Computing effective diffusivities in 3D time-dependent chaotic flows with a convergent Lagrangian numerical method

Wang, Zhongjian; Xin, Jack; Zhang, Zhiwen

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 5, pp. 1521-1544

In this paper, we study the convergence analysis for a robust stochastic structure-preserving Lagrangian numerical scheme in computing effective diffusivity of time-dependent chaotic flows, which are modeled by stochastic differential equations (SDEs). Our numerical scheme is based on a splitting method to solve the corresponding SDEs in which the deterministic subproblem is discretized using a structure-preserving scheme while the random subproblem is discretized using the Euler-Maruyama scheme. We obtain a sharp and uniform-in-time convergence analysis for the proposed numerical scheme that allows us to accurately compute long-time solutions of the SDEs. As such, we can compute the effective diffusivity for time-dependent chaotic flows. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method in computing effective diffusivity for the time-dependent Arnold-Beltrami-Childress (ABC) flow and Kolmogorov flow in three-dimensional space.

Classification : 35B27, 37A30, 60H35, 65M12, 65M75
Keywords: Convection-enhanced diffusion, time-dependent chaotic flows, effective diffusivity, structure-preserving scheme, convergence analysis

[1] B. Afkham and J. Hesthaven, Structure preserving model reduction of parametric hamiltonian systems. SIAM J. Sci. Comput. 39 (2017) A2616–A2644. | MR | Zbl

[2] G. Ben Arous and H. Owhadi, Multiscale homogenization with bounded ratios and anomalous slow diffusion. Commun. Pure Appl. Math. 56 (2003) 80–113. | MR | Zbl

[3] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. Vol. 374. American Mathematical Soc. (2011). | Zbl | MR

[4] L. Biferale, A. Crisanti, M. Vergassola and A. Vulpiani, Eddy diffusivities in scalar transport. Phys. Fluids 7 (1995) 2725–2734. | MR | Zbl

[5] N. Brummell, F. Cattaneo and S. Tobias, Linear and nonlinear dynamo properties of time-dependent ABC flows. Fluid Dyn. Res. 28 (2001) 237.

[6] S. Childress and A. D. Gilbert, Stretch, Twist, Fold: the Fast Dynamo. Vol. 37. Springer Science & Business Media (1995). | Zbl

[7] A. Debussche and E. Faou, Weak backward error analysis for SDEs. SIAM J. Numer. Anal. 50 (2012) 1735–1752. | MR | Zbl

[8] T. Dombre, U. Frisch, J. Greene, M. Hénon, A. Mehr and A. Soward, Chaotic streamlines in the ABC flows. J. Fluid Mech. 167 (1986) 353–391. | MR | Zbl

[9] A. Fannjiang and G. Papanicolaou, Convection-enhanced diffusion for periodic flows. SIAM J. Appl. Math. 54 (1994) 333–408. | MR | Zbl

[10] K. Feng and Z. Shang, Volume-preserving algorithms for source-free dynamical systems. Numer. Math. 71 (1995) 451–463. | MR | Zbl

[11] D. Galloway and M. Proctor, Numerical calculations of fast dynamos in smooth velocity fields with realistic diffusion. Nature 356 (1992) 691.

[12] J. Garnier, Homogenization in a periodic and time-dependent potential. SIAM J. Appl. Math. 57 (1997) 95–111. | MR | Zbl

[13] R. Gilmore, Baker-Campbell-Hausdorff formulas. J. Math. Phys. 15 (1974) 2090–2092. | MR | Zbl

[14] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations. Springer Science and Business Media (2006). | MR | Zbl

[15] J. Hong, H. Liu and G. Sun, The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs. Math. Comput. 75 (2006) 167–181. | MR | Zbl

[16] V. V. Jikov, S. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994). | MR

[17] C. Kao, Y.-Y. Liu and J. Xin, A Semi-Lagrangian Computation of Front Speeds of $G$-equation in $ABC$ and Kolmogorov Flows with Estimation via Ballistic Orbits. SIAM J. Multiscale Model. Simul. 20 (2022) 107–117. | MR | Zbl

[18] T. Kato, Perturbation Theory for Linear Operators. Vol. 132. Springer Science & Business Media (2013). | Zbl

[19] N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, In Vol. 12 of Graduate Studies in Mathematics. American Mathematical Soc. (1996). | MR | Zbl

[20] C. Landim, S. Olla and H. T. Yau, Convection–diffusion equation with space–time ergodic random flow. Probab. Theory Relat. Fields. 112 (1998) 203–220. | MR | Zbl

[21] T. Lelièvre and G. Stoltz, Partial differential equations and stochastic methods in molecular dynamics. Acta Numer. 25 (2016) 681–880. | MR | Zbl

[22] J. Lyu, J. Xin and Y. Yu, Computing residual diffusivity by adaptive basis learning via spectral method. Numer. Math.: Theory Methods Appl. 10 (2017) 351–372. | MR | Zbl

[23] A. J. Majda and P. R. Kramer, Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena. Phys. Rep. 314 (1999) 237–574. | MR

[24] T. Mcmillen, J. Xin, Y. Yu and A. Zlatos, Ballistic orbits and front speed enhancement for ABC flows. SIAM J. Appl. Dyn. Syst. 15 (2016) 1753–1782. | MR | Zbl

[25] I. Mezić, J. F. Brady and S. Wiggins, Maximal effective diffusivity for time-periodic incompressible fluid flows. SIAM J. Appl. Math. 56 (1996) 40–56. | MR | Zbl

[26] G. Milstein, Y. Repin and M. Tretyakov, Symplectic integration of Hamiltonian systems with additive noise. SIAM J. Numer. Anal. 39 (2002) 2066–2088. | MR | Zbl

[27] B. Øksendal, Stochastic Differential Equations: an Introduction with Applications. Springer Science and Business Media (2013). | Zbl | MR

[28] G. Pavliotis and A. Stuart, Multiscale Methods: Averaging and Homogenization. Springer Science and Business Media (2008). | MR | Zbl

[29] G. Pavliotis, A. Stuart and K. Zygalakis, Calculating effective diffusivities in the limit of vanishing molecular diffusion. J. Comput. Phys. 228 (2009) 1030–1055. | MR | Zbl

[30] S. Reich, Backward error analysis for numerical integrators. SIAM J. Numer. Anal. 36 (1999) 1549–1570. | MR | Zbl

[31] M. Tao, H. Owhadi and J. Marsden, Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging. Multiscale Model. Simul. 8 (2010) 1269–1324. | MR | Zbl

[32] Z. Wang, J. Xin and Z. Zhang, Computing effective diffusivity of chaotic and stochastic flows using structure-preserving schemes. SIAM J. Numer. Anal. 56 (2018) 2322–2344. | MR | Zbl

[33] Z. Wang, J. Xin and Z. Zhang, Sharp error estimates on a stochastic structure-preserving scheme in computing effective diffusivity of 3D chaotic flows. SIAM Multiscale Model. Simul. 19 (2021) 1167–1189. | MR | Zbl

[34] J. Xin, Y. Yu and A. Zlatoš, Periodic orbits of the $ABC$ flow with $A=B=C=1$. SIAM J. Math. Anal. 48 (2016) 4087–4093. | MR | Zbl