Structure of an adaptive grid computational method from the viewpoint of dynamic chaos, Part II: grid addition and probability distribution

Further advancement has been made in understanding the spatial-temporal dynamic structure of an adaptive grid computational method based on the concept of equidistribution. A one-dimensional convection-diffusion equation with source term is used as a model problem. The effects of the number of grid points on the behavior of the grid adaptation are studied. The intermittency scenario of the route to chaos is found to be identifiable to the variations of total number of grid points and free parameters in the weighting function. The spatial evolution of the structure of grid distribution can be observed from a collection of one-dimensional maps. The increase of the total number of grid points does not necessarily stabilize the grid adaptation procedure; the probability distribution function of grid locations indicate that the pattern of grid adaptation can become more complicated, from single-modal to multiple-modal shapes, with increasing number of total grid points. The histograms also reveal that the oscillations of grid locations can appear either as a series of short bursts or as continuously unsettling grid movements. The results presented show some success in unifying the scattered observations to form a more coherent understanding of the phenomena.