "mpi_gmp.h" and "mpi_mpfr.h": the number of significant digits and the order of Taylor expansion can be set at Lines 15 & 16, and the settings of other numerical parameters are annotated. One can compile the main program via the command mpicc such as that written in "makefile", and run the compiled executable file via the command mpirun such as that written in "run.sh".
In 2023, according to the exponentially growing property of noise in numerical simulations of chaos, Qin and Liao [11] proposed a modified strategy of the CNS, called the "self-adaptive CNS", to significantly increase the computational efficiency of the CNS algorithm. Using this self-adaptive CNS strategy, a reproducible/convergent numerical simulation of the above-mentioned Lorenz system in the temporal interval [0, 10000] with the critical predictable time Tc=10000 is obtained successfully, by means of the parallel CNS algorithm using 1200 CPUs of the National Supercomputer TH-2 at Guangzhou, China. Note that it took only 37.2 hours (i.e. about 1 day and 13 hours), just about 17% of the CPU time of the previous CNS algorithm applied by Liao & Wang [4] who likewise used a supercomputer.
For the code, please refer to the website:
"Lorenz.c" is the main program written in C language using the MPFR library and MPI parallel technique, which needs two header files "mpi_gmp.h" and "mpi_mpfr.h": critical predictable time, safety factor, temporal interval of changing precision, and so on can be set at Lines 15-19. One can compile the main program via the command mpicc such as that written in "makefile", and run the compiled executable file via the command mpirun such as that written in "run.sh".
[1] S. Liao, On the reliability of computed chaotic solutions of non-linear differential equations, Tellus Ser. A-Dyn. Meteorol. Oceanol. 61 (4) (2009) 550–564.
[2] S. Liao, On the numerical simulation of propagation of micro-level inherent uncertainty for chaotic dynamic systems, Chaos Solitons Fractals 47 (2013) 1–12.
[3] S. Liao, Physical limit of prediction for chaotic motion of three-body problem, Commun. Nonlinear Sci. Numer. Simul. 19 (3) (2014) 601–616.
[4] S. Liao, P. Wang, On the mathematically reliable long-term simulation of chaotic solutions of lorenz equation in the interval [0, 10000], Sci. China-Phys. Mech. Astron. 57 (2) (2014) 330–335.
[5] T. Hu, S. Liao, On the risks of using double precision in numerical simulations of spatio-temporal chaos, J. Comput. Phys. 418 (2020) 109629.
[6] S. Qin, S. Liao, Influence of numerical noises on computer-generated simulation of spatio-temporal chaos, Chaos Solitons Fractals 136 (2020) 109790.
[7] T. Xu, J. Li, Z. Li, S. Liao, Accurate predictions of chaotic motion of a free fall disk, Phys. Fluids 33 (3) (2021) 037111.
[8] S. Liao, S. Qin, Ultra-chaos: an insurmountable objective obstacle of reproducibility and replication, Adv. Appl. Math. Mech. 14 (4) (2022) 799–815.
[9] S. Qin, S. Liao, Large-scale influence of numerical noises as artificial stochastic disturbances on a sustained turbulence, J. Fluid Mech. 948 (2022) A7.
[10] Y. Yang, S. Qin, S. Liao, Ultra-chaos of a mobile robot: a higher disorder than normal-chaos, Chaos Solitons Fractals 167 (2023) 113037.
[11] S. Qin, S. Liao, A self-adaptive algorithm of the clean numerical simulation (CNS) for chaos, Adv. Appl. Math. Mech. (Accepted).