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 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 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

Abstract. In this paper a multilevel-like ILU preconditioner is introduced. The ILU factorization generates its own ordering during the elimination process. Both ordering and dropping depend on the size of the entries. The method can handle structured and unstructured problems. Results are presented for some important classes of matrices and for several well-known test examples. The results illustrate the efficiency of the method and show in several cases near grid independent convergence. Key words. multilevel methods, preconditioning, ILU, dropping strategies, Krylov-subspace methods AMS subject classifications. 65F10, 65N06PII. S0895479897319301 1. Introduction. Solving large sparse systems of equations continues to be a major research area. This attention is caused by the fact that solving such equations forms the bottleneck in many practical problems. For really large systems direct methods become too expensive in CPU time and storage requirements, and therefore an iterative approach is needed. In particular the use of preconditioned CG-type methods has proved to be very competitive. It is also widely recognized that the quality of the preconditioner determines the success of the iterative method. With a proper preconditioner the choice of the CG-like accelerator is not that critical.The preconditioner presented in this paper is a special multilevel-like incomplete factorization. In this introduction we briefly describe the various incomplete decomposition approaches available today and their relation to the approach presented here.The history of ILU factorizations is amongst others described in [15]. Moreover, historical notes are to be found in the textbooks of Axelsson [1], Hackbusch [31], and Saad [50]. The first roots of the approach lie in the 1960s [13,42,43] and since then the method has become applicable to a wide class of problems. Furthermore, analyses for important classes of matrices could be made. Today, ILU factorizations are an important tool for solving large-scale problems.Classical ILU approach. The classical approach is to allow only fill entries in the L and U factors, where the original matrix A has nonzeros. This simple approach allows for a very efficient implementation by Eisenstat [25] and is still very popular.As observed by Dupont, Kendall, and Rachford [23], an important improvement in the convergence of the classical approach can be obtained by lumping the dropped elements onto the diagonal. With this modification, the factorization, called MILU, is made exact for a constant vector. For a more general matrix A Gustafsson [30] found a similar result. For many second-order elliptic problems, the preconditioning with the classical ILU gives asymptotically the same condition number as with diagonal scaling, i.e., O(h −2 ). After this simple modification this improves to O(h −1 ). For Mmatrices the existence of ILU factorizations can be proved [35], but this is not the case

In this paper, we present a new linear system solver for use in a fully-implicit ocean model. The new solver allows to perform bifurcation analysis of relatively high-resolution primitive-equation ocean-climate models. It is based on a block-ILU approach and takes special advantage of the mathematical structure of the governing equations. In implicit models Jacobian matrices have to be constructed. Analytical construction is hard for complicated but more realistic representations of mixing. This is overcome by evaluating the Jacobian in part numerically. The performance of the new implicit ocean model is demonstrated using (i) a high-resolution model of the wind-forced double-gyre flow problem in a (relatively small) midlatitude spherical basin, and (ii) a medium-resolution model of thermohaline and wind-driven flows in an Atlantic size single-hemispheric basin.

The Atlantic Meridional Overturning Circulation (AMOC) is considered to be a tipping element of the climate system. As it cannot be excluded that the AMOC is in a multiple regime, transitions can occur due to atmospheric noise between the present-day state and a weaker AMOC state. For the first time, we here determine estimates of the transition probability of noise-induced transitions of the AMOC, within a certain time period, using a methodology from large deviation theory. We find that there are two types of transitions, with a partial or full collapse of the AMOC, having different transition probabilities. For the present-day state, we estimate the transition probability of the partial collapse over the next 100 years to be about 15%, with a high sensitivity of this probability to the surface freshwater noise amplitude.

In this paper, a fully implicit numerical model of the three-dimensional thermoha-