Error Analysis of Full-Discrete Invariant Energy Quadratization Schemes for the Cahn–Hilliard Type Equation
Document Type
Article
Journal/Book Title/Conference
Journal of Computational and Applied Mathematics
Volume
372
Publisher
Elsevier BV * North-Holland
Publication Date
1-18-2020
Award Number
NSF, Division of Mathematical Sciences 1816783
Funder
NSF, Division of Mathematical Science
Abstract
In this paper, we present the error analysis for a fully discrete scheme of the Cahn–Hilliard type equation, along with numerical verifications. The numerical schedule is developed by first transforming the Cahn–Hilliard type equation into an equivalent form using the invariant energy quadratization (IEQ) technique. Then the equivalent form is discretized by using the linear-implicit Crank–Nicolson method for the time variable and the Fourier pseudo-spectral method for the spatial variables. The resulted full-discrete scheme is linear and unconditionally energy stable, which makes it easy to implement. By constructing an appropriate interpolation equation, the uniform boundedness of the numerical solution is obtained theoretically. Then, we prove that the numerical solutions converge with the order O((δt) 2+hm)" role="presentation">, where δt" role="presentation"> is the temporal step and h" role="presentation"> is the spatial step with m" role="presentation"> the regularity of the exact solution. Several numerical examples are presented to confirm the theoretical results and demonstrate the effectiveness of the presented linear scheme. The numerical strategies and analytical tools in this paper could be readily applied to study other phase field models or gradient flow problems.
Recommended Citation
Zhang, Jun, Jia Zhao, and Yuezheng Gong. “Error Analysis of Full-Discrete Invariant Energy Quadratization Schemes for the Cahn–Hilliard Type Equation.” Journal of Computational and Applied Mathematics 372.C (2020): n. pag. Web. doi:10.1016/j.cam.2020.112719.
