TY - JOUR

T1 - Parallel pseudo-transient Newton-Krylov-Schwarz continuation algorithms for bifurcation analysis of incompressible sudden expansion flows

AU - Huang, Chiau Yu

AU - Hwang, Feng Nan

N1 - Funding Information:
The work was supported in part by the National Science Council of Taiwan, 96-2115-M-008-007-MY2. The authors thank Prof. C.-Y. Soong for useful discussion and Prof. X.-C. Cai for constructive comments used to improve the presentation of the manuscript. The authors are also grateful to the Center for Scientific Computing at the National Central University for providing computing resources.

PY - 2010/7

Y1 - 2010/7

N2 - We propose a parallel pseudo-transient continuation algorithm, in conjunction with a Newton-Krylov-Schwarz (NKS) algorithm, for the detection of the critical points of symmetry-breaking bifurcations in sudden expansion flows. One classical approach for examining the stability of a stationary solution to a system of ordinary differential equations (ODEs) is to apply the so-called a method-of-line approach, beginning with some perturbed stationary solution to a system of ODEs and then to investigate its time-dependent response. While the time accuracy is not our concern, the adaptability of time-step size is a key ingredient for the success of the algorithm in accelerating the time-marching process. To allow large time steps, unconditionally stable time integrators, such as the backward Euler's method, are often employed. As a result, the price paid is that at each time step, a large sparse nonlinear system of equations needs to be solved. The NKS is a good candidate solver for a system. Our numerical results obtained from a parallel machine show that our algorithm is robust and efficient and also verify, qualitatively, the bifurcation prediction with published results. Furthermore, imperfect pitchfork bifurcations are observed, especially for the case with a small expansion ratio, in which the occurrence of bifurcation points is delayed due to the stabilization terms in Galerkin/Least squares finite elements on asymmetric, unstructured meshes.

AB - We propose a parallel pseudo-transient continuation algorithm, in conjunction with a Newton-Krylov-Schwarz (NKS) algorithm, for the detection of the critical points of symmetry-breaking bifurcations in sudden expansion flows. One classical approach for examining the stability of a stationary solution to a system of ordinary differential equations (ODEs) is to apply the so-called a method-of-line approach, beginning with some perturbed stationary solution to a system of ODEs and then to investigate its time-dependent response. While the time accuracy is not our concern, the adaptability of time-step size is a key ingredient for the success of the algorithm in accelerating the time-marching process. To allow large time steps, unconditionally stable time integrators, such as the backward Euler's method, are often employed. As a result, the price paid is that at each time step, a large sparse nonlinear system of equations needs to be solved. The NKS is a good candidate solver for a system. Our numerical results obtained from a parallel machine show that our algorithm is robust and efficient and also verify, qualitatively, the bifurcation prediction with published results. Furthermore, imperfect pitchfork bifurcations are observed, especially for the case with a small expansion ratio, in which the occurrence of bifurcation points is delayed due to the stabilization terms in Galerkin/Least squares finite elements on asymmetric, unstructured meshes.

KW - Bifurcation

KW - Domain decomposition

KW - Incompressible sudden expansion flow

KW - Newton-Krylov-Schwarz

KW - Parallel computing

KW - Pseudo-transient continuation

UR - http://www.scopus.com/inward/record.url?scp=77953137950&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2010.03.014

DO - 10.1016/j.apnum.2010.03.014

M3 - 期刊論文

AN - SCOPUS:77953137950

SN - 0168-9274

VL - 60

SP - 738

EP - 751

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

IS - 7

ER -