Extending BACOLI to solve multi-scale problems

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.