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.

这项工作的前半部分针对具有可变系数的两点边值问题开发了Shortley-Weller方案（或采用二次外推法的Ghost-Fluid方法）并进行了分析，其中边界点不在均匀网格上。我们证明了局部截断误差在边界附近是一阶收敛的，但是解在边界附近是三阶精确的，而在边界附近则是二阶精确的。这项工作的后半部分应用此数值方案来研究异质微环境中的肿瘤生长问题。我们发现经典的达西定律肿瘤模型可以使用可变的营养物扩散速率和通过可变的细胞外基质（ECM）通透性的触觉机理来捕获趋化特性。特别，