A Spectral Deferred Correction Method for Solving Cardiac Models

Abstract

<p>Many numerical approaches exist to solving models of electrical activity in the heart. These models consist of a system of stiff nonlinear ordinary differential equations for the voltage and other variables governing channels, with the voltage coupled to a diffusion term. In this work, we propose a new algorithm that uses two common discretization methods, operator splitting and finite elements. Additionally, we incorporate a temporal integration process known as spectral deferred correction. Using these approaches,</p><p>we construct a numerical method that can achieve arbitrarily high order in both space and time in order to resolve important features of the models, while gaining accuracy and efficiency over lower order schemes.</p><p>Our algorithm employs an operator splitting technique, dividing the reaction-diffusion systems from the models into their constituent parts. </p><p>We integrate both the reaction and diffusion pieces via an implicit Euler method. We reduce the temporal and splitting errors by using a spectral deferred correction method, raising the temporal order and accuracy of the scheme with each correction iteration.</p><p> </p><p>Our algorithm also uses continuous piecewise polynomials of high order on rectangular elements as our finite element approximation. This approximation improves the spatial discretization error over the piecewise linear polynomials typically used, especially when the spatial mesh is refined. </p><p>As part of these thesis work, we also present numerical simulations using our algorithm of one of the cardiac models mentioned, the Two-Current Model. We demonstrate the efficiency, accuracy and convergence rates of our numerical scheme by using mesh refinement studies and comparison of accuracy versus computational time. We conclude with a discussion of how our algorithm can be applied to more realistic models of cardiac electrical activity.</p>