The Poisson equation on the unit disk: a multigrid solver using polar coordinates
Applied Mathematics and Computation
Multigrid methods for the solution of Poisson's equation in a thick spherical shell
SIAM Journal on Scientific and Statistical Computing
A front-tracking method for viscous, incompressible, multi-fluid flows
Journal of Computational Physics
Total variation diminishing Runge-Kutta schemes
Mathematics of Computation
Three-Dimensional Front Tracking
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
An immersed boundary method with formal second-order accuracy and reduced numerical viscosity
Journal of Computational Physics
A multigrid tutorial (2nd ed.)
A multigrid tutorial (2nd ed.)
Geometric Level Set Methods in Imaging,Vision,and Graphics
Geometric Level Set Methods in Imaging,Vision,and Graphics
Theory of Self-Reproducing Automata
Theory of Self-Reproducing Automata
Efficient semi-implicit schemes for stiff systems
Journal of Computational Physics
The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains (Frontiers in Applied Mathematics)
A front tracking algorithm for limited mass diffusion
Journal of Computational Physics
Applied Numerical Mathematics
A numerical algorithm for viscous incompressible interfacial flows
Journal of Computational Physics
Journal of Computational Physics
Multigrid anisotropic diffusion
IEEE Transactions on Image Processing
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Growth of developing and regenerative biological tissues of different cell types is usually driven by stem cells and their local environment. Here, we present a computational framework for continuum tissue growth models consisting of stem cells, cell lineages, and diffusive molecules that regulate proliferation and differentiation through feedback. To deal with the moving boundaries of the models in both open geometries and closed geometries (through polar coordinates) in two dimensions, we transform the dynamic domains and governing equations to fixed domains, followed by solving for the transformation functions to track the interface explicitly. Clustering grid points in local regions for better efficiency and accuracy can be achieved by appropriate choices of the transformation. The equations resulting from the incompressibility of the tissue is approximated by high-order finite difference schemes and is solved using the multigrid algorithms. The numerical tests demonstrate an overall spatiotemporal second-order accuracy of the methods and their capability in capturing large deformations of the tissue boundaries. The methods are applied to two biological systems: stratified epithelia for studying the effects of two different types of stem cell niches and the scaling of a morphogen gradient with the size of the Drosophila imaginal wing disc during growth. Direct simulations of both systems suggest that that the computational framework is robust and accurate, and it can incorporate various biological processes critical to stem cell dynamics and tissue growth.