Energy dissipation laws of time filtered BDF methods up to fourth-order for the molecular beam epitaxial equation. (English) Zbl 1537.65109
Summary: This report presents the time filtered BDF-\(k\) (FiBDF-\(k\)) methods up to fourth-order time accuracy for the molecular beam epitaxial equation with no-slope selection. The new \((k + 1)\)-order methods are developed by introducing an inexpensive post-filtering step to the variable-step BDF-\(k\) (\(k=1,2,3\)) methods. We show that the FiBDF-\(k\) methods are uniquely solvable and volume conservative. Some novel discrete gradient structures of the FiBDF-\(k\) formulas are derived such that we can build up the discrete energy dissipation laws for the associated time-steppings. Numerical examples are included to show the mesh-robustness and effectiveness of the variable-step methods especially in the long-time simulations.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35Q99 | Partial differential equations of mathematical physics and other areas of application |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
Keywords:
time filtered BDF method; nonuniform time meshes; discrete gradient structure; energy dissipation law; adaptive time-stepping algorithmReferences:
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