Scientific American are reporting that “a team of mathematicians has for the first time succeeded in simulating a panoply of snowflake shapes using basic conservation laws, such as preserving the number of water molecules in the air”.

This explains that previous simulations often simulate the crystal surface using interlocking triangles, but that:

the triangles often deform and collapse in simulations, leading to singularities that bring the simulation to an abrupt halt… Garcke’s team got around this difficulty by devising a method to describe the curvature and other geometric information about the snowflake surface so that it could be appropriately encoded into a computer.

Moreover, the research does what previous attempts to model snowflakes using a similar approach could not, model the two main types of snowflake growth simultaneously:

faceted growth, in which flat plates, such as hexagons and triangles, dominate the process, and dendritic growth, in which the flakes form treelike branches that themselves beget branches, just as dendrites extend out from nerve cells.

The full article goes into some further detail and history of attempts to solve the problem.

Source: Snowflake Growth Successfully Modeled from Physical Laws.

Paper: Numerical computations of facetted pattern formation in snow crystal growth.