No abstract

In this article, we present the so-called optimal, nearly analytic, discrete method (ONADM), which is an improved version of the NADM proposed recently (Yang et al., 2003a). We compare numerically the error of the ONADM with those of the NADM and other finite-difference methods for 1D and 2D cases, and give wavefield modeling in 2D isotropic media. We also discuss the validity of the ntimes absorbing boundary condition, when absorbing boundary conditions are incorporated in the ONADM. We show that, compared with the original NADM, the ONADM for the 2D case can significantly reduce storage space and computational cost. The temporal accuracy of the optimal method is also increased from second order in the original NADM to fourth order, and spatial accuracy remains the same as that of the original. Promising numerical results suggest that the ONADM is suitable for large-scale numerical modeling, as it can suppress effectively numerical dispersion caused by discretizing the wave equations when too coarse grids are used.

We present a new numerical method for elastic wave modeling in 3D isotropic and anisotropic media, which is called the 3D optimal nearly analytic discrete method (ONADM) in this article. This work is an extension of the 2D ONADM (Yang et al., 2004) that models acoustic and elastic waves propagating in 2D media. The formulation is derived by using a multivariable truncated Taylor series expansion and high-order interpolation approximations. Our 3D ONADM enables wave propagation to be simulated in three dimensions through isotropic and anisotropic models. Promising numerical results show that the error of the ONADM for the 3D case is less than those of the conventional finite-difference (FD) method and the so-called Lax-Wendroff correction (LWC) schemes, measured quantitatively by the rootmean-square deviation from analytical solution. The seismic wave fields in the 3D isotropic and anisotropic media are simulated and compared with those obtained by using the fourth-order LWC method and exact solutions for the acoustic wave case. We show that, compared with the conventional FD method and the LWC schemes, the ONADM for the 3D case can reduce significantly the storage space and computational costs. Numerical experiments illustrate that the ONADM provides a useful tool for the 3D large-scale isotropic and anisotropic problems and it can suppress effectively numerical dispersions caused by discretizing the 3D wave equations when too-coarse grids are used, which is the same as the 2D ONADM. Numerical modeling also implies that simultaneously using both the wave displacement and its gradients to approximate the high-order derivatives is important for both decreasing the numerical dispersion caused by the discretization of wave equations and compensating the important wave-field information included in wave-displacement gradients.

We study the existence of periodic solutions for the one-dimensional prescribed mean curvature delay equation(d/dt)(x'(t)/1+x't2) +∑i=1naitgxt-τit=pt. By using Mawhin's continuation theorem, a new result is obtained. Furthermore, the nonexistence of periodic solution for the equation is investigated as well.

A direct coupling method is developed for coupling meshless methods such as the MLPG and FEM. The radial point interpolation method with polynomial terms (RPIMp) which lead to a shape function that indeed obeys the Kronecker delta property are used to approximate the trial functions in the MLPG. This important property make the MLPG method could couple the FEM directly as other coupling method must use additional techniques. An electromagnetic problem governed by the Poisson's equation and nonlinear ionized field examples are given in this paper. The results, which are compared with both the exact solutions and solutions obtained from the FEM, show that the method in this paper could couple the MLPG FEM directly without any additional numerical techniques. The paper offer an coupling method by making full use of the MLPG and FEM advantages to deal with engineering problems and more difficult problem geometries.