  Phase-field models of microstructural pattern formation during alloy
solidification are commonly solved numerically using the finite-difference
method, which is ideally suited to carry out computationally efficient
simulations on massively parallel computer architectures such as Graphic
Processing Units. However, one known drawback of this method is that the
discretization of differential terms involving spatial derivatives introduces a
spurious lattice anisotropy that can influence the solid-liquid interface
dynamics. We find that this influence is significant for the case of
polycrystalline dendritic solidification, where the crystal axes of different
grains do not generally coincide with the reference axes of the
finite-difference lattice. In particular, we find that with the commonly used
finite-difference implementation of the quantitative phase-field model of
binary alloy solidification, both the operating state of the dendrite tip and
the dendrite growth orientation are strongly affected by the lattice
anisotropy. To circumvent this problem, we use known methods in both real and
Fourier space to derive finite-difference approximations of leading
differential terms in 2D and 3D that are isotropic at order $h^2$ of the
lattice spacing $h$. Importantly, those terms include the divergence of the
anti-trapping current that is found to have a critical influence on pattern
selection. The 2D and 3D discretizations use an approximated form of the
anti-trapping current that facilitates the Fourier-space derivation of the
associated isotropic differential operator at $O(h^2)$, but we also derive a 2D
discretization of the standard form of this current. Finally, we present 2D and
3D phase-field simulations of alloy solidification, showing that the isotropic
finite-difference implementations can dramatically reduce spurious lattice
anisotropy effects.
