Petra Csomós

Error analysis of the numerical solution of split differential equations

The operator splitting method is a widely used approach for solving partial differential equations describing physical processes. Its application usually requires the use of certain numerical methods in order to solve the different split sub-problems. The error analysis of such a numerical approach is a complex task. In the present paper we show that an interaction error appears in the numerical solution when an operator splitting procedure is applied together with a lower-order numerical method. The effect of the interaction error is investigated by an analytical study and by numerical experiments made for a test problem.

Operator splitting for delay equations

Operator splitting methods are widely used for partial differential equations. Up until now, they have not been used for delay differential equations. In this paper we introduce splitting methods for delay equations in an abstract setting. We then prove the convergence of the method and discuss the results of some numerical experiments.

Operator Splitting with Spatial-temporal Discretization

Continuing earlier investigations, we analyze the convergence of operator splitting procedures combined with spatial discretization and rational approximations.

Operator splitting for nonautonomous delay equations

We provide a general product formula for the solution of nonautonomous abstract delay equations. After having shown the convergence we obtain estimates on the order of convergence for differentiable history functions. Finally, the theoretical results are demonstrated by some typical numerical examples.

Analysis of a transport model applying operator splitting and semi-Lagrangian method

In the present paper, a simple 2D air pollution transport model is investigated applying the sequential and the Strang Marchuk operator splitting procedures. A comparison is done between the cases when the upwind scheme and the semi-Lagrangian scheme are used for solving the advection sub-problem.

Computational complexity of weighted splitting schemes on parallel computers

In models of complicated physical–chemical processes operator splitting is very often applied in order to achieve sufficient accuracy as well as efficiency of the numerical solution. The recently rediscovered weighted splitting schemes have the great advantage of being parallelizable on operator level, which allows us to reduce the computational time if parallel computers are used. In this paper, the computational times needed for the weighted splitting methods are studied in comparison with the sequential (S) splitting and the Marchuk–Strang (MSt) splitting and are illustrated by numerical experiments performed by use of simplified versions of the Danish Eulerian model (DEM).

Stability and Convergence of Product Formulas for Operator Matrices

We present easy to verify conditions implying stability estimates for operator matrix splittings which ensure convergence of the associated Trotter, Strang and weighted product formulas. The results are applied to inhomogeneous abstract Cauchy problems and to boundary feedback systems.

Some Aspects of Interaction Between Operator Splitting Procedures and Numerical Methods

For solving numerically the partial differential equations describing air pollution transport, numerical methods are usually applied together with operator splitting procedures. In the present paper the interaction between operator splitting procedures and numerical methods is investigated by analytical and numerical study of the order of the error oppressing the numerical solution.

Innovative Integrators for Computing the Optimal State in LQR Problems

We consider the numerical approximation of linear quadratic optimal control problems for partial differential equations where the dynamics is driven by a strongly continuous semigroup. For this problems, the optimal control is given in feedback form, i.e., it relies on solving the associated Riccati equation and the optimal state. We propose innovative integrators for solving the optimal state based on operator splitting procedures and exponential integrators and prove their convergence. We illustrate the performance of our approach in numerical experiments.

An Invitation to Meteorological Data Assimilation

The chapter introduces the basic data assimilation methods used in meteorological modelling. After briefly recalling the mathematical notions and tools needed, we present the optimal interpolation, the variational methods, and the Kalman Filter techniques in one and more dimensions. In order to illustrate the use of the methods introduced, we present the results of numerical experiments done for simple models.