As a joint initative of researchers from functional analysis and numerical mathematics a bilateral German-Hungarian Project CSITI – Coupled Systems and Innovative Time Integrators was granted in December 2018. It will run for two years (01/2019 – 12/2020) and is jointly coordinated by Bálint Farkas (University of Wuppertal) and Petra Csomós (Eötvös Loránd University, Budapest). This project is financed by DAAD and Magyar Ösztöndíj Bizottság (MÖB).
Coupled systems and the underlying equations provide the mathematically tractable model for many phenomena that occur in physics, biology, chemistry, social sciences or economics. Hence the efficient numerical treatment of such problems is at the basis of our civilization and therefore has been attracting an increasing amount of attention parallel to the technological development. We will carry out the mathematical study of three mathematical procedures for the numerical solution of such coupled problems:
- Operator splitting
- Magnus integrators
- Exponential Runge-Kutta methods
In this project we will further develop these, and provide the theoretical basis for their applicability. For testing the developed methods we propose three problems to be solved numerically, each of which addresses real-life questions of vital importance and is mathematically modelled by a coupled system:
- (plastic) contamination of Arctic
- urban smog
- early phase of planet formation.
The coupling highly depends on the actual problem. Due to the complicated structure of each problem, we need to apply an innovative time integrator (or even their combination) to obtain an efficient numerical time integration method.
Since coupled systems of our interest contain various types of equations (linear, semilinear, non-autonomous linear, nonlinear), in the present project we study first each innovative integrator’s (splitting, exponential Runge-Kutta, and Magnus) numerical features (such as convergence, stability, preservation of qualitative properties, etc.) separately for problems formulated as abstract Cauchy problems. Then we focus on the three real-life problems outlined above. Since each of these coupled systems has a rather complicated structure, we need to combine the innovative time integrators to derive effective numerical methods for solving them. Thus, in the computer implementation phase we will study the theoretical and numerical properties of the combined methods.
As a first action of CSITI we organized a joint workshop with short courses in ELTE Budapest during April 12-18, 2019.
Further information can be found on the project homepage.