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Bifurcation tracking algorithms and software for large scale applications. (English) Zbl 1076.65118

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.

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
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References:

[1] Burroughs E. A., Technical Report SAND2001–0113 (2001)
[2] DOI: 10.1108/09615530410544328 · Zbl 1086.76511 · doi:10.1108/09615530410544328
[3] DOI: 10.1017/S002211207900015X · doi:10.1017/S002211207900015X
[4] DOI: 10.1063/1.868768 · Zbl 0838.76028 · doi:10.1063/1.868768
[5] K. Cliffe, A. Spence and S. Tavener, The Numerical Analysis of Bifurcation with Application to Fluid Mechanics, Acta Numerica (Cambridge University Press, 2000) pp. 39–131. · Zbl 1005.65138
[6] DOI: 10.1002/(SICI)1097-0363(20000130)32:2<175::AID-FLD912>3.0.CO;2-5 · Zbl 0966.76025 · doi:10.1002/(SICI)1097-0363(20000130)32:2<175::AID-FLD912>3.0.CO;2-5
[7] DOI: 10.1017/S0022112099006941 · Zbl 0957.76014 · doi:10.1017/S0022112099006941
[8] DOI: 10.1137/S1064827500372262 · Zbl 0992.65020 · doi:10.1137/S1064827500372262
[9] DOI: 10.1145/779359.779362 · Zbl 1070.65574 · doi:10.1145/779359.779362
[10] Doedel E. J., Technical Report (1997)
[11] DOI: 10.1145/513001.513002 · Zbl 1070.65556 · doi:10.1145/513001.513002
[12] DOI: 10.1063/1.1558314 · doi:10.1063/1.1558314
[13] DOI: 10.1063/1.1518685 · doi:10.1063/1.1518685
[14] DOI: 10.1007/s004660000170 · Zbl 0989.74023 · doi:10.1007/s004660000170
[15] DOI: 10.1137/1.9780898719543 · Zbl 0935.37054 · doi:10.1137/1.9780898719543
[16] DOI: 10.1093/imanum/3.3.295 · Zbl 0521.65070 · doi:10.1093/imanum/3.3.295
[17] DOI: 10.1142/S0218127402004498 · Zbl 1044.37053 · doi:10.1142/S0218127402004498
[18] Hendrickson B., Technical Report SAND94–2692 (1995)
[19] DOI: 10.1137/0916028 · Zbl 0816.68093 · doi:10.1137/0916028
[20] Heroux M., Technical Report SAND2003–2927 (2003)
[21] DOI: 10.1016/0045-7825(86)90025-3 · Zbl 0622.76077 · doi:10.1016/0045-7825(86)90025-3
[22] DOI: 10.1016/0045-7825(89)90111-4 · Zbl 0697.76100 · doi:10.1016/0045-7825(89)90111-4
[23] Hutchinson S. A., Technical Report SAND95-1559 (1995)
[24] H. B. Keller, Applications of Bifurcation Theory, ed. P. H. Rabinowitz (Academic Press, NY, 1977) pp. 159–384.
[25] DOI: 10.1002/fld.135 · Zbl 1037.76036 · doi:10.1002/fld.135
[26] DOI: 10.1002/fld.135 · Zbl 1037.76036 · doi:10.1002/fld.135
[27] DOI: 10.1137/1.9780898719628 · Zbl 0901.65021 · doi:10.1137/1.9780898719628
[28] DOI: 10.1137/S1064827594277673 · Zbl 0915.65088 · doi:10.1137/S1064827594277673
[29] DOI: 10.1063/1.868730 · Zbl 0836.76033 · doi:10.1063/1.868730
[30] Maschhoff K. J., Lecture Notes in Computer Science 1184, in: Applied Parallel Computing in Industrial Problems and Optimization (1996) · Zbl 0886.65034 · doi:10.1007/3-540-62095-8_51
[31] DOI: 10.1137/0717048 · Zbl 0454.65042 · doi:10.1137/0717048
[32] Muratov C., Physica 186 pp 93–
[33] DOI: 10.1063/1.869442 · Zbl 1185.76640 · doi:10.1063/1.869442
[34] Nore C., J. Fluid Mech. 477 pp 51–
[35] Pawlowski R. P., J. Phys. IV 11 pp 197–
[36] Pawlowski R. P., Technical Report SAND2004 (2004)
[37] DOI: 10.1063/1.1558313 · doi:10.1063/1.1558313
[38] DOI: 10.1002/fld.392 · Zbl 1047.76053 · doi:10.1002/fld.392
[39] Salinger A. G., Technical Report SAND2002-0396 (2002)
[40] DOI: 10.1016/S0022-0248(99)00140-2 · doi:10.1016/S0022-0248(99)00140-2
[41] Shadid J. N., IJCFD 12 pp 199–
[42] Shepherd J. F., Technical Report SAND2000-2647 (2000)
[43] DOI: 10.1063/1.869489 · Zbl 1185.76871 · doi:10.1063/1.869489
[44] DOI: 10.1063/1.868781 · Zbl 1026.76521 · doi:10.1063/1.868781
[45] DOI: 10.1137/0721029 · Zbl 0554.65045 · doi:10.1137/0721029
[46] DOI: 10.1017/S0022112070002070 · doi:10.1017/S0022112070002070
[47] DOI: 10.1002/fld.399 · Zbl 1025.76033 · doi:10.1002/fld.399
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