A least squares based finite volume method for the Cahn–Hilliard and Cahn–Hilliard-reaction equations

A vertex-centred finite volume method (FVM) for the Cahn–Hilliard (CH) and recently proposed Singh et al. (2008) and Burch (2009) Cahn–Hilliard-reaction (CHR) equations is presented. Information at control volume faces is computed using a high-order least-squares approach based on Taylor series approximations. This least-squares problem explicitly includes the variational boundary condition (VBC) that ensures that the discrete equations satisfy all of the boundary conditions. We use this approach to solve the CH and CHR equations in one and two dimensions and show that our scheme satisfies the VBC to at least second order. For the CH equation we show evidence of conservative, gradient stable solutions, however for the CHR equation, strict gradient-stability is more challenging to achieve.