Verification of the order of accuracy of the discretization error in the simulation of tumor growth

Resumo

In Engineering there are several problems to be studied, such as applications in biomedicine, which
are typical problems in Computational Fluid Dynamics (CFD). To solve these problems, numerical methods are
used regardless of complexity, geometry, physical parameters, boundary and initial conditions. Linear or nonlinear
models can be considered to assess both temporal and spatial evolution. However, one of the main disadvantages
of numerical methods is the determination of computational errors associated with their use, in which numerical
solutions can be affected by truncation, iteration, rounding, and programming errors. Although numerical errors

cannot be eliminated, they must be controlled or minimized. The discretization error is considered the most signif-
icant among the sources of numerical error, requiring its analysis. Therefore, this work aims to verify the accuracy

of the discretization error of a one-dimensional model of tumour growth, using a priori and a posteriori estimates
of numerical solutions. We predict the asymptotic behaviour of the discretization error in the a priori estimation.
We estimate the magnitude of the error based on multiple meshes using the Richardson estimator in the a posteriori
estimation. The model used in this work is described by a system of partial differential equations in a transient

regime, with four variables involved in the process of tumour cell invasion, resulting in the description and evolu-
tion of cancer cell density, extracellular matrix (ECM) density, the concentration of matrix degradative enzymes

(MDE) and tissue inhibitors of metalloproteinases (TIMP). To discretize the mathematical model, we used the
finite difference method with Central Difference Scheme (CDS) for spatial discretization and the Crank-Nicolson
method for temporal discretization. The nonlinear terms involved in the model were linearized by applying the

Taylor series expansion. To advance in time, this discretization procedure results in the resolution of a set of alge-
braic equations to be solved with the aid of the iterative Gauss-Seidel method. The simulations are performed with

Dirichlet boundary conditions. We use the manufactured solutions method for code verification and error analysis.