A Jacobian-Free Newton–Krylov Method to Solve Tumor Growth Problems with Effective Preconditioning Strategies

Abstract

A Jacobian-free Newton–Krylov (JFNK) method with effective preconditioning strategies is introduced to solve a diffusion-based tumor growth model, also known as the Fisher–Kolmogorov partial differential equation (PDE). The time discretization of the PDE is based on the backward Euler and the Crank–Nicolson methods. Second-order centered finite differencing is used for the spatial derivatives. We introduce two physics-based preconditioners associated with the first- and second-order temporal discretizations. The theoretical time and spatial accuracies of the numerical scheme are verified through convergence tables and graphs that correspond to different computational settings. We present efficiency studies with and without using the preconditioners. Our numerical findings indicate the excellent performance of the newly proposed preconditioning strategies. In other words, when we turn the preconditioners on, the average number of GMRES and the Newton iterations are significantly reduced.