Numerical Bifurcation Methods and Their Application to Fluid Dynamics: Analysis Beyond Simulation (Communications in Computational Physics, 2014)
We provide an overview of current techniques and typical applications ofnumerical bifurcation analysis in fluid dynamical problems. Many of these problemsare characterized by high-dimensional dynamical systems which undergo transitionsas parameters are changed. The computation of the critical conditions associated with these transitions, popularly referred to as 'tipping points', is important for understanding the transition mechanisms. We describe the two basic classes of methods of numerical bifurcation analysis, which differ in the explicit or implicit use of the Jacobianmatrix of the dynamical system. The numerical challenges involved in both methodsare mentioned and possible solutions to current bottlenecks are given. To demonstratethat numerical bifurcation techniques are not restricted to relatively low-dimensionaldynamical systems, we provide several examples of the application of the moderntechniques to a diverse set of fluid mechanical problems.