Extending BACOLI to solve multi-scale problems
Date
2014-11-12
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Degree Level
Masters
Abstract
The BACOLI package is a numerical software package for solving parabolic partial differential
equations in one spatial dimension. It implements a B-spline collocation method for the spatial
discretization of a system of partial differential equations. The resultant ordinary differential equations
together with the boundary conditions form a system of differential-algebraic equations. The
differential-algebraic equations are then solved using the DASSL solver. The BACOLI software package
features adaptive error control in the temporal and spatial domains. The estimate of the temporal
error is controlled through the DASSL solver. The estimate of the spatial error is controlled based
on the difference between two solutions computed in the BACOLI software package. This difference
gives an estimation of the error. If this error estimate does not meet the user-supplied tolerance,
then the spatial mesh is changed.
The BACOLI software package can only solve parabolic partial differential equations that depend
on spatial derivatives. In this thesis, the BACOLI software package is modified to solve a broader
spectrum of problems. In fact, after some modifications, the extended BACOLI software package can
solve systems of parabolic partial differential equations and time-dependent equations that do not
depend on spatial derivatives. We apply this extended software package to solve the monodomain
model of cardiac electrophysiology.
The monodomain model is a multi-scale mathematical model for the evolution of the electrical
potential in cardiac tissue that couples the ionic currents at the cellular scale with their propagation
at the tissue scale. Because of their local nature, the mathematical models of a single cell have no
dependency on spatial derivatives whereas the models at the tissue level do.
The heart models considered in our numerical experiments use various cardiac cell models. We
find that solving the heart models through the extended BACOLI software package, in some cases,
leads to a speed-up in comparison with the Chaste software package, which is a powerful, widely
used, and well-respected software package for heart simulation.
Description
Keywords
BACOLI, mathematical modelling
Citation
Degree
Master of Science (M.Sc.)
Department
Mathematics and Statistics
Program
Mathematics