Convergence and Numerical Solution of a Model for Tumor Growth

Abstract

In this paper, we show the application of the meshless numerical method called “Generalized Finite Diference Method” (GFDM) for solving a model for tumor growth with nutrient density, extracellular matrix and matrix degrading enzymes, [recently proposed by Li and Hu]. We derive the discretization of the parabolic–hyperbolic–parabolic–elliptic system by means of the explicit formulae of the GFDM. We provide a theoretical proof of the convergence of the spatial–temporal scheme to the continuous solution and we show several examples over regular and irregular distribution of points. This shows the feasibility of the method for solving this nonlinear model appearing in Biology and Medicine in complicated and realistic domains.

A technique for generating adapted discretizations to solve partial differential equations with the generalized finite difference method

“…The influence of key parameters of the method was studied by Benito et al 6 In the last years, the development of researches involving higher‐order approximations has gained strength 7–11 . The different applications of the GFDM in current engineering problems show the versatility of this method 12–20 …”

“…[7][8][9][10][11] The different applications of the GFDM in current engineering problems show the versatility of this method. [12][13][14][15][16][17][18][19][20] The discretization of the domain when applying GFDM to solve a partial differential equation has been addressed in many ways. It can be discretized regularly whenever possible and irregularly only in regions where it cannot be done otherwise.…”