Wenfan Yi

Weizhu Bao, Yue Feng, Wenfan Yi

We establish error bounds of the finite difference time domain (FDTD) methods for the long time dynamics of the nonlinear Klein-Gordon equation (NKGE) with a cubic nonlinearity, while the nonlinearity strength is characterized by $\varepsilon^2$ with $0 <\varepsilon \leq 1$ a dimensionless parameter. When $0 < \varepsilon \ll 1$, it is in the weak nonlinearity regime and the problem is equivalent to the NKGE with small initial data, while the amplitude of the initial data (and the solution) is at $O(\varepsilon)$. Four different FDTD methods are adapted to discretize the problem and rigorous error bounds of the FDTD methods are established for the long time dynamics, i.e. error bounds are valid up to the time at $O(1/\varepsilon^β)$ with $0 \le β\leq 2$, by using the energy method and the techniques of either the cut-off of the nonlinearity or the mathematical induction to bound the numerical approximate solutions. In the error bounds, we pay particular attention to how error bounds depend explicitly on the mesh size $h$ and time step $τ$ as well as the small parameter $\varepsilon\in (0,1]$, especially in the weak nonlinearity regime when $0 < \varepsilon \ll 1$. Our error bounds indicate that, in order to get ``correct'' numerical solutions up to the time at $O(1/\varepsilon^β)$, the $\varepsilon$-scalability (or meshing strategy) of the FDTD methods should be taken as: $h = O(\varepsilon^{β/2})$ and $τ= O(\varepsilon^{β/2})$. Extensive numerical results are reported to confirm our error bounds and to demonstrate that they are sharp.

Zhaoxing Chen, Wei Liu, Ziqing Xie et al.

This paper presents a class of efficient manifold optimization algorithms for computing the ground state solutions of a semilinear elliptic system, which are unstable saddle points of the variational functional. Variational arguments show that these unstable saddle points can be characterized as the local minimizers of the variational functional constrained to the Nehari manifold $\mathcal{N}$. The Nehari manifold optimization method (NMOM) proposed in [Z. Chen, W. Liu, Z. Xie, and W. Yi. SIAM J. Sci. Comput., 47(4): A2098-A2126, 2025] provides a Riemannian gradient descent framework on $\mathcal{N}$ for such constrained minimization problems. To deal with both the intrinsic instability of the solutions and the increased computational complexity introduced by the coupling between components, we combine the ideas from the NMOM and the Nesterov-type acceleration to develop a new efficient Riemannian accelerated gradient algorithm on $\mathcal{N}$ (RAG-$\mathcal{N}$). The key idea is to perform an easy-to-implement nonlinear extrapolation step on $\mathcal{N}$, followed by a Riemannian steepest-descent update at the extrapolated point. To enhance the robustness, we further incorporate a nonmonotone step-size search strategy into the RAG-$\mathcal{N}$ algorithm, obtaining a variant with improved stability. Numerical experiments show that the RAG-$\mathcal{N}$ algorithms substantially reduce the number of iterations compared with the Riemannian steepest descent algorithm of NMOM. Finally, we apply the RAG-$\mathcal{N}$ algorithms to compute the ground state solutions of semilinear elliptic systems with two, three and four components, and investigate their behavior under different coupling coefficients and various settings, including Gaussian-type external potentials and singular diffusion coefficients.