Most implicit methods for the Swift–Hohenberg (SH) equation with quadratic–cubic nonlinearity require costly iterative solvers at each time step. In this paper, a non-iterative method for obtaining approximate solutions of the SH equation which is based on the convex splitting idea is presented. By regularizing the cubic–quartic function in the energy for the SH equation and adding an extra linear stabilizing term, we arrive at a non-iterative convex splitting method, where the operator involved is linear and positive and has constant coefficients. We further prove the unconditional energy stability of the method. Numerical examples illustrating the accuracy, efficiency, and energy stability of the proposed method are provided.

大多数具有二次三次非线性的 Swift-Hohenberg (SH) 方程的隐式方法在每个时间步都需要昂贵的迭代求解器。本文提出了一种基于凸分裂思想的非迭代求解SH方程近似解的方法。通过对 SH 方程的能量中的三次四次函数进行正则化并添加一个额外的线性稳定项，我们得出了一种非迭代凸分裂方法，其中所涉及的算子是线性的、正的并且具有常数系数。我们进一步证明了该方法的无条件能量稳定性。提供了说明所提出方法的准确性、效率和能量稳定性的数值例子。