| In this work, I have developed accurate and efficient numerical schemes for partial differential equations arising from physical and biological free boundary problems.;We present a new, adaptive boundary integral method to simulate solid tumor growth in 3-d. We use a reformulation of a classical model that accounts for cell-proliferation, apoptosis, cell-to-cell and cell-to-matrix adhesion. We present simulations of the nonlinear evolution of growing tumors and we compare the results of a linear stability analysis and nonlinear simulations. Numerical results show that nonlinearity results in mode creation and interaction that leads to the formation of dimples and the tumor surface buckles inwards.;To account for additional biophysical processes, we develop a thermodynamically consistent mixture model for avascular solid tumor growth which takes into account the effects of cell-to-cell adhesion, chemotaxis, and haptotaxis. The mixture model is well-posed and the governing equations are of Cahn-Hilliard type. Numerical results illustrate that this model is capable of describing complex invasive patterns observed in experiments.;Finally, we adapt our numerical methods to study solid-liquid phase transitions. We study a crystal growing in an undercooled liquid with isotropic surface tension, with isotropic and anisotropic interface kinetics. We develop an adaptive boundary integral method to simulate the morphological evolution of the growing crystal in 3-d. Numerical simulations reveal that with isotropic interface kinetics, evolution is found to depend strongly on the azimuthal wavenumber in the initial condition. With anisotropic interface kinetics, numerical results suggest a deterministic mechanism for the generation and development of side-branches in dendritic growth. |
