Yongyong Cai

Yongyong Cai, Jie Xu, Haixin Zhang

In tensor dynamics for liquid crystals derived from molecular models, a common problem is closure approximation. For rod-like molecules, the Bingham closure has proved to outperform other methods because it inherits the gradient flow structure of the molecular model, but is difficult to achieve efficient computations maintaining the gradient flow structure. We propose a closure approximation by the quasi-entropy that has been successfully applied to the free energy, based on which we construct the tensor gradient flow. The quasi-entropy closure has the same symmetry properties as the Bingham closure. The resulting tensor gradient flow is able to constrain the eigenvalues of the tensor within the physical range, guaranteeing the positive definiteness of the dissipation operator given by the higher-order tensors. The quasi-entropy closure is easy to implement since it can be reduced to minimizing an elementary function of three variables. As a result, we construct a numerical scheme preserving the eigenvalue constraints and energy dissipation, with the closure approximation decoupled from solving the scheme. Numerical simulations are carried out for the interface between the isotropic and the uniaxial nematic phase, as well as the defect evolutions, where the higher-order tensors indeed make a difference.

Weizhu Bao, Yongyong Cai, Xiaowei Jia et al.

We propose and rigourously analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation with a dimensionless parameter $\varepsilon\in(0,1]$ which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the solution exhibits highly oscillatory propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. Due to the rapid temporal oscillation, it is quite challenging in designing and analyzing numerical methods with uniform error bounds in $\varepsilon\in(0,1]$. We present the MTI-FP method based on properly adopting a multiscale decomposition of the solution of the Dirac equation and applying the exponential wave integrator with appropriate numerical quadratures. By a careful study of the error propagation and using the energy method, we establish two independent error estimates via two different mathematical approaches as $h^{m_0}+\frac{τ^2}{\varepsilon^2}$ and $h^{m_0}+τ^2+\varepsilon^2$, where $h$ is the mesh size, $τ$ is the time step and $m_0$ depends on the regularity of the solution. These two error bounds immediately imply that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $O(τ)$ for all $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(τ^2)$ in the regimes when either $\varepsilon=O(1)$ or $0<\varepsilon\lesssim τ$. Numerical results are reported to demonstrate that our error estimates are optimal and sharp. Finally, the MTI-FP method is applied to study numerically the convergence rates of the solution of the Dirac equation to those of its limiting models when $\varepsilon\to0^+$.

Weizhu Bao, Yongyong Cai, Xiaowei Jia et al.

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size $h$ and time step $τ$ as well as the small parameter $0<\varepsilon\le 1$. Based on the error bound, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the CNFD method requests the $\varepsilon$-scalability: $τ=O(\varepsilon^3)$ and $h=O(\sqrt{\varepsilon})$. Then we propose and analyze two numerical methods for the discretization of the nonlinear Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability is improved to $τ=O(\varepsilon^2)$ and $h=O(1)$ when $0<\varepsilon\ll 1$ compared with the CNFD method. Extensive numerical results are reported to confirm our error estimates.

Weizhu Bao, Yongyong Cai, Xiaowei Jiao et al.

We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with several frequently used finite difference time domain (FDTD) methods and obtain rigorously their error estimates in the nonrelativistic limit regime by paying particular attention to how error bounds depend explicitly on mesh size $h$ and time step $τ$ as well as the small parameter $\varepsilon$. Based on the error bounds, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the FDTD methods share the same $\varepsilon$-scalability on time step: $τ=O(\varepsilon^3)$. Then we propose and analyze two numerical methods for the discretization of the Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability on time step is improved to $τ=O(\varepsilon^2)$ when $0<\varepsilon\ll 1$. Extensive numerical results are reported to support our error estimates.