Tracing the solution surface with folds of a two-parameter system. (English) Zbl 1092.65503
Summary: We describe a special Gauss-Newton method for tracing solution manifolds with singularities of multiparameter systems. First we choose one of the parameters as the continuation parameter, and fix the others. Then we trace one-dimensional solution curves by using continuation methods. Singularities such as folds, simple and multiple bifurcations on each solution curve can be easily detected. Next, we choose an interval for the second continuation parameter, and trace one-dimensional solution curves for certain values in this interval. This constitutes a two-dimensional solution surface. The procedure can be generalized to trace a \(k\)-dimensional solution manifold. Numerical results in 1D, 2D and 3D second-order semilinear elliptic eigenvalue problems given by Lions in a previous paper are reported.
MSC:
| 65H20 | Global methods, including homotopy approaches to the numerical solution of nonlinear equations |
| 35G15 | Boundary value problems for linear higher-order PDEs |
| 47J15 | Abstract bifurcation theory involving nonlinear operators |
Software:
PITCONReferences:
| [1] | DOI: 10.1137/0722020 · Zbl 0567.65029 · doi:10.1137/0722020 |
| [2] | DOI: 10.1137/0724033 · Zbl 0618.65006 · doi:10.1137/0724033 |
| [3] | E. L. Allgower and K. Georg, Numerical Path Following, Handbook of Numerical Analysis 5, eds. P. G. Ciarlet and J. L. Lions (North Holland, Amsterdam, 1996) pp. 3–207. |
| [4] | DOI: 10.1137/1.9780898719154 · Zbl 1036.65047 · doi:10.1137/1.9780898719154 |
| [5] | DOI: 10.1137/0721050 · Zbl 0572.65012 · doi:10.1137/0721050 |
| [6] | DOI: 10.1137/0905009 · Zbl 0536.65028 · doi:10.1137/0905009 |
| [7] | DOI: 10.1142/S0218127403007175 · Zbl 1062.65117 · doi:10.1142/S0218127403007175 |
| [8] | Chien C.-S., Chinese J. Math. 15 pp 181– |
| [9] | DOI: 10.1016/S0898-1221(98)00042-X · Zbl 1001.65095 · doi:10.1016/S0898-1221(98)00042-X |
| [10] | DOI: 10.1137/S1064827596307324 · Zbl 0985.74021 · doi:10.1137/S1064827596307324 |
| [11] | Doedel E. J., Congr. Numer. 30 pp 265– |
| [12] | DOI: 10.1137/0906055 · Zbl 0589.65075 · doi:10.1137/0906055 |
| [13] | Golub G. H., Matrix Computations (1996) · Zbl 0865.65009 |
| [14] | DOI: 10.1007/978-1-4612-5034-0 · doi:10.1007/978-1-4612-5034-0 |
| [15] | DOI: 10.1142/S0218127495000181 · Zbl 0887.65053 · doi:10.1142/S0218127495000181 |
| [16] | DOI: 10.1137/S0036142994272167 · Zbl 0874.65041 · doi:10.1137/S0036142994272167 |
| [17] | DOI: 10.1137/1.9780898719543 · Zbl 0935.37054 · doi:10.1137/1.9780898719543 |
| [18] | DOI: 10.1142/S0218127402004498 · Zbl 1044.37053 · doi:10.1142/S0218127402004498 |
| [19] | DOI: 10.1137/0722021 · Zbl 0576.65052 · doi:10.1137/0722021 |
| [20] | Keller H. B., Lectures on Numerical Methods in Bifurcation Problems (1987) |
| [21] | DOI: 10.1137/1026033 · Zbl 0574.65116 · doi:10.1137/1026033 |
| [22] | DOI: 10.1137/1024101 · Zbl 0511.35033 · doi:10.1137/1024101 |
| [23] | DOI: 10.1007/978-3-0348-8709-0 · doi:10.1007/978-3-0348-8709-0 |
| [24] | DOI: 10.1137/0717048 · Zbl 0454.65042 · doi:10.1137/0717048 |
| [25] | DOI: 10.1137/0719046 · Zbl 0489.65033 · doi:10.1137/0719046 |
| [26] | Rheinboldt W. C., Numerical Analysis of Parametrized Nonlinear Equations (1986) · Zbl 0582.65042 |
| [27] | DOI: 10.1007/BF01395883 · Zbl 0676.65047 · doi:10.1007/BF01395883 |
| [28] | DOI: 10.1006/jcph.2002.7072 · Zbl 1130.76401 · doi:10.1006/jcph.2002.7072 |
| [29] | DOI: 10.1090/S0002-9947-1988-0933322-5 · doi:10.1090/S0002-9947-1988-0933322-5 |
| [30] | Schaaf R., Lectures in Applied Mathematics 26, in: Computational Solution of Nonlinear Systems of Equations (1990) |
| [31] | DOI: 10.1007/BF01398649 · Zbl 0396.65023 · doi:10.1007/BF01398649 |
| [32] | Wang S.-H., Proc. Amer. Math. Soc. |
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