Research
Fourier Spectral Methods for Numerical Solving of the Kuramoto-Sivashinsky Equation
In this paper I presented 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 Furie space, we get a system. In this case, the exponential time differencing methods integrate the system 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 theâ€Cauchy integralâ€. All computational work is done with Matlab package.
An Exponential Time Differencing Method for the Kuramoto-Sivashinsky Equation
The spectral methods offer very high spatial resolution for a wide range of nonlinear wave equations, so, for the best computational efficiency, it should be desirable to use also high order methods in time but without very strict restrictions on the step size by reason of numerical stability. In this paper we study the exponential time differencing fourthorder Runge–Kutta (ETDRK4) method; this scheme was derived by Cox and Matthews in [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, J. Comp. Phys. 176 (2002) 430–455] and was modified by Kassam and Trefethen in [A. Kassam, L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comp. 26 (2005) 1214–1233]. We compute its amplification factor and plot its stability region, which gives us an explanation of its good behavior for dissipative and dispersive problems. We apply this method to the Kuramoto-Sivashinsky Equation obtaining excellent results.
