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We review the application of advanced numerical techniques such as adaptive mesh refinement, implicit time stepping, multigrid solvers and massively parallel implementations as a route to obtaining solutions to the three-dimensional phase-field problem with a domain size and interface resolution previously possible only in two dimensions. Using such techniques it is shown that such models are tractable even as the interface width approaches the solute capillary length.

Towards a Three-Dimensional Phase-Field Model of Dendritic Solidification with Physically Realistic Interface Width