It is well-known that artificial boundary conditions are crucial for the efficient and accurate computations of wavefields on unbounded domains. In this paper, we investigate stability analysis for the wave equation coupled with the first and the second order absorbing boundary conditions. The computational scheme is also developed. The approach allows the absorbing boundary conditions to be naturally imposed, which makes it easier for us to construct high order schemes for the absorbing boundary conditions. A third-order Lagrange finite element method with mass lumping is applied to obtain the spatial discretization of the wave equation. The resulting scheme is stable and is very efficient since no matrix inversion is needed at each time step. Moreover, we have shown both abstract and explicit conditional stability results for the fully-discrete schemes. The results are helpful for designing computational parameters in computations. Numerical computations are illustrated to show the efficiency and accuracy of our method. In particular, essentially no boundary reflection is seen at the artificial boundaries.

The aim of this paper is to solve numerically the inverse problem of reconstructing small amplitude perturbations in the magnetic permeability of a dielectric material from partial or total dynamic boundary measurements. Our numerical algorithm is based on the resolution of the time-dependent Maxwell equations, an exact controllability method and Fourier inversion for localizing the perturbations. Two-dimensional numerical experiments illustrate the performance of the reconstruction method for different configurations even in the case of limited-view data.

We propose a non-traditional finite element method with non-body-fitting grids to solve the matrix coefficient elliptic equations with sharp-edged interfaces. All possible situations that the interface cuts the grid are considered. Both Dirichlet and Neumann boundary conditions are discussed. The coefficient matrix data can be given only on the grids, rather than an analytical function. Extensive numerical experiments show that this method is second order accurate in the $L^∞$ norm.

In this paper, we analyze a compact finite difference scheme for computing a coupled nonlinear Schrödinger equation. The proposed scheme not only conserves the total mass and energy in the discrete level but also is decoupled and linearized in practical computation. Due to the difficulty caused by compact difference on the nonlinear term, it is very hard to obtain the optimal error estimate without any restriction on the grid ratio. In order to overcome the difficulty, we transform the compact difference scheme into a special and equivalent vector form, then use the energy method and some important lemmas to obtain the optimal convergent rate, without any restriction on the grid ratio, at the order of $O(h^4+τ^2)$ in the discrete $L^∞$-norm with time step $τ$ and mesh size $h$. Finally, numerical results are reported to test our theoretical results of the proposed scheme.

In this paper, a new operator splitting scheme is introduced for the numerical solution of the incompressible Navier-Stokes equations. Under some mild regularity assumptions on the PDE solution, the stability of the scheme is presented, and error estimates for the velocity and the pressure of the proposed operator splitting scheme are given.

A new framework based on the curved Riemannian manifold is proposed to calculate the numerical solution of the Lyapunov matrix equation by using a natural gradient descent algorithm and taking the geodesic distance as the objective function. Moreover, a gradient descent algorithm based on the classical Euclidean distance is provided to compare with this natural gradient descent algorithm. Furthermore, the behaviors of two proposed algorithms and the conventional modified conjugate gradient algorithm are compared and demonstrated by two simulation examples. By comparison, it is shown that the convergence speed of the natural gradient descent algorithm is faster than both of the gradient descent algorithm and the conventional modified conjugate gradient algorithm in solving the Lyapunov equation.