A second-order BDF scheme for the Swift-Hohenberg gradient flows with quadratic-cubic nonlinearity and vacancy potential. (English) Zbl 1499.65382
Summary: In this paper, we propose a second-order BDF time marching scheme for the Swift-Hohenberg gradient flows (Swift-Hohenberg equation and phase-field crystal equation) with quadratic-cubic nonlinearity and vacancy potential. Based on the multiple scalar auxiliary variables (MSAV) approach and stabilization technique, an efficient, second-order accurate, and unconditionally energy stable numerical scheme is constructed. In this scheme, two new scalar auxiliary variables are introduced to deal with the nonlinear terms, and a linear stabilization term is added to enhance the stability and keep the required accuracy while using the large time steps. The unique solvability and unconditional energy stability are proved. Some numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed scheme and show the effect of quadratic term and vacancy potential on pattern formation.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65N35 | Spectral, collocation and related methods for 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 |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35R09 | Integro-partial differential equations |
Keywords:
Swift-Hohenberg equation; phase field crystal equation; vacancy; unique solvability; energy stableReferences:
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