Lack of robustness and accuracy of many numerical schemes for phase-field simulations

In this paper, we study the stability, accuracy and convergence behavior of various numerical schemes for phase-field modeling through a simple ODE model. Both theoretical analysis and numerical experiments are carried out on this ODE model to demonstrate the limitation of most numerical schemes that have been used in practice. One main conclusion is that the first-order fully implicit scheme is the only robust algorithm for phase-field simulations while all other schemes (that have been analyzed) may have convergence issue if the time step size is not exceedingly small. More specifically, by rigorous analysis in most cases, we have the following conclusions: (i) The first-order fully implicit scheme converges to the correct steady state solution for all time step sizes. In the case of multiple solutions, one of the solution branches always converges to the correct steady state solution. (ii) The first-order convex splitting scheme, which is equivalent to the first-order fully implicit scheme with a different time scaling, always converges to the correct steady state solution but may seriously lack numerical accuracy for transient solutions. (iii) For the second-order fully implicit and convex splitting schemes, for any time step size > 0, there exists an initial condition u0, with |u0| > 1, such that the numerical solution converges to the wrong steady state solution. (iv) For |u0|≤ 1, all second-order schemes studied in this paper converge to the correct steady state solution although severe numerical oscillations occur for most of them if the time step size is not sufficiently small. (v) An unconditionally energy-stable scheme (such as the modified Crank-Nicolson scheme) is not necessarily better than a conditionally energy-stable scheme (such as the Crank-Nicolson scheme). Most, if not all, of the above conclusions are expected to be true for more general Allen-Cahn and other phase-field models.