Error Estimates of Energy Stable Numerical Schemes for Allen–Cahn Equations with Nonlocal Constraints
- 138 Downloads
We present error estimates for four unconditionally energy stable numerical schemes developed for solving Allen–Cahn equations with nonlocal constraints. The schemes are linear and second order in time and space, designed based on the energy quadratization (EQ) or the scalar auxiliary variable (SAV) method, respectively. In addition to the rigorous error estimates for each scheme, we also show that the linear systems resulting from the energy stable numerical schemes are all uniquely solvable. Then, we present some numerical experiments to show the accuracy of the schemes, their volume-preserving as well as energy dissipation properties in a drop merging simulation.
KeywordsEnergy stable schemes Energy quadratization methods Scalar auxiliary variable methods Error estimates Finite difference methods
Qi Wang’s research is partially supported by NSF awards DMS-1517347, DMS-1815921 and OIA-1655740, and NSFC awards #11571032, #91630207 and NSAF-U1530401.
- 8.Gong, Y., Zhao, J., Wang, Q.: Linear second order in time energy stable schemes for hydrodynamic models of binary mixtures based on a spatially pseudospectral approximation. Adv. Comput. Math. (2018). https://doi.org/10.1007/s10444-018-9597-5
- 12.Jing, X., Li, J., Zhao, X., Wang, Q.: Second order linear energy stable schemes for Allen–Cahn equations with nonlocal constraints. arXiv preprint arXiv:1810.05311 (2018)
- 27.Yang, X., Forest, M.G., Wang, Q.: Near equilibrium dynamics and one-dimensional spatial–temporal structures of polar active liquid crystals. Chin. Phys. B 23(11), 75–100 (2014)Google Scholar
- 28.Yang, X., Zhang, G.: Numerical approximations of the Cahn–Hilliard and Allen–Cahn equations with general nonlinear potential using the invariant energy quadratization approach. arXiv preprint arXiv:1712.02760 (2017)