Unconditionally Stable Finite Difference, Nonlinear Multigrid Simulation of the Cahn-Hilliard-Hele-Shaw System of Equations

• Published in: Journal of scientific computing 2010-07, Vol.44 (1), p.38-68
• Main Author: Wise, S. M.
• Format: Article
• Language: English
• Subjects: Algorithms; Binary fluids; Computational Mathematics and Numerical Analysis; Computer simulation; Finite difference method; Gradient flow; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Nonlinearity; Norms; Porous media; Sorting; Spinodal decomposition; Theoretical; Tumors; Two phase flow
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-010-9363-4

We present an unconditionally energy stable and solvable finite difference scheme for the Cahn-Hilliard-Hele-Shaw (CHHS) equations, which arise in models for spinodal decomposition of a binary fluid in a Hele-Shaw cell, tumor growth and cell sorting, and two phase flows in porous media. We show that the CHHS system is a specialized conserved gradient-flow with respect to the usual Cahn-Hilliard (CH) energy, and thus techniques for bistable gradient equations are applicable. In particular, the scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step-size. Owing to energy stability, we show that the scheme is stable in the norm, and, assuming two spatial dimensions, we show in an appendix that the scheme is also stable in the norm. We demonstrate an efficient, practical nonlinear multigrid method for solving the equations. In particular, we provide evidence that the solver has nearly optimal complexity. We also include a convergence test that suggests that the global error is of first order in time and of second order in space.