A problem involving the solidification of a solid seed in an undercooled melt is posed. A sharp interface between solid and liquid is assumed, and the equilibrium solidification temperature at this interface is controlled by both curvature and kinetic undercooling. Numerical solutions based on a fixed grid and on a deforming grid are developed. In the limits of a vanishing surface energy and large molecular mobility, these solutions are verified by comparing with known analytical solutions. In general applications, with curvature and kinetic undercooling present, predictions from the fixed and deforming grid solutions are in close agreement. The results obtained highlight the role of the surface curvature and kinetic undercooling in controlling solidification speed and acceleration. In addition, both the fixed and deforming solution methods provide verification tools for more general 2-D and 3-D crystal growth simulations.