An efficient second-order linear scheme for the phase field model of corrosive dissolution

Abstract

In this paper we propose an efficient numerical scheme for solving the phase field model (PFM) of corrosive dissolution that is linear and second-order accurate in both time and space. The PFM of corrosion is based on the gradient flow of a free energy functional depending on a phase field variable and a single concentration variable. While classic backward differentiation formula (BDF) schemes have been used for time discretization in the literature, they require very small time step sizes owing to the strong numerical stiffness and nonlinearity of the parabolic partial differential equation (PDE) system defining the PFM. Based on the observation that the governing equation corresponding to the phase field variable is very stiff due to the reaction term, the key idea of this paper is to employ an exponential time integrator that is more effective for stiff dynamic PDEs. By combining the exponential integrator based Rosenbrock–Euler scheme with the classic Crank–Nicolson scheme for temporal integration of the spatially semi-discretized system, we develop a decoupled linear numerical scheme that alleviates the time step size restriction due to high stiffness. Several numerical examples are presented to demonstrate accuracy, efficiency and robustness of the proposed scheme in two-dimensions, and we find that a time step size of 10−3 second for meshes with the typical spatial resolution 1 μm is stable. Additionally, the proposed scheme is robust and does not suffer from any convergence issues often encountered by nonlinear Newton methods.