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We extend the diffuse interface model developed in Wise et al. (2008) to study nonlinear tumor growth in 3-D. Extensions include the tracking of multiple viable cell species populations through a continuum diffuse-interface method, onset and aging of discrete tumor vessels through angiogenesis, and incorporation of individual cell movement using a hybrid(More)
We develop a thermodynamically consistent mixture model for avascular solid tumor growth which takes into account the effects of cell-to-cell adhesion, and taxis inducing chemical and molecular species. The mixture model is well-posed and the governing equations are of Cahn-Hilliard type. When there are only two phases, our asymptotic analysis shows that(More)
Globally convergent, probability-one homotopy methods have proven to be very effective for finding all the isolated solutions to polynomial systems of equations. After many years of development, homotopy path trackers based on probability-one homotopy methods are reliable and fast. Now, theoretical advances reducing the number of homotopy paths that must be(More)
We present efficient, second-order accurate and adaptive finite-difference methods to solve the regularized, strongly anisotropic Cahn–Hilliard equation in 2D and 3D. When the surface energy anisotropy is sufficiently strong, there are missing orientations in the equilibrium level curves of the diffuse interface solutions, corresponding to those missing(More)
We construct unconditionally stable, unconditionally uniquely solvable, and second-order accurate (in time) schemes for gradient flows with energy of the form Ω F (∇φ(x)) + 2 2 |∆φ(x)| 2 dx. The construction of the schemes involves the appropriate combination and extension of two classical ideas: (i) appropriate convex-concave decomposition of the energy(More)
We present unconditionally stable and convergent numerical schemes for gradient flows with energy of the form R Ω " F (∇φ(x)) + ǫ 2 2 |∆φ(x)| 2 " dx. The construction of the schemes involves an appropriate extension of Eyre's idea of convex-concave decomposition of the energy functional. As an application , we derive unconditionally stable and convergent(More)