Local exponential divergence plot and optimal embedding of a chaotic time-series

we propose here a local exponential divergence plot which is capable of providing a new means of characterizing chaotic time series. The suggested plot defines a time dependent exponent LAMBDA and a ''plus'' exponent LAMBDA+ which serves as a criterion for estimating simultaneously the minimal acceptable embedding dimension, the proper delay time and the largest Lyapunov exponent.

A perturbational h4 exponential finite-difference scheme for the convective diffusion equation

A perturbational h4 compact exponential finite difference scheme with diagonally dominant coefficient matrix and upwind effect is developed for the convective diffusion equation. Perturbations of second order are exerted on the convective coefficients and source term of an h2 exponential finite difference scheme proposed in this paper based on a transformation to eliminate the upwind effect of the convective diffusion equation. Four numerical examples including one- to three-dimensional model equations of fluid flow and a problem of natural convective heat transfer are given to illustrate the excellent behavior of the present exponential schemes, the h4 accuracy of the perturbational scheme is verified using double precision arithmetic.

H4 exponential finite-difference scheme for convective diffusion-equations

Perturbations are applied to the convective coefficients and source term of a convection-diffusion equation so that second-order corrections may be applied to a second-order exponential scheme. The basic Structure of the equations in the resulting fourth-order scheme is identical to that for the second order. Furthermore, the calculations are quite simple as the second-order corrections may be obtained in a single pass using a second-order scheme. For one to three dimensions, the fourth-order exponential scheme is unconditionally stable. As examples, the method is applied to Burgers' and other fluid mechanics problems. Compared with schemes normally used, the accuracies are found to be good and the method is applicable to regions with large gradients.

A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations.

The effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods. The problem with this representation is that the volume force exhibits non-physical oscillations in moving boundary simulations. A smoothing technique for discrete delta functions has been developed in this paper to suppress the non-physical oscillations in the volume forces. We have found that the non-physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions. It has been shown that the smoothed discrete delta functions constructed in this paper have one-order higher derivative than the regular ones. Moreover, not only the smoothed discrete delta functions satisfy the first two discrete moment conditions, but also their derivatives satisfy one-order higher moment condition than the regular ones. The smoothed discrete delta functions are tested by three test cases: a one-dimensional heat equation with a moving singular force, a two-dimensional flow past an oscillating cylinder, and the vortex-induced vibration of a cylinder. The numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non-physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations.