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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Semi-implicit Krylov deferred correction methods for differential algebraic equations


Authors: Sunyoung Bu, Jingfang Huang and Michael L. Minion
Journal: Math. Comp. 81 (2012), 2127-2157
MSC (2010): Primary 65B10, 65F10, 65L20, 65L80, 65N35
Published electronically: April 26, 2012
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Abstract: In the recently developed Krylov deferred correction (KDC) methods for differential algebraic equation initial value problems (Huang, Jia,
Minion, 2007), a Picard-type collocation formulation is preconditioned using low-order time integration schemes based on spectral deferred correction (SDC), and the resulting system is solved efficiently using Newton-Krylov methods. KDC methods have the advantage that methods with arbitrarily high order of accuracy can be easily constructed which have similar computational complexity as lower order methods. In this paper, we investigate semi-implicit KDC (SI-KDC) methods in which the stiff component of the preconditioner is treated implicitly and the non-stiff parts explicitly. For certain types of problems, such a semi-implicit treatment can significantly reduce the computational cost of the preconditioner compared to fully implicit KDC (FI-KDC) methods. Preliminary analysis and numerical experiments show that the convergence of Newton-Krylov iterations in the SI-KDC methods is similar to that in FI-KDC, and hence the SI-KDC methods offer a reduction in overall computational cost for such problems.


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Additional Information

Sunyoung Bu
Affiliation: Department of Mathematics, University of North Carolina, CB #3250, Phillips Hall, Chapel Hill North Carolina 27599-3250
Address at time of publication: Institute of Mathematical Sciences, Ewha Womans University, Seoul 120-750, Korea
Email: syboo@ewha.ac.kr

Jingfang Huang
Affiliation: Department of Mathematics, University of North Carolina, CB #3250, Phillips Hall, Chapel Hill North Carolina 27599-3250
Email: huang@email.unc.edu

Michael L. Minion
Affiliation: Department of Mathematics, University of North Carolina, CB #3250, Phillips Hall, Chapel Hill North Carolina 27599-3250
Email: minion@email.unc.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02564-6
PII: S 0025-5718(2012)02564-6
Keywords: Differential algebraic equations, Krylov deferred correction, semi-implicit schemes, preconditioner
Received by editor(s): July 21, 2010
Received by editor(s) in revised form: March 18, 2011
Published electronically: April 26, 2012
Additional Notes: The work of Bu was partially supported by the Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0028298). Part of the work was done while Bu was a visiting member of the Institute for Mathematics and Its Applications at the University of Minnesota.
The work of Huang and Bu was supported by NSF grants 0811130 and 0941235
The work of Minion was supported by NSF grant 0854961
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.