Application of locally one-dimensional semi-implicit scheme in phase-field equations

A locally one-dimensional (LOD) semi-implicit scheme is proposed for improving the numerical efficiency in the solving of parabolic partial differential equations in phase-field simulations. With LOD splitting, multi-dimensional parabolic problems can be numerically approximated by treating each of the spatial variables individually in single cycles. Additionally, each spatial variable can be treated in either real or Fourier space, allowing equations to be solved across a range of boundary conditions, including periodic, non-periodic, and even partial periodic. The proposed LOD semi-implicit scheme exhibits noticeable advantages over both explicit and implicit traditional schemes in terms of computational efficiency and accuracy, as demonstrated by two standard numerical tests. It is anticipated that future large-scale phase-field simulations will benefit greatly from the use of this LOD scheme.