The project is funded by TWF-Projekt Nr. UNI-0404/880 (Tiroler Wissenschaftsfonds) and the main coordinator is Dr. Stefan Rainer
Many problems in physics, biology, chemistry can be described by a time
dependent partial differential equation (PDE). Often the solution of this
equation is compactly supported. For the numerical computation of these
solutions a huge quantity of integrators exists. Especially for problems with
dominating advection part, the use of exponential integrators is a recommended
choice (see e.g. Hochbruck, Ostermann and Schweitzer (2009)) and will be
the main focus of this project. However, for large domains, one has to use
many grid points, ending up with a very sti and memory consuming problem
to be solved, in order to get the desired accuracy.
On the other hand, since the solution has compact support within a possible
large region of interest, meshfree methods, such as moving meshes, moving
grids and methods based on radial basis functions (RBF), are also a suitable
choice. RBF methods can achieve high accuracy. They are
flexible and easy to implement. Therefore the first aim of this project is to combine
exponential integrators with RBF methods.
State of the art exponential integrators make use of variable time stepping
in order to control the local error. But this error estimate only helps to get
rid of the error in time propagation. It is neither a means to estimate the
error in space, nor the total error.
The use of meshfree methods to represent the solution at every time step
opens the possibility to check for the spatial error, too.
Another aim of the project is to get a spatial error estimation during time
evolution in order to control not only the time error but both, temporal and
spatial error for every step. Our approach will be based on ideas presented
by Driscoll and Heryudono (2007) and Behrens, Iske and Käser (2003)
The general aim of this project is to develop a meshfree exponential integrator
that has both high accuracy in time and space and takes advantage of the
local support of the solution.