高阶平均向量场方法在Allen-Cahn方程中的应用

High Order Average Vector Field Method of the Allen-Cahn Equation

Allen-Cahn方程是材料科学中描述流体动力学问题和反应扩散问题中的一类重要方程。Allen-Cahn方程的能量具有散逸性,即能量会随着时间的增长会逐渐降低。在数值模拟中,设计精确地保持Allen-Cahn方程能量散逸性的格式对模拟方程的演化具有显著的优点。目前,保Allen-Cahn方程能量散逸性的数值格式都是低阶的。最近有人构造了保持常微分方程能量散逸特性的高阶平均向量场方法,是一种有效的离散梯度法。国内外还少有人把保能量散逸性的高阶离散梯度方法应用于能量散逸性的偏微分方程。利用高阶离散梯度方法构造了Allen-Cahn方程的高阶格式。新的高阶格式能很好地长时间模拟Allen-Cahn方程数值解的演化,并长时间保持Allen-Cahn方程的内在特性。

Allen-Cahn equation is a class of important equation describing fluid dynamics and reaction diffusion problems in material science. The energy of the Allen-Cahn equation has the dissipation property. That is to say, the energy of the Allen-Cahn equation will gradually diminish with time. In numerical simulations, it is significant to design a numerical format which can accurately preserve the energy dissipation property of the Allen-Cahn equation in simulating evolution of the equation. The current numerical formats which can preserve the energy dissipation property of the Allen-Cahn equation are low-order. Recently, the high-order average vector field method which can preserve the energy dissipative property of the differential equations is constructed, which is a kind of efficient discrete gradient method. However, few people apply the high-order discrete gradient method to solve the energy-dissipative partial differential equation at home and abroad. In this paper, a high order scheme of the Allen-Cahn equation is proposed by the high-order discrete gradient method and Fourier pseudospectral method. The new high-order scheme can well simulate the evolution behaviors of numerical solutions of the Allen-Cahn equation with long time. Moreover, the new scheme can also well preserve the intrinsic property of the Allen-Cahn equation with long time.