In this paper we present a numerical technique for solving Kuramoto-Sivashinsky equation, based on spectral Fourier methods. This equation describes reaction diffusion problems, and the dynamics of viscous-fuid films flowing along walls. After we wrote the equation in Fourier space, we get a system. In this case, the exponential time differencing methods integrate the system very much more accurately than other methods since the exponential time differencing methods assume in their derivation that the solution varies slowly in time. When evaluating the coefficients of the exponential time differencing and the exponential time differencing Runge Kutta methods via the”Cauchy integral”. All computational work is done with Matlab package.
| [1] | G. Beylkin, J. M. Keiser, and L. Vozovoi. A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs. J. Comput. Phys., 147:362-387, 1998.View Article |
| [2] | J. Certaine. The Solution of Ordinary Dierential Equations with Large Time Constants. In Mathematical Methods for Digital Computers, A. Ralston and H. S. Wilf, eds.:128-132, Wiley, New York, 1960. |
| [3] | A. Friedli. Generalized Runge-Kutta Methods for the Solution of Stiff Dierential Equations. In Numerical Treatment of Dierential Equations, R. Burlirsch, R. Grigorie, and J. Schröder, eds., 631, Lecture Notes in Mathematics:35-50, Springer, Berlin, 1978. |
| [4] | S. P. Norsett. An A-Stable Modication of the Adams-Bashforth Methods. In Conf. on Numerical Solution of Dierential Equations, Lecture Notes in Math. 109/1969: 214-219, Springer-Verlag, Berlin, 1969.View Article |
| [5] | C. Klein. Fourth Order Time-Stepping for Low Dispersion Korteweg-de Vries and Nonlinear Schrödinger Equations. Electronic Transactions on Numerica Analysis, 29: 116-135, 2008. |
| [6] | A. K. Kassam and L. N. Trefethen. Fourth-Order Time Stepping for Stiff PDEs. SIAM J. Sci. Comput., 26: 1214-1233, 2005.View Article |
| [7] | D. Armbruster, J. Guckenheimer, P. Holmes, "Kuramoto–Sivashinsky dynamics on the center-unstable manifold" SIAM J. Appl. Math. 49 (1989) pp. 676-691View Article |
| [8] | D. Michelson, "Steady solutions of the Kuramoto-Sivashinsky equation" Physica D, 19 (1986) pp. 89-111.View Article |
| [9] | B. Nicolaenko, B. Scheurer, R. Temam, "Some global dynamical properties of the Kuramoto-Sivashinsky equations: Nonlinear stability and attractors" Physica D, 16 (1985) pp. 155-183.View Article |
| [10] | R. L. Burden and J. D. Faires. Numerical Analysis. Wadsworth Group, seventh edition, 2001. |
| [11] | M. Hochbruck and A. Ostermann.Explicit Exponential Runge-Kutta Methods for Semi-linear Parabolic Problems. SIAM J. Numer. Anal., 43: 1069-1090, 2005.View Article |
| [12] | S. M. Cox and P. C. Matthews. Exponential Time Differencing for Stiff Systems. J. Comput. Phys., 176: 430-455, 2002.View Article |
| [13] | A. K. Kassam. High Order Time stepping for Stiff Semi-Linear Partial Differential Equations. PhD thesis, Oxford University, 2004. |
