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<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>
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