Numerical analysis and simulation of mathematical models for the rice blast fungus

In this thesis we present numerical analysis and simulations of mathematical models relating to the model for the rice blast fungus proposed in [102]. We begin with a computational study of a diffuse interface approximation of surface advection-diffusion equations on evolving surfaces. We study the experimental order of convergence produced by the finite element approximation presented in [44, 63] with added streamline diffusion from [63] and the stability term introduced in [44]. Furthermore we study the instabilities caused by an advection-dominated advection-diffusion equation and we introduce a finite volume approximation of the diffuse interface approximation. We then extend the computational study to include models in which the velocity law of the surface satisfies curve shortening ow. Next we prove optimal error bounds for a semi-discrete finite element approximation for a system consisting of the evolution of a curve evolving by forced curve shortening flow coupled to a reaction-diffusion equation on the evolving curve, such that the curve evolves in a given domain O ? R² and meets the boundary, ?O, orthogonally. We compliment this analysis with error bounds for a fully discrete finite element approximation of curve shortening flow in the same fixed boundary configuration without the reaction-diffusion coupling. We also present numerical experiments and show the experimental order of convergence of the approximations that we analysed. Finally we derive a diffuse interface approximation to the mathematical model of the rice blast fungus presented in [102] and present numerical simulations that are consistent with the simulations presented in [102].