Numerical methods and error analysis for the nonlinear Sivashinsky equation

Abstract Our purpose is to solve numerically the solution of the nonlinear evolutionary problem called Sivashinsky equation. First we have derived error estimates of semidiscrete finite element method for the approximation of the Sivashinsky problem. A completely discrete scheme based on the backward Galerkin scheme, a linearized backward Euler method and Crank–Nicolson–Galerkin scheme have been developed. Second, we analyze a linearized finite difference scheme. We show existence and uniqueness of the approximate solutions and we derive second-order error estimates.

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