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A non-iterative and unconditionally energy stable method for the Swift-Hohenberg equation with quadratic-cubic nonlinearity. (English) Zbl 1524.65664

Summary: Most implicit methods for the Swift-Hohenberg (SH) equation with quadratic-cubic nonlinearity require costly iterative solvers at each time step. In this paper, a non-iterative method for obtaining approximate solutions of the SH equation which is based on the convex splitting idea is presented. By regularizing the cubic-quartic function in the energy for the SH equation and adding an extra linear stabilizing term, we arrive at a non-iterative convex splitting method, where the operator involved is linear and positive and has constant coefficients. We further prove the unconditional energy stability of the method. Numerical examples illustrating the accuracy, efficiency, and energy stability of the proposed method are provided.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q35 PDEs in connection with fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
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