Finite difference discretization of the Kuramoto-Sivashinsky equation (Q1203421)

The Kuramoto-Sivashinsky equation \(u_ t+uu_ x+u_{xx}+\nu u_{xxxx}=0\) arising in fluid mechanics is investigated in one dimension \((x\in\mathbb{R})\) with 1-periodic initial condition \(u(x,0)=u_ 0\). The classical Crank-Nicolson finite difference scheme is applied to this problem, and the existence, convergence and uniqueness of the approximate solutions are obtained employing the Brouwer fixed point theorem and Gronwall's discrete inequality. Some second order error estimates representing the stability of the scheme are also considered. Finally, the author proposes to solve the resulting nonlinear discrete algebraic equations by Newton's linearization method.