Numerical solution of parabolic Cauchy problems in planar corner domains

An iterative method for the parabolic Cauchy problem in planar domains having a finite number of corners is implemented based on boundary integral equations. At each iteration, mixed well-posed problems are solved for the same parabolic operator. The presence of corner points renders singularities of the solutions to these mixed problems, and this is handled with the use of weight functions together with, in the numerical implementation, mesh grading near the corners. The mixed problems are reformulated in terms of boundary integrals obtained via discretization of the time-derivative to obtain an elliptic system of partial differential equations. To numerically solve these integral equations a Nyström method with super-algebraic convergence order is employed. Numerical results are presented showing the feasibility of the proposed approach. © 2014 IMACS.

Computation of rare transitions in the barotropic quasi-geostrophic equations

We investigate the theoretical and numerical computation of rare transitions in simple geophysical turbulent models. We consider the barotropic quasi-geostrophic and two-dimensional Navier–Stokes equations in regimes where bistability between two coexisting large-scale attractors exist. By means of large deviations and instanton theory with the use of an Onsager–Machlup path integral formalism for the transition probability, we show how one can directly compute the most probable transition path between two coexisting attractors analytically in an equilibrium (Langevin) framework and numerically otherWe adapt a class of numerical optimization algorithms known as minimum action methods to simple geophysical turbulent models. We show that by numerically minimizing an appropriate action functional in a large deviation limit, one can predict the most likely transition path for a rare transition between two states. By considering examples where theoretical predictions can be made, we show that the minimum action method successfully predicts the most likely transition path. Finally, we discuss the application and extension of such numerical optimization schemes to the computation of rare transitions observed in direct numerical simulations and experiments and to other, more complex, turbulent systems.

Langevin dynamics, large deviations and instantons for the quasi-geostrophic model and two-dimensional Euler equations

We investigate a class of simple models for Langevin dynamics of turbulent flows, including the one-layer quasi-geostrophic equation and the two-dimensional Euler equations. Starting from a path integral representation of the transition probability, we compute the most probable fluctuation paths from one attractor to any state within its basin of attraction. We prove that such fluctuation paths are the time reversed trajectories of the relaxation paths for a corresponding dual dynamics, which are also within the framework of quasi-geostrophic Langevin dynamics. Cases with or without detailed balance are studied. We discuss a specific example for which the stationary measure displays either a second order (continuous) or a first order (discontinuous) phase transition and a tricritical point. In situations where a first order phase transition is observed, the dynamics are bistable. Then, the transition paths between two coexisting attractors are instantons (fluctuation paths from an attractor to a saddle), which are related to the relaxation paths of the corresponding dual dynamics. For this example, we show how one can analytically determine the instantons and compute the transition probabilities for rare transitions between two attractors.