Tobias Schäfer

The space-fractional diffusion-advection equation: Analytical solutions and critical assessment of numerical solutions

The present work provides a critical assessment of numerical solutions of the space-fractional diffusion-advection equation, which is of high significance for applications in various natural sciences. In view of the fact that, in contrast to the case of normal (Gaussian) diffusion, no standard methods and corresponding numerical codes for anomalous diffusion problems have been established yet, it is of importance to critically assess the accuracy and practicability of existing approaches. Three numerical methods, namely a finite-difference method, the so-called matrix transfer technique, and a Monte-Carlo method based on the solution of stochastic differential equations, are analyzed and compared by applying them to three selected test problems for which analytical or semi-analytical solutions were known or are newly derived. The differences in accuracy and practicability are critically discussed with the result that the use of stochastic differential equations appears to be advantageous.

The instanton method and its numerical implementation in fluid mechanics

A precise characterization of structures occurring in turbulent fluid flows at high Reynolds numbers is one of the last open problems of classical physics. In this review we discuss recent developments related to the application of instanton methods to turbulence. Instantons are saddle point configurations of the underlying path integrals. They are equivalent to minimizers of the related Freidlin-Wentzell action and known to be able to characterize rare events in such systems. While there is an impressive body of work concerning their analytical description, this review focuses on the question on how to compute these minimizers numerically. In a short introduction we present the relevant mathematical and physical background before we discuss the stochastic Burgers equation in detail. We present algorithms to compute instantons numerically by an efficient solution of the corresponding Euler-Lagrange equations. A second focus is the discussion of a recently developed numerical filtering technique that allows to extract instantons from direct numerical simulations. In the following we present modifications of the algorithms to make them efficient when applied to two- or three-dimensional fluid dynamical problems. We illustrate these ideas using the two-dimensional Burgers equation and the three-dimensional Navier-Stokes equations.

Quickest Detection in Coupled Systems

This work considers the problem of quickest detection of signals in a coupled system of N sensors, which receive continuous sequential observations from the environment. It is assumed that the signals, which are modeled a general Itô processes, are coupled across sensors, but that their onset times may differ from sensor to sensor. The objective is the optimal detection of the first time at which any sensor in the system receives a signal. The problem is formulated as a stochastic optimization problem in which an extended average Kullback-Leibler divergence criterion is used as a measure of detection delay, with a constraint on the mean time between false alarms. The case in which the sensors employ cumulative sum (CUSUM) strategies is considered, and it is proved that the minimum of N CUSUMs is asymptotically optimal as the mean time between false alarms increases without bound.

Propagation of ultra-short solitons in stochastic Maxwell's equations

We study the propagation of ultra-short short solitons in a cubic nonlinear medium modeled by nonlinear Maxwells equations with stochastic variations of media. We consider three cases: variations of (a) the dispersion, (b) the phase velocity, (c) the nonlinear coefficient. Using a modified multi-scale expansion for stochastic systems, we derive new stochastic generalizations of the short pulse equation that approximate the solutions of stochastic nonlinear Maxwells equations. Numerical simulations show that soliton solutions of the short pulse equation propagate stably in stochastic nonlinear Maxwells equations and that the generalized stochastic short pulse equations approximate the solutions to the stochastic Maxwells equations over the distances under consideration. This holds for both a pathwise comparison of the stochastic equations as well as for a comparison of the resulting probability densities.

Instanton filtering for the stochastic Burgers equation

We address the question of whether one can identify instantons in direct numerical simulations of the stochastically driven Burgers equation. For this purpose, we first solve the instanton equations using the Chernykh–Stepanov method (2001 Phys. Rev. E 64 026306). These results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations. Using this approach we can extract the entire time history of the instanton evolution, which allows us to identify the different phases predicted by the direct method of Chernykh and Stepanov with remarkable agreement.

Arclength Parametrized Hamilton's Equations for the Calculation of Instantons

A method is presented to compute minimizers (instantons) of action functionals using arclength parametrization of Hamiltons equations. This method can be interpreted as a local variant of the geometric minimum action method introduced to compute minimizers of the Freidlin--Wentzell action functional that arises in the context of large deviation theory for stochastic differential equations. The method is particularly well suited to calculate expectations dominated by noise-induced excursions from deterministically stable fixpoints. Its simplicity and computational efficiency are illustrated here using several examples: a finite-dimensional stochastic dynamical system (an Ornstein--Uhlenbeck model) and two models based on stochastic partial differential equations: the

Higher-order corrections to the short-pulse equation

\phi^4

Averaged dynamics of time-periodic advection diffusion equations in the limit of small diffusivity

-model and the stochastically driven Burgers equation.

Long Term Effects of Small Random Perturbations on Dynamical Systems: Theoretical and Computational Tools

Using renormalization group techniques, we derive an extended short-pulse equation as an approximation to a nonlinear wave equation. We investigate the new equation numerically and show that the new equation captures efficiently higher-order effects on pulse propagation in cubic nonlinear media. We illustrate our findings using one- and two-soliton solutions of the first-order short-pulse equation as initial conditions in the nonlinear wave equation.

Relevance of instantons in Burgers turbulence

We study the effect of advection and small diffusion on passive tracers. The advecting velocity field is assumed to have mean zero and to possess time-periodic stream lines. Using a canonical transform to action-angle variables followed by a Lie transform, we derive an averaged equation describing the effective motion of the tracers. An estimate for the time validity of the first-order approximation is established. For particular cases of a regularized vortical flow we present explicit formulas for the coefficients of the averaged equation both at first and at second order. Numerical simulations indicate that the validity of the above first-order estimate extends to the second order.