The Shortley-Weller scheme for variable coefficient two-point boundary value problems and its application to tumor growth problem with heterogeneous microenvironment. (English) Zbl 1436.65160
Summary: 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.
MSC:
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 92C17 | Cell movement (chemotaxis, etc.) |
| 92C37 | Cell biology |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 76Z05 | Physiological flows |
| 76S05 | Flows in porous media; filtration; seepage |
Keywords:
Shortley-Weller scheme; variable coefficient; tumor growth; heterogeneous microenvironment; chemotaxis; haptotaxisReferences:
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