Sistemas do Tipo Difusão-Reação e Preservação de Pontos Singulares

ABSTRACT

Motivated by recent computer graphics applications, this work presents a theoretical and computational study of diffusion-reaction systems based on Gradient Vector Flow (GVF), focusing on the behavior of GVF concerning the singularities of the initial vector field. The theoretical study starts from a local analysis, regardless of boundary conditions. Then, the boundary condition at infinity is assumed, and Fourier analysis is used to establish sufficient conditions for preserving the singular point. Finally, a compact domain with rectangular geometry is assumed. The preservation of a singular point concerning the boundary condition is analyzed using a method for solving partial differential equations (PDEs) based on Haar wavelets. We have also developed an implementation of a direct method for the GVF stationary equation based on finite differences (DF) to compare with the traditional explicit Euler solution with respect to singularity. The influence of vorticity on the problem of interest is discussed using the streamlines function and the Helmholtz equation. In the computational experiments, we consider two boundary conditions, two types of singularities, and the three numerical methods (explicit Euler, finite differences for the stationary equation, and wavelets) to verify the theoretical results obtained.

Keywords:
diffusion-reaction; Gradient Vector Flow; singularities; Haar wavelets