Non-iterative compact operator splitting scheme for Allen-Cahn equation

Abstract

The compact method guarantees high order accuracy and comprise point stencils; however, the alternating direction implicit (ADI)-type operator splitting method is difficult to implement. Here, a non-iterative compact ADI-type operator splitting scheme with stability for the Allen-Cahn equation is presented. Operator splitting comprises temporal and spatial splitting. Temporal splitting is based on the hybrid method, which combines numerical and analytical methods to resolve the nonlinearity of the equation, and the maximum principle is maintained unconditionally. In addition, the temporal accuracy can be easily extended to the second or higher order. Spatial splitting is considered for a compact operator with a simple ADI-type implementation. This allows the Thomas algorithm, which is simple and fast, to be applied, including for solving two- and three-dimensional problems. The stability and accuracy proofs of the proposed scheme are presented. The numerical results show that the accuracy, stability, efficiency, and dynamics are consistent with theory.