Tumor growth simulation using the multigrid method

Abstract

A multigrid method is applied to accelerate the numerical solution of a nonlinear coupled system of partial differential equations modeling avascular tumor growth. The model describes the interaction between tumor cell density, extracellular matrix, matrix-degrading enzymes, and their inhibitors, incorporating diffusion, proliferation, degradation, and haptotaxis mechanisms. The governing system consists of reaction–diffusion–haptotaxis equations defined in a two-dimensional domain, resulting in large-scale sparse linear systems after discretization. The equations are discretized using the finite difference method in space and the Crank–Nicolson scheme in time, and the resulting systems are solved using Gauss–Seidel Red–Black smoother within a V-cycle multigrid framework. Code verification is performed using the method of manufactured solutions, confirming second-order convergence. Computational results show that the multigrid approach significantly reduces processing time compared to single-grid methods, exhibiting near-linear computational complexity and increasing speed-up factors with mesh refinement. Simulations of a realistic tumor growth scenario reproduce key features such as tumor invasion, extracellular matrix degradation, enzyme dynamics, and haptotaxis-driven migration. These results demonstrate that the multigrid approach enables efficient tumor growth simulations on refined meshes with significantly reduced computational time.