Stability Effects of Finite Difference Methods on a Mathematical Tumor Growth Model

Abstract

Numerical methods used for solving differential equations should be chosen with great care. Not considering numerical aspects such as stability, consistency and wellposed-ness results in erroneous solutions, which in turn will result in incorrect judgments. One of the most important aspects that should be considered is the stability of the numerical method. In this paper, we discuss stability problems of some of the so far proposed finite difference methods for solving the anisotropic diffusion equation, a second order parabolic equation. This equation is used in a variety of applications in physics and image processing. Here, we focus on its usage in formulating brain tumor growth using the Diffusion Weighted Imaging (DWI) technique. Our study shows that the commonly used chain rule method to discretize the diffusion equation is unstable. We propose a new 3D stable discretization method with its stability conditions to solve the diffusion equation. The new method uses directional discretization and forward differences. We also extend standard discretization method to 3D. The theoretical and practical comparisons of the three methods both on synthetic and real patient data show that while chain rule model is always unstable and standard discretization is unstable in theory, our proposed directional discretization is stable both in theory and practice.