Numerical methods for stiff reaction-diffusion equations with applications to cardiological modeling

Sophisticated numerical methods must be implemented to solve the large, high dimensional ODE and PDE problems that arise in cardiac electrophysiology. This dissertation examines the nature of these problems and exactly what make them so difficult to solve as well as practical solution strategies that can be successfully applied to the cardiac model and to other problems with similar characteristics.; A basic description of the cardiac model is presented along with an overview of some of the currently used solution techniques. The drawbacks of the currently used methods will motivate the advanced schemes developed here. The general framework for solving the reaction-diffusion PDE is operator splitting which gives two subproblems: a system of ODEs and a parabolic PDE. Although others have used this method, the splitting error was unknown and the splitting time step chosen certainly was not optimal. An error analysis will be made for the cardiac model to show the accuracy even with a large splitting step to allow for flexibility in choosing ODE and PDE time steps. In particular, it will allow small adaptive time stepping to be used for the ODE while a much larger time step is used for the PDE.; It will be shown that the cardiac ODE suffers from two serious complications: stiffness and steep gradients in the solution. A highly efficient implicit adaptive solver is developed to deal with these issues. The adaptive time step will be based on an error estimate that is efficient for implicit schemes and measured using a novel norm that is weighted in the components that change the most rapidly.; Reduced manifold methods are explored as another way to solve the cardiac model and stiff systems of ODEs in general. Stiff ODES define a slow invariant manifold (SIM) in phase space which can be exploited to approximate the solution. A practical reduced manifold method is the intrinsic low dimensional manifold (ILDM) method which approximates the SIM. The full ODE is solved until the solution approaches the manifold; then, an algebraic equation describing the manifold along with a reduced ODE on the manifold are solved. A novel, modified ILDM approach is developed which generalizes the ILDM method to any stiff non-linear system.