| Abstract: |
| Mechanical-based tumor growth models derived from the incompressible limit face significant challenges in capturing the emergence of a necrotic core. This talk presents a unified free boundary model that resolves this by formulating the internal pressure as an obstacle problem, where the necrotic core is naturally defined as the coincidence set. First, we will establish the analytical foundations of this model, detailing semi-analytical solutions that quantify transitional phases of necrotic core development and proving the existence of traveling wave solutions that incorporate non-zero outer cell densities. Second, we will address the fundamental numerical challenge of simulating this system: the inner necrotic interface lacks the explicit advection structure seen on the outer boundary. To overcome this, we introduce a stabilized predictor-corrector strategy coupled with Boundary Integral (BI) and Kernel-Free Boundary Integral (KFBI) solvers on Cartesian grids. Together, these analytical and numerical frameworks allow us to accurately capture complex geometric evolution and the topological transitions of necrotic core nucleation. |
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