Iterative Solvers for Diffusion Equations Arising in Models of Calcium Dynamics

Abstract

Reaction-Diffusion equations arise in many models in biomedical computing. One of these areas are models for the calcium dynamic in the heart cells. As the terms in reaction-diffusion equations can be solved separately using operator splitting it is important to have efficient and accurate ways of solving these parts. The focus of this thesis is solving the diffusion equation, particularly diffusion equations arising in models of calcium dynamics. We study a simple test problem and present alternative ways of discretizing this problem. Our focus has been on solving this test problem using an implicit scheme and iterative solvers. In this thesis we present the theory behind these solvers. We have also implemented and tested these solvers using a serial implementation in MATLAB and compared their performance to an explicit scheme. Based on the results of our serial tests we have implemented two of our iterative solvers using parallel programming in C++. Finally we have attempted to find which of the discussed iterative solvers perform best for our diffusion equation using parallel implementation. It is shown that implicit schemes combined with iterative equation solvers can be an interesting option for solving the diffusion equation.