We study the (2+1)-dimensional nonlocal Fokas system by using the Hirota’s bilinear method. Firstly, a general tau-function of Kadomtsev-Petviashvili (KP) hierarchy satisfied with the bilinear equation under nonzero boundary condition is derived by considering differential relations and a variable transformation. Secondly, two Gram-type solutions are utilized to the construction of multi-breather, high-order rogue wave, and multi-bright-dark soliton solutions. Then the corresponding parameter restrictions of these solutions are given to satisfy with the complex conjugation symmetry. Furthermore, we find that if the parameter $p_{iI}$ takes different values, the rogue wave solution can be classified as three types of states, such as dark-dark, four-peak and bright-bright high-order rogue wave. If the parameter $c_i$ takes different values, the soliton solution can be classified as three type of states, including the multi-dark, multi-bright-dark and multi-bright solitons. By considering third-type of reduced tau-function to the Hirota’s bilinear equations, we give the collisions between the high-order rogue wave and the multi-bright-dark solitons on constant ($N$ is positive even) or periodic background ($N$ is positive odd). In order to understand the dynamics behaviors of the obtained solutions better, the various rich patterns are theoretically and graphically analyzed in detail.

The fluid particle dynamics method is employed to study the fluid-solid interaction problem, which can avoid the explicit implementation of fluid particle boundary condition and capture the hydrodynamic interaction well. We solve the incompressible Navier-Stokes equation coupled with the rigid body motion equation in polar or cylindrical coordinates. A pressure stabilization scheme is used to solve the system in polar coordinates for two dimensional case and cylindrical coordinates for three dimensional case. Our objective is to understand numerically the fluid-solid interaction in rotational extrusion flow. We numerically verify the correctness of method presented here and give comparative analysis for different parameters. We present the Jeffery orbit formulation in annular region. Numerical experiments show that the fluid particle dynamics method is reliable and efficient for numerical simulation of particulate flow in cylindrical coordinate system.

In this paper, we extend a diagonalization-based parallel-in-time (PinT) algorithm to the viscoelastic equation. The central difference method is used for spatial discretization, while for temporal discretization, we use the Crank-Nicolson scheme. Then an all-at-once system collecting all the solutions at each time level is formed and solved using a fixed point iteration preconditioned by an $α$-circulant matrix in parallel. Via a rigorous analysis, we find that the spectral radius of the iteration matrix is uniformly bounded by $α/(1 − α),$ independent of the model parameters (the damping coefficient $\varepsilon$ and the wave velocity $\sqrt{\gamma}$) and the discretization parameters (the time step $\tau$ and the spatial mesh size $h$). Unlike the classical wave equation with Dirichlet boundary condition where the upper bound $α/(1 − α)$ is very sharp, we find that the occurrence of the damping term $−\varepsilon∆y_t,$ as well as the large final time $T,$ leads to even faster convergence of the algorithm, especially when $α$ is not very small. We illustrate our theoretical findings with several numerical examples.

Numerical methods for the nonlinear Dirac equation (NDE) in the massless nonrelativistic regime are considered. In this regime, the equation contains a small dimensionless parameter $0 <\varepsilon≤ 1,$ and its solution is highly oscillatory in time. We present and analyze traditional numerical schemes for the NDE, including finite difference methods, time-splitting methods and exponential integrators. Error analysis indicates that all these methods require an $\varepsilon$-dependent time-step size to achieve an optimal convergence order. Utilizing an operator splitting technique, we propose a uniformly accurate (UA) scheme. The scheme enables first-order convergence in time for all $\varepsilon ∈ (0, 1]$ without restrictions on time-step size. Error estimates for the UA scheme are rigorously established and numerical results confirm the properties of the method.

Although physics-informed neural networks (PINNs) have been successfully applied in a wide variety of science and engineering fields, they can fail to accurately predict the underlying solution in slightly challenging convection-diffusion-reaction problems. In this paper, we investigate the reason of this failure from a domain distribution perspective, and identify that learning multi-scale fields simultaneously makes the network unable to advance its training and easily get stuck in poor local minima. We show that the widespread experience of sampling more collocation points in high-loss regions hardly help optimize and may even worsen the results. These findings motivate the development of a novel curriculum learning method that encourages neural networks to prioritize learning on easier non-layer regions while downplaying learning on harder regions. The proposed method helps PINNs automatically adjust the learning emphasis and thereby facilitates the optimization procedure. Numerical results on typical benchmark equations show that the proposed curriculum learning approach mitigates the failure modes of PINNs and can produce accurate results for very sharp boundary and interior layers. Our work reveals that for equations whose solutions have large scale differences, paying less attention to high-loss regions can be an effective strategy for learning them accurately.

A non-stationary Stokes equation coupled with an evolution equation of temperature field is studied. Boundary conditions for velocity and temperature fields contain the generalized Clarke gradient. The corresponding variational formulation is governed by a system of hemivariational inequalities. The existence and uniqueness of a weak solution is proved by employing Banach fixed point theorem and hemivariational inequalities. Besides, a fully-discrete problem for this system of hemivariational inequalities is given and error estimates are derived.

We consider a shape reconstruction inverse problem constrained by a semi-linear elliptic interface problem in reaction diffusion. The existence of the model is shown. We perform shape sensitivity analysis and propose two numerical optimization algorithms based on the distributed shape gradient. The first algorithm allows shape changes and the second algorithm uses a level set method allowing shape and topological changes. Numerical results are presented to verify effectiveness of the algorithms.

A new delay estimation model using a round trip time as a weight of the asymmetry of each host pair is developed. It improves the estimation accuracy and is suitable for complex wide area network architecture. For large-scale scenarios in practice, we design a symmetric Gauss-Seidel alternating direction method of multipliers. It significantly reduces memory consumption and computational cost. Numerical experiments demonstrate the accuracy and efficiency of the model and algorithm.

In this paper, we are concerned with solving variational inequalities on Hadamard manifolds, the curvature of which is bounded from below. The underlying vector field is assumed to be continuous and pseudomonotone. By combining the hyperplane projection method and the inertial extrapolation technique, a Halpern-type method is proposed. Under some mild assumptions, global convergence of the proposed algorithm is established. Numerical experiments are reported to show the efficiency of the proposed algorithm.