Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
Shape Modeling with Front Propagation: A Level Set Approach
IEEE Transactions on Pattern Analysis and Machine Intelligence
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
A simple level set method for solving Stefan problems
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
Numerical Linear Algebra for High Performance Computers
Numerical Linear Algebra for High Performance Computers
A second-order-accurate symmetric discretization of the Poisson equation on irregular domains
Journal of Computational Physics
Modelling the role of cell-cell adhesion in the growth and development of carcinomas
Mathematical and Computer Modelling: An International Journal
A level-set method for interfacial flows with surfactant
Journal of Computational Physics
Journal of Computational Physics
A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth
Journal of Scientific Computing
Journal of Computational Physics
Computational Modeling of Solid Tumor Growth: The Avascular Stage
SIAM Journal on Scientific Computing
Journal of Computational Physics
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We develop an algorithm for the evolution of interfaces whose normal velocity is given by the normal derivative of a solution to an interior Poisson equation with curvature-dependent boundary conditions. We improve upon existing techniques and develop new finite difference, ghost fluid/level set methods to attain full second-order accuracy for the first time in the context of a fully coupled, nonlinear moving boundary problem with geometric boundary conditions (curvature). The algorithm is capable of describing complex morphologies, including pinchoff and merger of interfaces. Our new methods include a robust, high-order boundary condition-capturing Poisson solver tailored to the interior problem, improved discretizations of the normal vector and curvature, a new technique for extending variables beyond the zero level set, a new orthogonal velocity extension technique that is both faster and more accurate than traditional PDE-based approaches, and a new application of Gaussian filter technology ordinarily associated with image processing. While our discussion focuses on two-dimensional problems, the techniques presented can be readily extended to three dimensions. We apply our techniques to a model for tumor growth and present several 2D simulations. Our algorithm is validated by comparison to an exact solution, by resolution studies, and by comparison to the results of a spectrally accurate method boundary integral method (BIM). We go beyond morphologies that can be described by the BIM and present accurate simulations of complex, evolving tumor morphologies that demonstrate the repeated encapsulation of healthy tissue in the primary tumor domain - an effect seen in the growth of real tumors.