Numerical simulation of Swift–Hohenberg equation by the fourth-order compact scheme

  • Jian Su
  • Weiwei Fang
  • Qian Yu
  • Yibao LiEmail author


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.


Swift–Hohenberg equation Fourth-order compact scheme Nonlinear stabilized splitting scheme Adaptive time step method 

Mathematics Subject Classification

65M06 65M55 35G20 



J. Su is supported by National Natural Science Foundation of China (no. 91630206, no. 11771348). Y. B. Li is supported by National Natural Science Foundation of China (no. 11601416, no. 11631012) and by the China Postdoctoral Science Foundation (no. 2018M640968).


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Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anChina
  2. 2.Department of Computational Science and EngineeringYonsei UniversitySeoulRepublic of Korea

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