Abstract
B-spline collocation methods have been shown to be effective for solving systems of parabolic partial differential equations (PDEs). Using B-spline bases for spatial discretization and backward differentiation formulae for temporal discretization, software can be developed that allows full spatio-temporal error control throughout the solution. The software package eBACOLI, which uses \({\mathscr {C}^1}\)-continuous B-splines, has been extended for this approach to work with PDEs in one spatial dimension that have a multi-scale structure like the bidomain model, i.e., parabolic and elliptic PDEs at the macro-scale coupled with ordinary differential equations at the micro-scale. We present numerical results of cardiac bidomain simulations, validating them through comparison with solutions obtained from the software package Nektar++. The performance of eBACOLI and Nektar++ simulations are compared by considering solution times with comparable error with respect to a reference solution, showing that in addition to automatically controlling the error (a feature unavailable in Nektar++), eBACOLI is generally more than an order of magnitude faster than Nektar++ for a given error level.
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Green, K.R., Spiteri, R.J. (2021). Solving Cardiac Bidomain Problems with B-spline Adaptive Collocation. In: Kilgour, D.M., Kunze, H., Makarov, R., Melnik, R., Wang, X. (eds) Recent Developments in Mathematical, Statistical and Computational Sciences. AMMCS 2019. Springer Proceedings in Mathematics & Statistics, vol 343. Springer, Cham. https://doi.org/10.1007/978-3-030-63591-6_28
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