Finite-difference discretization of a one-dimensional diffusion equation with logistic linear reaction for bounded solutions

Abstract

This Thesis focuses on the application of the finite difference method, combined with the Lyapunov method, to solve linear partial differential equations (PDEs) with homogeneous Dirichlet boundary conditions. PDEs are fundamental in the modeling of many physical and engineering phenomena, but their analytical resolution is often complex, if not impossible. Thus, digital methods become essential. We start with an introduction to the basic concepts of PDEs and homogeneous Dirichlet boundary conditions. Next, we detail the finite difference method, a technique that discretizes the continuous domain into a lattice of points and approximates the partial derivatives by finite differences. This method is known for its simplicity of implementation and its effectiveness for problems defined on regular domains. In order to improve the stability and convergence of the finite difference method, we integrate the Lyapunov method. The Lyapunov method, formulated by the matrix equation AX+XB=C, is used to analyze the stability of solutions and optimize numerical schemes. This approach allows the discrete PDE problem to be reformulated into a Lyapunov equation, thereby facilitating numerical solution while ensuring desirable properties such as stability and precision. We then develop the practical implementation of this combined approach, detailing the algorithms and steps required to solve linear PDEs. Case studies and numerical simulations are presented to illustrate the effectiveness of our method. These examples show how the finite difference method, enhanced by Lyapunov analysis, can efficiently solve typical problems while respecting the constraints of homogeneous Dirichlet boundary conditions. Finally, we discuss the advantages and limitations of this combined approach. We also discuss future perspectives for the improvement and extension of these techniques, including the adaptation to nonlinear problems