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Numerical simulation of Swift-Hohenberg equation by the fourth-order compact scheme. (English) Zbl 1438.65192

Summary: A high-order accurate compact scheme for the Swift-Hohenberg equation is presented in this paper. We discretize the Swift-Hohenberg equation by a fourth-order compact finite difference formula in space and a backward differentiation with second-order accurate in time, respectively. A stabilized splitting scheme is presented and a Newton-type iterative method is introduced to deal with the nonlinear term. Therefore, a large time step can be used. The resulting discrete systems are solved by a fast and efficient nonlinear multigrid solver. Adaptive time step method is implemented to reduce the computational cost. Various numerical simulations including a convergence test of the proposed scheme, comparison with second-order scheme, a test of the stability of the proposed scheme, extension of the adaptive time step method, comparison with the phase field crystal equation, a study of the effect of computational domain and boundary condition, and an evolution of Swift-Hohenberg equation in three dimensions, are performed to demonstrate the efficiency of our proposed method.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
35G20 Nonlinear higher-order PDEs
35Q35 PDEs in connection with fluid mechanics
65H10 Numerical computation of solutions to systems of equations
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
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