高阶能量稳定的数值方案用于二维向列液晶中非线性变分波方程
High-order energy stable numerical schemes for a nonlinear variational wave equation modeling nematic liquid crystals in two dimensions
AI总结：
本文提出高阶能量稳定的数值方案，用于求解二维向列液晶中非线性变分波方程，验证了方案的能量稳定性、收敛性和计算效率，并展示了解在有限时间后失去正则性的情况。
摘要：
我们考虑了一个非线性变分波方程，用于建模向列液晶中取向场的动力学，具有高分子旋转惯性。该方程源自能量原理，因此能量稳定性是该模型解的固有属性。在二维情况下，我们基于间断伽辽金框架设计了数值方案，这些方案要么守恒要么耗散离散的能量版本。进行了大量数值实验，验证了该方案的能量稳定性、收敛阶和计算效率。数值解与一个更简单的一阶哈密顿方案的解进行了比较。我们提供了数值证据表明，二维变分波方程的解在有限时间内失去正则性。之后，耗散和保守方案似乎收敛到不同的解。
英文摘要：
We consider a nonlinear variational wave equation that models the dynamics of the director field in nematic liquid crystals with high molecular rotational inertia. Being derived from an energy principle, energy stability is an intrinsic property of solutions to this model. For the two-dimensional case, we design numerical schemes based on the discontinuous Galerkin framework that either conserve or dissipate a discrete version of the energy. Extensive numerical experiments are performed verifying the scheme's energy stability, order of convergence and computational efficiency. The numerical solutions are compared to those of a simpler first-order Hamiltonian scheme. We provide numerical evidence that solutions of the 2D variational wave equation loose regularity in finite time. After that occurs, dissipative and conservative schemes appear to converge to different solutions.
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