Efficient Iterative Solution Algorithms For Numerical Models of Multiphase Flow

This thesis is concerned with the development and application of optimally efficient numerical methods for the simulation of vascular tumour growth, based upon the multiphase fluid model introduced by Hubbard and Byrne [57]. This multiphase model involves the flow and interaction of four different, but coupled, phases which are each treated as incompressible fluids. Following a short review of models for tumour growth we describe in detail the model of Hubbard and Byrne [57], and introduce the discretization schemes used. This involves a finite volume scheme to approximate mass conservation and conforming finite element schemes to approximate momentum conservation and a reaction-diffusion equation for the background nutrient concentration. The momentum conservation system is represented as a Stokes-like flow of each phase, with source terms that reflect the phase interactions. It will be demonstrated that the solution of these coupled momentum equations, approximated using a Taylor-Hood finite element method in two dimensions, is the most computationally intensive component of the solution algorithm. The nonlinear system arising from the nutrient equation is the second most computationally expensive component. The solvers presented in this work for the discretized systems are based on preconditioned Krylov methods. Algebraic multigrid (AMG) preconditioner and a novel block preconditioner are used with Krylov methods for solving the linear systems arising from the nutrient equation at each Newton step and from the momentum equation, respectively. In each case these are shown to be very efficient algorithms: when the preconditioning strategies are applied to practical problems, the CPU time and memory are demonstrated to scale almost linearly with the problem size. Finally, the basic multiphase tumour model is extended to consider drug delivery and the inclusion of additional phases. To solve this extended model our preconditioning strategy is extended to cases with more than four phases. This is again demonstrated to perform optimally.