Yongyong Cai

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 $\varepsilon\to0^+$.

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.

Yongyong Cai, Xingwei Yang

We propose and analyze a linearly implicit mass-lumped finite element method for the heat flow of harmonic maps into the unit sphere. The method consists of a linear predictor followed by a nodal projection and therefore preserves the unit-length constraint exactly at all finite element nodes. The predictor is derived from a cross-product reformulation of the equation and is shown to be equivalent to a mass-lumped discretization of the original formulation with a correction term enforcing nodal orthogonality, as well as to a tangent plane scheme. A key ingredient is the consistent use of the discrete inner product in both the mass and stiffness terms. This yields a nodal orthogonality relation implying that the auxiliary solution lies on or outside the unit sphere at every node. Consequently, the projection is well defined and the projected error satisfies a contraction property in the discrete \(L^2\)-norm. On Cartesian rectangular and cuboidal tensor-product meshes, the nodal projection is also nonexpansive in a discrete Dirichlet energy, which gives an unconditional discrete energy dissipation law. For sufficiently smooth solutions, we prove optimal error estimates without any coupling condition between the time step and the mesh size: the method converges with order \(O(Δt+h^2)\) in \(\ell^\infty(0,T;L^2)\) and order \(O(Δt+h)\) in \(\ell^2(0,T;H^1)\). The proof combines the projected-error contraction, quadrature consistency estimates, edge-based cancellation identities, and a bootstrap argument for controlling nonlinear terms. Numerical experiments confirm the predicted convergence rates and the discrete energy decay.