Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
A fast level set based algorithm for topology-independent shape modeling
Journal of Mathematical Imaging and Vision - Special issue on topology and geometry in computer vision
A variational level set approach to multiphase motion
Journal of Computational Physics
Total variation diminishing Runge-Kutta schemes
Mathematics of Computation
The fast construction of extension velocities in level set methods
Journal of Computational Physics
SIAM Journal on Scientific Computing
A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)
Journal of Computational Physics
A PDE-based fast local level set method
Journal of Computational Physics
A boundary condition capturing method for Poisson's equation on irregular domains
Journal of Computational Physics
Weighted ENO Schemes for Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
Level set methods: an overview and some recent results
Journal of Computational Physics
A second-order-accurate symmetric discretization of the Poisson equation on irregular domains
Journal of Computational Physics
From linear to nonlinear iterative methods
Applied Numerical Mathematics
A Level Set Approach for the Numerical Simulation of Dendritic Growth
Journal of Scientific Computing
Journal of Computational Physics
Discretization of Dirac delta functions in level set methods
Journal of Computational Physics
The computational efficiency of non-linear frequency domain methods
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Simulating the Hallmarks of Cancer
Artificial Life
Journal of Computational Physics
Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces
Journal of Computational Physics
Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces
Journal of Computational Physics
Journal of Computational Physics
An adaptive multigrid algorithm for simulating solid tumor growth using mixture models
Mathematical and Computer Modelling: An International Journal
A weak formulation for solving elliptic interface problems without body fitted grid
Journal of Computational Physics
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In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics--an effect observed in real tumor growth.