The Shortley–Weller scheme for variable coefficient two-point boundary value problems and its application to tumor growth problem with heterogeneous microenvironment

The first half of this work develops and analyzes the Shortley–Weller scheme (or Ghost-Fluid method with quadratic extrapolation) for a two-point boundary value problem with variable coefficients, where the boundary points are not on the uniform mesh. We prove that the local truncation error is first order convergent near the boundary, but the solution is third order accurate near the boundary and second order accurate away from the boundary. The second half of this work applies this numerical scheme to investigate the tumor growth problems in heterogeneous microenvironment. We discover that the classic Darcy's law tumor model can capture the chemotaxis property using variable nutrient diffusion rate and the haptotaxis mechanism through the variable extracellular matrix (ECM) permeability. Specifically, the tumor tends to move to the regions with higher diffusion rate or lower ECM permeability.