Abstract
A matrix-free monolithic homotopy continuation algorithm is developed which allows for approximate numerical solutions to nonlinear systems of equations without the need to solve a linear system, thereby avoiding the formation of any Jacobian or preconditioner matrices. The algorithm can converge from an arbitrary starting guess, under suitable conditions, and can give a sufficiently accurate approximation to the converged solution such that a rapid locally convergent method such as Newton’s method will converge successfully. Several forms of the algorithm are presented, as are augmentations to the algorithms which can lead to improved efficiency or stability. The method is validated and the stability and efficiency are investigated numerically based on a computational aerodynamics flow solver.
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Acknowledgements
The authors gratefully acknowledge financial assistance from the Natural Science and Engineering Research Council (NSERC), the Canada Research Chairs program, and the University of Toronto. Computations were performed on the GPC supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto.
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Brown, D.A., Zingg, D.W. Matrix-free monolithic homotopy continuation with application to computational aerodynamics. Numer Algor 78, 1303–1320 (2018). https://doi.org/10.1007/s11075-017-0424-8
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DOI: https://doi.org/10.1007/s11075-017-0424-8