  We employ adaptive mesh refinement, implicit time stepping, a nonlinear
multigrid solver and parallel computation, to solve a multi-scale, time
dependent, three dimensional, nonlinear set of coupled partial differential
equations for three scalar field variables. The mathematical model represents
the non-isothermal solidification of a metal alloy into a melt substantially
cooled below its freezing point at the microscale. Underlying physical
molecular forces are captured at this scale by a specification of the energy
field. The time rate of change of the temperature, alloy concentration and an
order parameter to govern the state of the material (liquid or solid) is
controlled by the diffusion parameters and variational derivatives of the
energy functional. The physical problem is important to material scientists for
the development of solid metal alloys and, hitherto, this fully coupled thermal
problem has not been simulated in three dimensions, due to its computationally
demanding nature. By bringing together state of the art numerical techniques
this problem is now shown here to be tractable at appropriate resolution with
relatively moderate computational resources.
