Numerical Bifurcation Methods and their Application to Fluid Dynamics: Analysis beyond Simulation

Dijkstra HA, Wubs FW, Cliffe AK, Doedel E, Dragomirescu IF, Eckhart B, Gelfgat AY, Hazel A, Lucarini V, Salinger AG, Phipps ET, Sanchez-Umbria J, Schuttelaars H, Tuckerman LS, Thiele U

Zusammenfassung

We provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems. Many of these problems are characterized by high-dimensional dynamical systems which undergo transitions as parameters are changed. The computation of the critical conditions associated with these transitions, popularly referred to as ‘tipping points’, is important for understand- ing the transition mechanisms. We describe the two basic classes of methods of nu- merical bifurcation analysis, which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system. The numerical challenges involved in both methods are mentioned and possible solutions to current bottlenecks are given. To demonstrate that numerical bifurcation techniques are not restricted to relatively low-dimensional dynamical systems, we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems.