A Nonstiff Euler Discretization of the Complex Ginzburg-Landau Equation in One Space Dimension

A nonstiff discretization method is applied to the complex Ginzburg--Landau (CGL) equation with periodic boundary conditions, \begin{eqnarray*} &\partial_t u = (1+i\nu) \partial_{xx}^2 u + Ru - (1+i\mu)|u|^2 u ; \\ u t\geq 0. \end{eqnarray*} This parabolic equation is discretized first in space by a truncated Fourier series (that is, a finite Fourier sum) and then in time by an explicit Euler method. The exponential decay of Fourier modes in both, the original CGL equation, and the resulting equation for the truncated Fourier series (independently from the number of Fourier modes in the truncation) is used in an essential way to prove the nonstiffness. Also dissipativity of the discretized equation is established and numerical results are discussed.

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