Efficient implementation of weighted ENO schemes
Journal of Computational Physics
A multigrid tutorial: second edition
A multigrid tutorial: second edition
Multigrid
Numerical Solution of Partial Differential Equations: An Introduction
Numerical Solution of Partial Differential Equations: An Introduction
Journal of Computational Physics
Simulating the Hallmarks of Cancer
Artificial Life
Mathematical Models of Avascular Tumor Growth
SIAM Review
A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth
Journal of Scientific Computing
Journal of Computational Physics
An Energy-Stable and Convergent Finite-Difference Scheme for the Phase Field Crystal Equation
SIAM Journal on Numerical Analysis
Journal of Scientific Computing
Individual-based approaches to birth and death in avascu1ar tumors
Mathematical and Computer Modelling: An International Journal
Mathematical and Computer Modelling: An International Journal
Journal of Scientific Computing
Three-dimensional simulation of unstable gravity-driven infiltration of water into a porous medium
Journal of Computational Physics
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In this paper we give the details of the numerical solution of a three-dimensional multispecies diffuse interface model of tumor growth, which was derived in [S.M. Wise, J.S. Lowengrub, H.B. Frieboes, V. Cristini, Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method, J. Theoret. Biol. 253 (2008) 524-543] and used to study the development of glioma in [H.B. Frieboes, J.S. Lowengrub, S.M. Wise, X. Zheng, P. Macklin, E.L. Bearer, V. Cristini, Computer simulation of glioma growth and morphology, NeuroImage 37 (2007) S59-S70] and tumor invasion in [E.L. Bearer, J.S. Lowengrub, Y.L. Chuang, H.B. Frieboes, F. Jin, S.M. Wise, M. Ferrari, D.B. Agus, V. Cristini, Multiparameter computational modeling of tumor invasion, Cancer Res. 69 (2009) 4493-4501; H.B. Frieboes, F. Jin, Y.L. Chuang, S.M. Wise, J.S. Lowengrub, V. Cristini, Three-dimensional multispecies nonlinear tumor growth-II: tissue invasion and angiogenesis, J. Theoret. Biol. 264 (2010) 1254-1278]. The model has a thermodynamic basis, is related to recently developed mixture models, and is capable of providing a detailed description of tumor progression. It utilizes a diffuse interface approach, whereby sharp tumor boundaries are replaced by narrow transition layers that arise due to differential adhesive forces among the cell-species. The model consists of fourth-order nonlinear advection-reaction-diffusion equations (of Cahn-Hilliard-type) for the cell-species coupled with reaction-diffusion equations for the substrate components. Computing numerical solutions of the model is challenging because the equations are coupled, highly nonlinear, and numerically stiff. In this paper we describe a fully adaptive, nonlinear multigrid/finite difference method for efficiently solving the equations. We demonstrate the convergence of the algorithm and we present simulations of tumor growth in 2D and 3D that demonstrate the capabilities of the algorithm in accurately and efficiently simulating the progression of tumors with complex morphologies.