Bilateral GermanHungarian Project
CSITI  Coupled Systems and Innovative Time Integrators
(01/2019  12/2020)

Scientific goals
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 carry out the mathematical study of three mathematical procedures for the
numerical solution of such coupled problems:
 Operator splitting
 Magnus integrators
 Exponential RungeKutta methods
We 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
reallife 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.
Main objective:
Since coupled systems of our interest contain various types of equations (linear, semilinear, nonautonomous
linear, nonlinear), in the present project we study first each innovative integrator's (splitting, exponential
RungeKutta, 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 reallife 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.
German team:
Hungarian team:
 Ágnes Bodó
 Szilvia Császár
 Petra Csomós (Leader)
 Bálint Máté Takács
German institutions:
Hungarian institutions:
Publications related to the Project
2019
 Lorenc Kapllani,
Multistep Schemes for Solving Backward Stochastic Differential Equations on GPU,
Master Thesis, University of Wuppertal, March 2019, (Supervisors: Long Teng, Matthias Ehrhardt).
 L. Kapllani, L. Teng and M. Ehrhardt,
A Multistep Scheme to solve Backward Stochastic Differential Equations for Option
Pricing on GPUs,
Preprint 19/20, June 2019.
 P. Csomós, B. Takács,
Operator splitting for spacedependent epidemic model,
submitted to Applied Numerical Mathematics
 S. Kovács, Sz. György,
On the qualitative behaviour of parameter dependent systems,
Miskolc Mathematical Notes, to appear.
 S. Treibert, H. Brunner and M. Ehrhardt,
Compartment models for vaccine effectiveness and nonspecific effects for Tuberculosis,
Mathematical Biosciences and Engineering, 2019, 16(6): 72507298.
DOI:10.3934/mbe.2019364 (open access)
 B. Érdi, E. Kövári,
The kite central configurations of four bodies with three equal masses,
Symmetry, submitted.
 C. Budde,
Positive MiyaderaVoigt perturbations of bicontinuous semigroups,
arXiv:1912.07334, submitted, Dec. 2019
 C. Budde and R. Heymann,
Extrapolation of operatorvalued multiplication operators,
arXiv:1911.08175, submitted, Nov. 2019.
 C. Budde and M. Kramar Fijavž,
Bicontinuous semigroups for flows in infinite networks,
arXiv:1901.10292, submitted, Jan. 2019.
 S. Kovács, Sz. György, N. Gyúró,
On an invasive species model with harvesting,
BIOMAT series, submitted, 2019.
 C. Budde,
General extrapolation spaces and perturbations of bicontinuous semigroups,
Ph.D. thesis, University of Wuppertal, 2019, (Supervisor: Bálint Farkas).
 E. Kövári,
Linearized stability of the central fourbody problem with axial symmetry
(in Hungarian), Master's thesis, Loránd Eötvös University, Budapest, 2019, (Supervisor: Bálint Érdi).
2020
 L. Teng, X. Wu, M. Günther and M. Ehrhardt,
A new methodology to create valid timedependent correlation matrices via isospectral flows,
ESAIM: Mathematical Modelling and Numerical Analysis (M2AN), Volume 54, Number 2,
(MarchApril 2020), 361371.
DOI:10.1051/m2an/2019064
 M. Muniz, M. Ehrhardt and M. Günther,
Approximating correlation matrices using stochastic Lie group methods,
Preprint 20/05, March 2020.
 A. Clevenhaus, M. Ehrhardt, M. Günther and D. Sevcovic,
Pricing American Options with a Nonconstant Penalty Parameter,
J. Risk Financial Manag. 2020, 13(6), 124. DOI 10.3390/jrfm13060124
 P. Csomós, M. Ehrhardt and B. Farkas,
Operator splitting for abstract Cauchy problems with dynamical boundary condition,
Preprint 20/11, April 2020.
Talks related to the Project
2019
 Bálint Farkas,
Analytic Semigroups, Budapest, April 15, 2019.
 Christian Budde,
An Introduction to nonautonomous problems, Budapest, April 15, 2019.
 Szilvia György,
Exponential dichotomy and stability of delay differential equations, Budapest, April 15, 2019.
 Michelle Muniz,
Lie group methods for matrix ODEs and SDEs, Budapest, April 15, 2019.
 Matthias Ehrhardt,
Operator Splitting Schemes, Budapest, April 16, 2019.
 Petra Csomós,
Exponential and Magnus integrators, Budapest, April 16, 2019.
 Anna Clevenhaus,
ADI Schemes for option pricing in the Heston model with stochastic correlation, Budapest, April 17, 2019.
 Bálint Takács,
Numerical solution of spacedependent SIR models, Budapest, April 17, 2019.
 Emese Kövári,
Central configurations of four bodies with an axis of symmetry, Wuppertal, December 12, 2019.
 Michelle Muniz,
Using stochastic Lie group methods to approximate correlation matrices in financial markets, Wuppertal, December 12, 2019.
 Anna Clevenhaus,
Pricing American Options with an adaptive Penalization Strategy and the Combination Technique, Wuppertal, December 12, 2019.
 Lorenc Kapllani,
A Multistep Scheme to solve Backward Stochastic Differential Equations for Option Pricing on GPUs, Wuppertal, December 12, 2019.
2020
Staff Exchange
2019
Activities related to the Project