Department of MathematicsProject leader: Professor Saulo Orizaga
Project manager: Lihan Wang
Team members: Riley Fisher, Mckenzie Garcia, Naga Rudrapatna
This project has produced the following published paper:
Riley Fisher, Mckenzie Garcia, Nagaprasad Rudrapatna, Comparing Numerical Solution Methods for the Cahn-Hilliard Equation, SIAM Undergraduate Research Online volume 14
The Cahn-Hilliard equation is a higher-order parabolic and nonlinear partial differential equation (PDE) used to model many phase separation phenomena in physics, engineering, and medicine. As this PDE contains nonlinear terms, we explore highly accurate numerical methods to estimate a solution. These methods must retain the energy decreasing property since the Cahn-Hilliard equation is structured as a gradient flow problem. This summer, we further developed, analyzed, and compared four methods and their variations–convexity splitting (CS), linear extrapolation (LINX) with iterations, second-order backwards difference formula (BDF) with iterations, and implicit-explicit Runge-Kutta (IMEX) with different sets of coefficients–to find a preferred numerical method, considering its robustness, efficiency, and stability. We determined that the strong, stability-preserving IMEX method–initially proposed by Song (2015)–is the most accurate given these considerations. We also found the first-order LINX method to be modestly accurate in solving this PDE.
Orizaga & Glasner (2016) observed instability when analyzing the BDF scheme with the convexity splitting parameter of a = 1.3. We found that, by setting the convexity splitting parameter to a = 2.5, this observed instability can be significantly reduced and controlled. Our error plots for the BDF scheme confirm that our selection of the convexity splitting parameter improves upon the results and investigations of previous researchers. Moreover, we explored the effect of varying the convexity splitting parameter a on error, specifically the classical restriction a > 2, and concluded that this restriction can be slightly ignored for the purpose of accuracy. However, concerns with numerical instability prevent us from completely disregarding this condition.