Lorentz Center

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## Tipping Points in Complex Flows -
Numerical Methods for Bifurcation Analysis of Large-Scale Systems |

In many natural and industrial systems transitions between different qualitative types of behavior may occur when external conditions are changed. This change in behavior is usually associated with loss of stability of a particular equilibrium state. In popular terms, these critical phenomena are referred to as `tipping points' but in mathematics they are known as bifurcation points. The study of bifurcation phenomena is done within the field of dynamical systems. In application areas such as fluid dynamics, climate physics, plasma- and astrophysics, the flows are described by systems of partial differential equations. When these equations are discretized by a spectral, finite difference or finite element method, finite dimensional systems with very large -- typically 10^6-10^9 -- dimensions result, which we refer below to as large-scale systems. Over the last decades much progress has been made in the application of the methodology of dynamical systems theory to these large-scale systems. Main computational bottlenecks of applying the methodology of dynamical systems to applications are the solution of giant-dimensional (in many cases ill-conditioned) linear systems of equations and generalized eigenvalue problems. Hence, targeted preconditioning techniques and eigenvalue solvers (where only part of the spectrum is calculated) are in many cases decisive on whether techniques can be applied to determine tipping points in a specific application. In this workshop our target audience is a mix of scientists who are at the forefront of developing new methods, users of these methods in a diverse range of applications and and new potential (in particular, young) users of these methods. The specific aims of this workshop are (i) to assess the state-of-the art of the methodology, (ii) to assess the remaining challenges and remaining numerical difficulties, (iii) to develop a road map for future research, (iv) to foster cooperation in research as well as in training of students. [Back] |