Analysis and Computation for a Fluid Mixture Model of Tissue Deformations

Abstract

A fluid mixture model of tissue deformations in one and two dimensions has been studied in this dissertation. The model is a mixed system of nonlinear hyperbolic and elliptic partial differential equations with interfaces. Both theoretical and numerical analysis are presented. We found the relationship between physical parameters and the resulting pattern of tissue deformations via linear stability analysis. Several numerical experiments support our theoretical analysis.

The solution of the system exhibits non-smoothness and discontinuities at the interfaces. The conventional high order finite difference methods (FDM), such as the WENO scheme and TVD Runge Kutta method, for the hyperbolic equation, coupled with the central FDM for the elliptic equation, give spurious oscillations near the interfaces in our problem. By enforcing the jump conditions across the interfaces, our approach, the immersed interface method (IIM), eliminates non-physical oscillations, improves the accuracy of the solution, and maintains the sharp interface as time evolves.

The IIM has been applied to solve a one dimensional linear advection equation with discontinuous initial conditions. By building the jump conditions into a conventional finite difference method, the Lax-Wendroff method, solutions of second order accuracy are observed. The IIM showed its robustness in solving the linear advection equation with nonhomogeneous jump conditions across the moving interface.

The two dimensional fluid mixture model has been derived asymptotically from the three dimensional model so that the thickness of the gel is taken into account. Many numerical examples have been completed using Clawpack and qualitatively reasonable numerical solutions have been obtained.