MBE
Mathematical model and its fast numerical method for the tumor growth
Hyun Geun Lee Yangjin Kim Junseok Kim
Mathematical Biosciences & Engineering 2015, 12(6): 1173-1187 doi: 10.3934/mbe.2015.12.1173
In this paper, we reformulate the diffuse interface model of the tumor growth (S.M. Wise et al., Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method, J. Theor. Biol. 253 (2008) 524--543). In the new proposed model, we use the conservative second-order Allen--Cahn equation with a space--time dependent Lagrange multiplier instead of using the fourth-order Cahn--Hilliard equation in the original model. To numerically solve the new model, we apply a recently developed hybrid numerical method. We perform various numerical experiments. The computational results demonstrate that the new model is not only fast but also has a good feature such as distributing excess mass from the inside of tumor to its boundary regions.
keywords: conservative Allen--Cahn equation Tumor growth operator splitting method multigrid method.
DCDS-B
An unconditionally stable numerical method for the viscous Cahn--Hilliard equation
Jaemin Shin Yongho Choi Junseok Kim
Discrete & Continuous Dynamical Systems - B 2014, 19(6): 1737-1747 doi: 10.3934/dcdsb.2014.19.1737
We present an unconditionally stable finite difference method for solving the viscous Cahn--Hilliard equation. We prove the unconditional stability of the proposed scheme by using the decrease of a discrete functional. We present numerical results that validate the convergence and unconditional stability properties of the method. Further, we present numerical experiments that highlight the different temporal evolutions of the Cahn--Hilliard and viscous Cahn--Hilliard equations.
keywords: finite-difference method multigrid method. Cahn--Hilliard equation viscous Cahn--Hilliard equation unconditionally stable scheme

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