Jürgen Geiser

Iterative operator-splitting methods for linear problems

The operator-splitting methods are based on splitting of the complex problem into a sequence of simpler tasks. A useful method is the iterative splitting method which ensures a consistent approximation in each step. In our paper, we suggest a new method which is based on the combination of the splitting time interval and the traditional iterative operator splitting. We analyse the local splitting error of the method. Numerical examples are given in order to demonstrate the method.

Numerical Simulation of a Model for Transport and Reaction of Radionuclides

We consider a mathematical model for the decay and sorption of radionuclides and their transport in a double porosity media. Such a model can describe transport and reaction processes in porous media, for examle, radioactive waste sites in the ground. We present the equations for a reduced model and apply an operator splitting method for computing the transport and reaction separately. We validate our numerical solutions by comparison with the analytical solutions of our particular test problem.

Higher order operator splitting methods via Zassenhaus product formula: Theory and applications

Abstract In this paper, we contribute higher order operator splitting methods improved by Zassenhaus product. We apply the contribution to classical and iterative splitting methods. The underlying analysis to obtain higher order operator splitting methods is presented. While applying the methods to partial differential equations, the benefits of balancing time and spatial scales are discussed to accelerate the methods. The verification of the improved splitting methods are done with numerical examples. An individual handling of each operator with adapted standard higher order time-integrators is discussed. Finally, we conclude the higher order operator splitting methods.

Computing Exponential for Iterative Splitting Methods: Algorithms and Applications

Iterative splitting methods have a huge amount to compute matrix exponential. Here, the acceleration and recovering of higher-order schemes can be achieved. From a theoretical point of view, iterative splitting methods are at least alternating Picards fix-point iteration schemes. For practical applications, it is important to compute very fast matrix exponentials. In this paper, we concentrate on developing fast algorithms to solve the iterative splitting scheme. First, we reformulate the iterative splitting scheme into an integral notation of matrix exponential. In this notation, we consider fast approximation schemes to the integral formulations, also known as 𝜙-functions. Second, the error analysis is explained and applied to the integral formulations. The novelty is to compute cheaply the decoupled exp-matrices and apply only cheap matrix-vector multiplications for the higher-order terms. In general, we discuss an elegant way of embedding recently survey on methods for computing matrix exponential with respect to iterative splitting schemes. We present numerical benchmark examples, that compared standard splitting schemes with the higher-order iterative schemes. A real-life application in contaminant transport as a two phase model is discussed and the fast computations of the operator splitting method is explained.

Model of PE-CVD Apparatus: Verification and Simulations

In this paper, we present a simulation of chemical vapor deposition with metallic bipolar plates. In chemical vapor deposition, a delicate optimization between temperature, pressure and plasma power is important to obtain homogeneous deposition. The aim is to reduce the number of real-life experiments in a given CVD plasma reactor. Based on the large physical parameter space, there are a hugh number of possible experiments. A detailed study of the physical experiments in a CVD plasma reactor allows to reduce the problem to an approximate mathematical model, which is the underlying transport-reaction model. Significant regions of the CVD apparatus are approximated and physical parameters are transferred to the mathematical parameters. Such an approximation reduces the mathematical parameter space to a realistic number of numerical experiments. The numerical results are discussed with physical experiments to give a valid model for the assumed growth and we could reduce expensive physical experiments.

Overlapping Schwarz wave form relaxation for the solution of coupled and decoupled system of convection diffusion reaction equation

In this article we study the convergence and the error bound for the solution of the convection diffusion reaction equation using overlapping Schwarz wave form relaxation method combined with the first order fractional splitting method (Strang’s splitting) as basic solver. We extended the study to solve decoupled and coupled system of equations of same class in order to demonstrate the effect of the coupling in the system, through the reaction term, on the convergence and error decay. The accuracy and the efficiency of the methods are investigated through the solution of different model problems of scalar, coupled and decoupled systems of convection diffusion reaction equations.

Recent Advances in Splitting Methods for Multiphysics and Multiscale: Theory and Applications

In this paper are presented some recent advances in multiscale splitting methods, based on additive and iterative schemes and applied to deterministic and stochastic differential equations. Several interesting algorithmic aspects of these novel splitting schemes will be discussed. For example, why a decomposed, or split, system may be the key to many important applications in multiply scaled subjects, and why iterative splitting methods can be powerful and more appropriate for well-balanced coupled nonlinear problems. Both theoretical and practical aspects of the recent advances in splitting methods for multiphysics and multiscale applications will be discussed.

Iterative operator-splitting methods for nonlinear differential equations and applications of deposition processes

In this article we consider iterative operator-splitting methods for nonlinear differential equations with bounded and unbounded operators. The main feature of the proposed idea is the embedding of Newton’s method for solving the split parts of the nonlinear equation at each step. The convergence properties of such a mixed method are studied and demonstrated. We confirm with numerical applications the effectiveness of the proposed scheme in comparison with the standard operator-splitting methods by providing improved results and convergence rates. We apply our results to deposition processes. Keyword numerical analysis, operator-splitting method, initial value problems, iterative solver method, stability analysis, convection-diffusion-reaction equation. AMS subject classifications. 35J60, 35J65, 65M99, 65N12, 65Z05, 74S10, 76R50.

Discretization methods with embedded analytical solutions for convection dominated transport in porous media

We will present a higher order discretization method for convection dominated transport equations. The discretisation for the convection-diffusion-reaction equation is based on finite volume methods which are vertex centered. The discretisation for the convection-reaction equation is improved with embedded analytical solutions for the mass. The method is based on the Godunovs-method, [10]. The exact solutions are derived for the one-dimensional convection- reaction equation with piecewise linear initial conditions. We introduce a special cases for the analytical solutions with equal reaction-parameters, confer [9]. We use operator-splitting for solving the convection-reaction-term and the diffusion-term. Numerical results are presented and compared the standard- with the modified- method. Finally we propose our further works on this topic.

Model order reduction for numerical simulation of particle transport based on numerical integration approaches

In this article, we present a non-linear model order reduction (MOR) method based on a linearization technique for a model of particle transport. Historically, non-linear differential equations have been applied to model particle transport. Such non-linear differential equations are expensive and time-consuming to solve. This is a motivation for reducing such a model, based on molecular collisions for heavy particle transport in plasma reactors. Here, we reduce the order by linearization with numerical integration approaches and obtain a controllable and calculable transport–reaction model. We linearize the transport model of the heavy particles with numerical fixed point schemes to a general linear control systems (GLCSs); see M.A. Lieberman and A.J. Lichtenberg [Principle of Plasma Discharges and Materials Processing, 2nd ed., Wiley-Interscience, 2005]. Such linearization allows modelling the collision of the plasma reactor by a system of ordinary differential equations; see the models in M. Ohring [Materials Science of Thin Films, 2nd ed., Academic Press, San Diego, CA, 2002]. Because of their non-linearity, we extend the linear splitting methods with linearization techniques to solve these non-linear equations. Numerical simulations are used to validate this modelling and linearization approach. The contribution is to reuse linear reaction models without neglecting the delicate extension to non-linear reaction models. With the help of higher-order quadrature rules, e.g. Simpson’s rule, we obtain sufficient accuracy and replace the non-linear models by a simpler lower-order linear model. Numerical simulations are presented to validate the coupling ideas of the linearized model.