Exponential time differencing schemes for phase-field modeling of solidification

We study the acceleration and accuracy resulting from the application of different Rosenbrock–Euler exponential integrators of orders 2, 3, and 4 for temporal integration of partial differential equations corresponding to the quantitative phase-field modeling of pure elements solidification. The exact Jacobian of differential operators is computed analytically using the continuous differentiation approach. After spatial discretization, we have large-scale nonlinear ordinary differential equations. Compared to linearly implicit time integration approaches that require at least solving one linear system of equations at each time step, our schemes do not include any implicit phase. Instead, it is essential to compute at least one matrix function vector product per time step, which is extremely computationally and memory demanding using direct approaches. An efficient implementation of the Krylov subspace method is exploited to approximate the action of a matrix function on a desired vector to address this challenge. The time step size is adaptively calculated based on posteriori error estimations. Numerical experiments confirm the convergence rates of our proposed schemes. According to our numerical results, these exponential time-differencing schemes reveal superior computational performance compared to the Euler explicit method, which is the most popular approach for time integration of phase-field solidification equations. Moreover, when a medium to high level of accuracy is desired, these schemes are orders of magnitude faster than this scheme. In addition, using exponential time differencing with adaptive time stepping allows us to use time step size of more than two orders of magnitude larger than that of the stable Euler explicit method, while having either accuracy of the same level or superior.