A Fully implicit, Fully adaptive multigrid method for multiscale phase-field modelling

A fully implicit, fully adaptive multigrid method for the simulation of free dendritic growth into an undercooled melt has been developed and applied to various phase-field models. The main aim of this work is to demonstrate that advanced numerical methods, such as mesh adaptivity, implicit time stepping, multigrid methods and variable time step control, are not only advantageous compared to widely used explicit time discretisation methods and uniformly refined meshes, but are also necessary to obtain fully-converged, steady-state solutions, especially for coupled thermal-solutal driven dendritic growth. Initially, a selection of linear and nonlinear elliptic and parabolic problems are used to compare: the finite difference and the finite element method; explicit and implicit time discretisation; solutions obtained on uniformly and adaptively refined meshes; and to test the convergence of the nonlinear multigrid solver that we have developed. Additionally, the proposed numerical scheme has been applied to a system of nonlinear PDEs for the simulation of chemical reactions; the Gray-Scott model. We showed that on geometrically simple domains the finite difference method outperforms the finite element method in all test cases and that the second-order implicit BDF2 method reproduces the same results as obtained with an explicit method, to the same or higher accuracy, but for much larger time steps. Furthermore, an h-independent convergence of the multigrid solver is demonstrated prior to applying our numerical methods to phase-field models. So far only a few attempts have been made to use a phase-field method to simulate the development of dendrites during solidification of an alloy system with high-undercoolings and at realistic Lewis numbers, for which both thermal and solutal diffusion need to be considered. In those investigations too small domain sizes, stability issues and step size restrictions have been major constraints. With our proposed method we are able to overcome all of these restrictions and solve problems with arbitrary ratios of solutal and thermal diffusion. As a result we are able to show the transition between purely solutal to coupled thermal-solutal to purely thermal controlled dendritic growth by increasing the thermal undercoolings which has been predicted by the LKT theory but has never before been observed with a phase-field method. As an example of the application of the model to real systems fully-converged, steady-state results are presented for a Ni10Cu90 system with a Lewis number of 10000 and a range of different undercoolings.