论文标题
用于求解非均匀强制的立方和五分之一的Swift-Hohenberg方程的数值方案严格尊重Lyapunov功能
Numerical scheme for solving the nonuniformly forced cubic and quintic Swift-Hohenberg equations strictly respecting the Lyapunov functional
论文作者
论文摘要
非平衡系统中模式形成的计算模型是研究生物学,化学,材料科学和工程中复杂现象的基本工具。在这些物理问题中,对某些人的理论描述的追求导致了Swift-Hohenberg方程(SH3),该方程描述了不稳定性附近的模式选择。有限差异方案被称为稳定校正(Christov&Pontes; 2001 doi:10.1016/s0895-7177(01)00151-0)在本文中审查并扩展了二维的立方Swift-Hohenberg方程。原始方案具有广义的Dirichlet边界条件(GDBC),具有控制参数的空间坡道的强制性,相关的Lyapunov功能的严格实现以及所有衍生物的二阶表示。现在,我们通过包括周期性边界条件(PBC),具有控制参数的高斯分布和Quintic Swift-Hohenberg(SH35)模型的强制扩展这些结果。本方案还具有对所有测试用例的功能的严格实现。完成了代码验证,显示了无条件的稳定性以及时间和空间的二阶准确性。测试用例证实了Lyapunov功能的单调衰减,所有数值实验表现出主要的物理特征:高度非线性行为,波长滤波器以及批量和边界效应之间的竞争。
Computational modeling of pattern formation in nonequilibrium systems is a fundamental tool for studying complex phenomena in biology, chemistry, materials science and engineering. The pursuit for theoretical descriptions of some among those physical problems led to the Swift-Hohenberg equation (SH3) which describes pattern selection in the vicinity of instabilities. A finite differences scheme, known as Stabilizing Correction (Christov & Pontes; 2001 DOI: 10.1016/S0895-7177(01)00151-0), developed to integrate the cubic Swift-Hohenberg equation in two dimensions, is reviewed and extended in the present paper. The original scheme features Generalized Dirichlet boundary conditions (GDBC), forcings with a spatial ramp of the control parameter, strict implementation of the associated Lyapunov functional, and second order representation of all derivatives. We now extend these results by including periodic boundary conditions (PBC), forcings with gaussian distributions of the control parameter and the quintic Swift-Hohenberg (SH35) model. The present scheme also features a strict implementation of the functional for all test cases. A code verification was accomplished, showing unconditional stability, along with second order accuracy in both time and space. Test cases confirmed the monotonic decay of the Lyapunov functional and all numerical experiments exhibit the main physical features: highly nonlinear behaviour, wavelength filter and competition between bulk and boundary effects.