An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System

Abstract

We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model and describes two phase very viscous flows in porous media. 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 time and space discrete ∞ 0, T ; H 1 h and 2 0, T ; H 2 h norms. We also present an efficient, practical nonlinear multigrid method -comprised of a standard FAS method for the CahnHilliard part, and a method based on the Vanka smoothing strategy for the Brinkman part -for solving these equations. In particular, we provide evidence that the solver has nearly optimal complexity in typical situations. The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium, as well as to the more general problems of buoyancy-and boundary-driven flows.

An efficient, unconditionally energy stable local discontinuous Galerkin scheme for the Cahn–Hilliard–Brinkman system

“…The time discretization method which is based on a convex splitting of the discrete CH energy is similar what is described by Collins et al in [12]. There is an important property that the convex splitting scheme generally inherit, unconditional energy stability.…”

“…Collins et al [12] presented an unconditionally energy stable and uniquely solvable finite difference scheme for the CHB system. Wise [28] introduced an unconditionally stable finite difference method for the Cahn-Hilliard-Hele-Shaw (CHHS) system.…”

Two‐phase flows in karstic geometry

“…A second order scheme that is unconditionally energy stable (with discrete energy law), and uniquely solvable for the Cahn-Hilliard-Stokes-Darcy system can be constructed by combining ideas from previous works, especially those from [59,55]. More specifically, we propose the following algorithm utilizing Onsager's extremum principle.…”

“…The main idea behind is the so-called convex splitting [85]. Applications to the Cahn-Hilliard-Darcy, Cahn-Hilliard-Stokes, and models of thin film epitaxial growth can be found at [55,59,86] among others.…”

“…For system that enjoys variational structure, a key idea in the development of unconditionally stable schemes is convex splitting (see [55,85,86,87] among many others). Indeed, the convex splitting idea has already been applied to the Cahn-Hilliard-Darcy model [55] and Cahn-Hilliard-Stokes model [59] to generate unconditionally stable (with discrete energy law) schemes. Here we combine the ideas and propose the following convex splitting scheme for the coupled Cahn-Hilliard-Stokes-Darcy system (3)- (4) with appropriate interface boundary conditions (5).…”