The Numeric of SPDEs with unbounded nonlinearities

Imagine a pond into which flows a chemical substance which reacts with water. This system can be described by a reaction diffusion equation. The pond however is not isolated as it is exposed to external conditions such as wind and rain which influences the behaviour of the system by e.g. advection and mixing. Both wind and rain are too complex to be described deterministically and, as every wind or rain has its own individual shape, it cannot be reproduced. One possible way to deal with the problem is to model them by means of stochastic processes, which show the same statistical properties as the wind and rain. The dynamics of the system can be described by a reaction diffusion equation with a Wiener process acting on the surface of the pond, which is a typical example of a so called nonlinear Stochastic Partial Differential Equation (SPDE). The presence of noise leads to new and important phenomena. E.g. there exist several examples, like the reaction diffusion equation with white noise forcing, where in the deterministic case, the invariant measure is not unique, and, in the stochastic case the system is uniquely ergodic. This new type of behaviour is often very useful in understanding real processes and leads often to a more realistic description of real systems than their deterministic counterpart. As in the deterministic case the numerical methods also are important. In line with the theory of Partial Differential Equations (PDEs), one can prove the existence and uniqueness of solutions; however only in a few cases it is possible to find analytical (i.e. in closed form) solutions. Hence those properties of the solution which cannot be found theoretically have to be studied by numerical simulations. Moreover, many properties cannot be found by experiments and proper measurement cannot be carried out on the physical system itself, e.g. fixing a measurement device on a wing of an aircraft would modify the airflow and lead to different results. The topic of the Project was the numerical approximation of such stochastic equations. The main emphasis will be on Stochastic Partial Differential Equations with unbounded nonlinearities. Since, the topic of research is still in the beginning, many problems or questions are open. Moreover, because of the peculiarities of the stochastic perturbation, many results from the Numerical Analysis of deterministic PDEs cannot be transferred. E.g. the Brownian motion is nowhere differentiable or a solution to an SPDE driven by Poisson random measure may have even infinitely many small jumps. Both are situations, which are barely covered by the theory of deterministic numeric, and, often one has to modify or even find new techniques to acquire error bounds. On the other hand, the implication of SPDEs in applied science is quite high - they are used in physical sciences (e.g. in plasmas turbulence, physics of growth phenomena, biology (e.g. bacteria growth and DNA structure), finance mathematics, just to mention some of the fields of applications. Progress in this field is therefore not only important, but also necessary.

SPDEs are quite a young research area; the first articles appeared in the mid 60`s. The presence of noise leads to new and important phenomena. E.g. there exist several examples, like the reaction diffusion equation with white noise forcing, where in the deterministic case the invariant measure is not unique, and in the stochastic case the system is uniquely ergodic. This new type of behaviour is often very useful in understanding real processes and leads often to a more realistic description of real systems than their deterministic counterpart. An example is provided by the Stochastic Navier-Stokes equations which in particular are used to model the airflow around a wing perturbed by the random state of the atmosphere and weather. Developments in such turbulence models lead questions about drag reduction and lift enhancement in aircraft, noise control and combustion control. The spread of an epidemic in some regions and the spatial spread of infectious diseases can be realistically modelled and mathematically described as a travelling front propagation for stochastic nonlinear parabolic wave equations. Investigating realistic epidemic models leads to understanding the development of illnesses like SARS and bird flu, and in turn results in new strategies in combating such major diseases. One should also mention here that SPDEs are used in one of the major models of finance mathematics, the so called Brace-Gatarek-Musiela model, whose aim is to study the interest rate dynamics. SPDEs are also used in physical sciences (e.g. in plasmas turbulence, physics of growth phenomena such as molecular beam epitaxy and fluid flow in porous media with applications to the production of semiconductors and oil industry) and biology (e.g. bacteria growth and DNA structure). Models related to the so called passive scalar equations have potential applications to the understanding of waste (e.g. nuclear) convection under the earths surface. The presence of noise leads to new and important phenomena. For example, the 2- dimensional Navier-Stokes equations with white noise forcing have a unique invariant measure and hence exhibit ergodic behaviour in the sense that the time average of a solution is equal to the average over all possible initial data. Such a property despite continuous efforts in the last thirty or so years has so far not been found for the deterministic counterpart of the equations. This property could lead to profound understanding of the nature of turbulence. What is really the difference between deterministic and stochastic systems the NS has also an invariant measure (theory of turbulence). That means, SPDEs are very different from PDEs. As in the deterministic case the numerical methods are also important. In line with the theory of Partial Differential Equations (PDEs), one can prove the existence and uniqueness of solutions; however it is only possible in a few cases to find analytical (i.e. in closed form) solutions. Hence those properties of the solution which cannot be found theoretically have to be studied by numerical simulations. Moreover, many properties cannot be found by experiments and proper measurement cannot be carried out on the physical system itself, e.g. fixing a measurement device on a wing of an aircraft would modify the airflow and lead to different results. In the project we worked on different aspects of SPDEs. First we showed existence of solutions for the stochastic Burgers equation. On one hand the Burgers appears if one want to describe traffic flows, on the other hand one can look at the Burgers equation as the one dimensional Navier-Stokes.Secondly we investigated the numerical approximation of the stochastic Navier-Stokes and investigated the exact rate of convergence for a certain scheme. Finally we considered the average problem. Here, one tries to found out the equation, if the noise tends to zero.