Salinger, A. G.; Burroughs, E. A.; Pawlowski, R. P.; Phipps, E. T.; Romero, L. A. Bifurcation tracking algorithms and software for large scale applications. (English) Zbl 1076.65118 Int. J. Bifurcation Chaos Appl. Sci. Eng. 15, No. 3, 1015-1032 (2005). Summary: We present the set of bifurcation tracking algorithms which have been developed in the LOCA software library to work with large scale application codes that use fully coupled Newton’s method with iterative linear solvers. Turning point (fold), pitchfork, and Hopf bifurcation tracking algorithms based on Newton’s method have been implemented, with particular attention to the scalability to large problem sizes on parallel computers and to the ease of implementation with new application codes. The ease of implementation is accomplished by using block elimination algorithms to solve the Newton iterations of the augmented bifurcation tracking systems. The applicability of such algorithms for large applications is in doubt since the main computational kernel of these routines is the iterative linear solve of the same matrix that is being driven singular by the algorithm. To test the robustness and scalability of these algorithms, the LOCA library has been interfaced with the MPSalsa massively parallel finite element reacting flows code. A bifurcation analysis of an 1.6 million unknown model of 3D Rayleigh-Bénard convection in a \(5\times 5\times 1\) box is successfully undertaken, showing that the algorithms can indeed scale to problems of this size while producing solutions of reasonable accuracy. Cited in 10 Documents MSC: 65P30 Numerical bifurcation problems 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems 37M20 Computational methods for bifurcation problems in dynamical systems 65Y15 Packaged methods for numerical algorithms 76R05 Forced convection 76M10 Finite element methods applied to problems in fluid mechanics Keywords:numerical examples; bifurcation tracking algorithms; LOCA software library; Newton’s method; Turning point; pitchfork; Hopf bifurcation; finite element; Rayleigh-Bénard convection Software:P - ARPACK; DDE-BIFTOOL; ARPACK; MATCONT; LOCA; AztecOO; Aztec PDFBibTeX XMLCite \textit{A. G. Salinger} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 15, No. 3, 1015--1032 (2005; Zbl 1076.65118) Full Text: DOI References: [1] Burroughs E. 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