Study of 4th order Kuramoto-Sivashinsky equation by septic Hermite collocation method

Abstract

In the current study, a robust septic Hermite collocation method (SHCM) is proposed to simulate the Kuramoto-Sivashinsky (KS) equation. To approximate the spatial derivatives, the collocation method with the 7th-degree Hermite interpolation polynomial is utilized and for the temporal derivative, the Crank-Nicolson (CN) technique is implemented. The nonlinear terms of the KS equation are linearized using the quasi-linearization process. Using the von-Neumann approach, it is demonstrated that the algorithm is unconditionally stable. The convergence analysis of the proposed technique is also given. In the temporal direction, the scheme is observed to be second-order convergent and in space direction, it is found to be sixth-order convergent. The proposed technique's robustness is demonstrated by solving six test problems. The L2, L∞, and global relative errors are determined and the findings are compared with other methods available in the literature. The behavior of few KS equations, for which their exact solution is not available, is also analyzed. The current findings are better than the results obtained from the other methods and are also matched well with the analytical solution.