Krylov methods and determinants for detecting bifurcations in one parameter dependent partial differential equations |
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| Authors: | Bosco García-Archilla Juan Sánchez Carles Simó |
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| Institution: | (1) Departamento de Matemática Aplicada II, Universidad de Sevilla, Escuela Superior de Ingenieros, Camino de los Descubrimientos, s/n, 41092 Sevilla, Spain;(2) Departament de Física Aplicada, Universitat Politècnica de Catalunya, Jordi Girona, 1–3, mòdul B4–B5, Campus Nord, 08034 Barcelona, Spain;(3) Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08071 Barcelona, Spain |
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| Abstract: | In this paper, we study the computation of the sign of the determinant of a large matrix as a byproduct of the preconditioned GMRES method when applied to solve the linear systems arising in the discretization of partial differential equations (PDEs). Numerical experiments are presented where the technique is applied to detect and locate pitchfork and transcritical bifurcation points on a one parameter dependent system. With an appropriate selection of the initial guess in the GMRES method, the technique is shown to detect and accurately locate bifurcation points. We present an analysis that helps to explain the good performance observed in practice. AMS subject classification (2000) 65F40, 15A60, 65P30, 35B60, 65N35, 65F10 |
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| Keywords: | determinants Arnoldi decomposition compact operators in Hilbert spaces spectral methods for PDEs continuation methods bifurcation location |
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