Efficient and Stable Exponential Time Differencing Runge-Kutta Methods for Phase Field Elastic Bending Energy Models

The Willmore flow formulated by phase field elastic bending energy models has been widely used to describe the shape transformation of biological lipid vesicles. In this talk, we will present some efficient and stable numerical methods for simulations of the unconstrained Willmore flow and the Willmore flow with fixed volume and surface area constraints. The proposed methods can be high-order accurate and are completely explicit in nature, by combining exponential time differencing Runge-Kutta approximations for time integration with spectral discretizations for spatial operators on regular meshes. We also incorporate novel linear operator splitting techniques into the numerical schemes to improve the discrete energy stability. Furthermore, in order to avoid extra numerical instability brought by use of large penalty parameters for solving the constrained Willmore flow problem, a modified augmented Lagrange multiplier approach is developed and adopted in the methods. Various numerical experiments are performed to demonstrate accuracy and stability of the proposed methods.