Positive-Preserving Implicit Difference Scheme for Solving Two-Dimensional Allen-Cahn Equation

本文研究求解二维Allen-Cahn方程的保正隐式差分格式。通过证明得到当网格比满足 1+τ−2 R x −2 R y ≥0 时，差分解具有保正性，且在无穷范数意义下有 O( τ+ h x 2 + h y 2 ) 的收敛阶，最后数值实验表明数值结果与理论结果相吻合。In this paper, we study the positive-preserving implicit difference scheme for solving two-dimensional Allen-Cahn equation. It is proved that when the grid ratio satisfies 1+τ−2 R x −2 R y ≥0 , the difference solution is positive-preserving and has the convergence order of O( τ+ h x 2 + h y 2 ) in the sense of infinite norm. Finally, numerical experiments show that the numerical results are consistent with the theoretical results.