In this paper, we generalize the so-called inverse Lax-Wendroff boundary treatment \cite{ilw} for the inflow boundary of a linear hyperbolic problem discretized by the recently introduced central compact schemes. The outflow boundary is treated by the classical extrapolation and a stability analysis for the resulting scheme is provided. To ensure the stability of the considered schemes provided with the chosen boundaries, the G-K-S theory is used, first in the semidiscrete case then in the fully discrete case with the third-order TVD Runge-Kutta time discretization. Afterwards, due to the high algebraic complexity of the G-K-S theory, the stability is analyzed by visualizing the eigenspectrum of the discretized operators. We show in this paper that the results obtained with these two different approaches are perfectly consistent. We also illustrate the high accuracy of the presented schemes on simple test problems.

This paper proposes a neural network approach to efficiently solve nonlinear convex programs with the second-order cone constraints. The neural network model is designed by the generalized FischerBurmeister function associated with second-order cone. We study the existence and convergence of the trajectory for the considered neural network. Moreover, we also show stability properties for the considered neural network, including the Lyapunov stability, the asymptotic stability and the exponential stability. Illustrative examples give a further demonstration for the effectiveness of the proposed neural network. Numerical performance based on the parameter being perturbed and numerical comparison with other neural network models are also provided. In overall, our model performs better than two comparative methods.

Hoogendoorn and Bovy (Transportation Research Part B, 2004, 38(7), 571--592) developed an approach for a pedestrian user-optimal dynamic assignment in continuous time and space. Although their model was proposed for pedestrian traffic, it can also be applied to urban cities. The model is very general, and consists of a conservation law (CL) and a Hamilton-Jacobi-Bellman (HJB) equation that contains a minimum value problem. However, only an isotropic application example was given in their paper. We claim that the HJB equation is difficult to compute numerically in an anisotropic case. To overcome this, we reformulate their model for a dense urban city that is arbitrary in shape and has a single central business district (CBD). In our model, the minimum value problem is only used in the CL portion, and the HJB equation reduces to a Hamilton-Jacobi (HJ) equation for easier computation. The dynamic path equilibrium of our model is proven in a different way from theirs, and a numerical algorithm is also provided to solve the model. Finally, we show a numerical example under the anisotropic case and compare the results with those of the isotropic case.

This is the first of a series of papers on the subject of projection methods for viscous incompressible flow calculations. The purpose of these papers is to provide a thorough understanding of the numerical phenomena involved in the projection methods, particularly when boundaries are present, and point to ways of designing more efficient, robust, and accurate numerical methods based on the primitive variable formulation. This paper contains the following topics: 1. convergence and optimal error estimates for both velocity and pressure up to the boundary; 2. explicit characterization of the numerical boundary layers in the pressure approximations and the intermediate velocity fields; 3. the effect of choosing different numerical boundary conditions at the projection step. We will show that a different choice of boundary conditions gives rise to different boundary layer structures. In particular, the straightforward Dirichlet boundary condition for the pressure leads to O(1) numerical boundary layers in the pressure and deteriorates the accuracy in the interior; and 4. postprocessing the numerical solutions to get more accurate approximations for the pressure.

In this paper, we propose a numerical method to solve isotropic elliptic equations on point cloud by generalizing the point integral method. The idea of the point integral method is to approximate the differential operators by integral operators and discretize the corresponding integral equation on point cloud. The key step is to get the integral approximation. In this paper, with proper kernel function, we get an integral approximation for the elliptic operators with isotropic coefficients. Moreover, the integral approximation has been proved to keep the coercivity of the original elliptic operator. The convergence of the point integral method is also proved.