Numerical Solution of Anisotropic Diffusion with Localized Source Using Euler Scheme and Finite Element Method

Abstract

This study investigates the numerical modeling of two-dimensional anisotropic diffusion processes involving a spatially localized and temporally limited energy or thermal source. The governing model is formulated as a parabolic partial differential equation, discretized in space using the Finite Element Method (FEM) with linear triangular elements, and in time using both explicit and implicit Euler integration schemes. To ensure spatial accuracy, a dense mesh configuration is employed, which has been shown to produce smooth and representative solution distributions. Simulation results demonstrate that the implicit Euler method exhibits superior numerical stability across various time step sizes, whereas the explicit method requires significantly smaller time steps to remain stable. Analysis of the transient regime reveals that the numerical solution gradually converges toward a steady-state configuration once the source is deactivated. These findings confirm that the combination of FEM with implicit time integration and dense meshing is effective in capturing the spatiotemporal dynamics of anisotropic diffusion processes with localized sources, a phenomenon relevant to thermal analysis, anisotropic materials, and environmental modeling.