Isaac Scientific Publishing

Journal of Advances in Applied Mathematics

Implementation of a Wiener Chaos Expansion Method for the Numerical Solution of the Stochastic Generalized Kuramoto-Sivashinsky Equation Driven by Brownian Motion Forcing

Download PDF (964.4 KB) PP. 119 - 142 Pub. Date: October 15, 2019

DOI: 10.22606/jaam.2019.44001


  • Victor Nijimbere*
    Carleton University, Ottawa, Ontario, Canada


Numerical computations based on the Wiener Chaos Expansion (WCE) are carried out to approximate the solutions of the stochastic generalized Kuramoto–Sivashinsky (SgKS) equation driven by Brownian motion forcing. In the assessment of the accuracy of the WCE based approximate numerical solutions, the WCE based solutions are contrasted with semi-analytical solutions, and the absolute and relative errors are evaluated. It is found that the absolute error is O(ςt), where ς is a small constant and t is the time variabe; and the relative error is order 10−2 or less. This demonstrates that numerical methods based on the WCE are powerful tools to solve the SgKS equation or other related stochastic evolution equations.


Wiener Chaos Expansion, semi-analytical solutions, stochastic Kuramoto-Sivashinsky equation, stochastic boundary conditions, error analysis.


[1] M. Alava, M. Dube and M. Rost, “Imbibition in disordered media”, Advances in Physics, Vol. 53, no. 2, 83–175, 2004.

[2] D. Blomker, C. Gugg, M. Raible, “Thin-film growth models: roughness and correlation functions”, European Journal Applied Mathematics, Vol. 13, pp. 385–402, 2002.

[3] E. Bouchbinder, I. Procaccia, S. Santucci and L. Vanel, “Fracture surfaces as multiscaling graphs”, Physical Review Letters, Vol. 96, 2006.

[4] J. Buceta, J. Pastor, M. A. Rubio and F. J. de la Rubia, “The stochastic Kuramoto–Sivashinsky equation: a model for compact electrodeposition growth”, Physical Letters A, Vol. 235, pp. 464–468, 1997.

[5] J. Buceta, J. Pastor, M. A. Rubio and F. J. de la Rubia, “Small scale properties of the stochastic stabilized Kuramoto–Sivashinsky equation”, Physica D, Vol. 113, pp. 166–171, 1998.

[6] R. H. Cameron and W. T. Martin, “The orthogonal development of non-linear functionals in series of Fourier– Hermite functionals”, Annals of Mathathematics, Vol. 48, pp.385–392, 1947.

[7] A. Cuerno, H. A. Makse, S. Tomassone, S. T. Harrington and H. E. Stanley, “Stochastic model for surface erosion via ion sputtering: Dynamical evolution from ripple morphology to rough morphology”, Physical Review Letters, Vol. 75, pp. 4464–4467, 1995.

[8] R. Cuerno and A. L. Barabasi, “Dynamic scaling of ion-sputtemaroon surfaces”, Physical Review Letters”, Vol. 74, no. 23, pp. 4746–4749, 1995.

[9] J. A. Diez and A. G. Gonzalez, “Metallic-thin-film instability with spatially correlated thermal noise”, Physical Review E, Vol. 93, 2016.

[10] P. Gao, C. J. Cai and X. Y. Liu, “Numerical Simulation of Stochastic Kuramoto-Sivashinsky Equation”. Journal of Applied Mathematics and Physics , Vol. 6, pp. 2363–2369, 2018.

[11] G. Grun, K. Mecke and M. Rauscher, “Thin-film flow influenced by thermal noise”, Journal of Statistical Physics, Vol. 122, no. 6, pp. 1261–1294, 2006.

[12] T. Y. Hou, W. Luo, B. Rozovskii and H. M. Zou, “Wiener chaos expansion and numerical solutions of randomly forced equations of fluid mechanics”, Journal of Computational Physics, Vol. 216, pp. 687–706, 2006.

[13] G. Hu, Y. Lou and P. D. Christofides, “Dynamic output feedback covariance control of stochastic dissipative partial differential equations”, Chemical Engineering Science, Vol. 63, pp. 4531–4542, 2008.

[14] G. Hu, G. Orkoulas and P. D. Christofides, “Stochastic modeling and simultaneous regulation of surface roughness and porosity in thin film deposition”, Industrial and Engineering Chemistry research, Vol. 48, pp. 6690– 6700, 2009.

[15] S. Kalliadasis, C. Ruyer-Quil, B. Scheid and M. G. Velarde, “Falling Liquid Films”, in Applied Mathematical Sciences, Vol. 176, Springer, 2012.

[16] R. Mikulevicius and B. L. Rozovskii, “Stochastic Navier-Stokes equations for turbulence”, SIAM Journal of Mathematical Analysis, Vol. 35, pp. 1250–1310, 2004 .

[17] T. Mikosch, “Elementary stochastic calculus with finance in view”, Vol. 6, World Scientific 2004.

[18] S. Nesic, R. Cuerno, E. Moro, L. Kondic, “Fully nonlinear dynamics of stochastic thin-film dewetting”, Physical Review E, Vol. 92, 2015.

[19] V. Nijimbere, “Ionospheric gravity wave interactions and their representation in terms of stochastic partial differential equations”, Ph.D. thesis, Carleton University, 2014.

[20] V. Nijimbere and L. J. Campbell, “A nonlinear time-dependent radiation condition for simulations of internal gravity waves in geophysical fluids”, Applied Numerical Mathematics, Vol. 110, pp. 75–92, 2016.

[21] M. Pradas and A. Hernandez-Machado, “Intrinsic versus superrough anomalous scaling in spontaneous imbibition”, Physical Review E, Vol. 74, 2006.

[22] J. Soriano, A. Mercier, R. Planet, A. Hernandez-Machado, M. A. Rodriguez and J. Ortin, “Anomalous roughening of viscous fluid fronts in spontaneous imbibition”, Physical Review Letters, Vol. 95, 2005.