István Faragó

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.

On the convergence and local splitting error of different splitting schemes

The convergence of different splitting methods – sequential, Strang, weighted sequential and weighted Strang splitting – is investigated in the semigroup context both for linear and m-dissipative operators, by use of the Trotter product formula and Laxs equivalence theorem. The local splitting errors of the Strang and weighted Strang schemes are analysed with the help of the Baker-Campbell-Hausdorff formula. It is shown that both methods, generally of second order, can be higher than second-order accurate if certain conditions are met. Our results are illustrated with examples.

Finite difference methods, theory and applications: 6th international conference, FDM 2014 Lozenetz, Bulgaria, June 18-23, 2014 revised selected papers

This volume is the Proceedings of the First Conference on Finite Difference Methods which was held at the University of Rousse, Bulgaria, 10--13 August 1997. The conference attracted more than 50 participants from 16 countries. 10 invited talks and 26 contributed talks were delivered. The volume contains 28 papers presented at the Conference. The most important and widely used methods for solution of differential equations are the finite difference methods. The purpose of the conference was to bring together scientists working in the area of the finite difference methods, and also people from the applications in physics, chemistry and other natural and engineering sciences.

Operator splitting and commutativity analysis in the Danish Eulerian model

In this paper the splitting error arising in the Danish Eulerian Model is investigated. Sufficient conditions under which the local splitting error vanishes are formulated for the continuous case. The numerical solution of the model problem introduces several other error sources, which makes the task of determining the effect of the splitting error more complicated. Therefore, we need numerical examples which will allow us to separate the splitting errors from the other errors in order to evaluate both the magnitude of these errors and the relationships between splitting errors and other errors for different values of the discretization parameters. Several such examples have been constructed and analysed. The appropriate conclusions were drawned. The experiences obtained from these experiments can be a starting step towards a total error analysis of the numerical solution of split systems of partial differential equations.

Splitting methods and their application to the abstract cauchy problems

In this paper we consider the interaction of the operator splitting method and applied numerical method to the solution of the different sub-processes. We show that the well-known fully-discretized numerical models (like Crank-Nicolson method, Yanenko method, sequential alternating Marchuk method, parallel alternating method, etc.), elaborated to the numerical solution of the abstract Cauchy problem can be interpreted in this manner. Moreover, on the base of this unified approach a sequence of the new methods can be defined and investigated.

Variable Preconditioning via Quasi-Newton Methods for Nonlinear Problems in Hilbert Space

The aim of this paper is to develop stepwise variable preconditioning for the iterative solution of monotone operator equations in Hilbert space and apply it to nonlinear elliptic problems. The paper is built up to reflect the common character of preconditioned simple iterations and quasi-Newton methods. The main feature of the results is that the preconditioners are chosen via spectral equivalence. The latter can be executed in the corresponding Sobolev space in the case of elliptic problems, which helps both the construction and convergence analysis of preconditioners. This is illustrated by an example of a preconditioner using suitable domain decomposition.

Testing weighted splitting schemes on a one-column transport-chemistry model

In many transport-chemistry models, a huge system of one-dimensional equations of the advection-diffusion-reaction type has to be integrated in time. Typically, this is done with the help of operator splitting. Operator splitting is attractive for complex large-scale transport-chemistry models because it can handle different processes separately in different parts of the computer program. Rosenbrock schemes combined with approximate matrix factorisation (ROS-AMF) are an alternative to operator splitting which does not suffer from splitting errors. However, since the ROS2-AMF schemes are not based on operator splitting, implementation of these methods often requires major changes in the code. In this paper we test another second-order splitting introduced by Strang in 1963, which seemed to be forgotten and rediscovered recently (partially due to its intrinsic parallelism). This splitting, called symmetrically weighted sequential (SWS) splitting, is simple and straightforward to apply, independent of the order of the operators and has an operator-level parallelism. In the experiments, the SWS scheme compares favourably with Strang splitting, but is less accurate than ROS-AMF.

The gradient-finite element method for elliptic problems

Abstract The coupling of the Sobolev space gradient method and the finite element method is developed. The Sobolev space gradient method reduces the solution of a quasilinear elliptic problem to a sequence of linear Poisson equations. These equations can be solved numerically by an appropriate finite element method. This coupling of the two methods will be called the gradient-finite element method (GFEM). Linear convergence of the GFEM is proved via suitable error control in the steps of the iteration. The GFEM defines an already preconditioned iteration in the sense that the theoretical ratio of convergence of the Sobolev space GM is preserved. Finally, a numerical example illustrates the method.

Efficient implementation of stable Richardson Extrapolation algorithms

Richardson Extrapolation is a powerful computational tool which can successfully be used in the efforts to improve the accuracy of the approximate solutions of systems of ordinary differential equations (ODEs) obtained by different numerical methods (including here combined numerical methods consisting of appropriately chosen splitting procedures and classical numerical methods). Some stability results related to two implementations of the Richardson Extrapolation (Active Richardson Extrapolation and Passive Richardson Extrapolation) are formulated and proved in this paper. An advanced atmospheric chemistry scheme, which is commonly used in many well-known operational environmental models, is applied in a long sequence of experiments in order to demonstrate the fact that (a)it is indeed possible to improve the accuracy of the numerical results when the Richardson Extrapolation is used (also when very difficult, badly scaled and stiff non-linear systems of ODEs are to be treated), (b)the computations can become unstable when the combination of the Trapezoidal Rule and the Active Richardson Extrapolation is used, (c)the application of the Active Richardson Extrapolation with the Backward Euler Formula is leading to a stable computational process, (d)experiments with different algorithms for solving linear systems of algebraic equations are very useful in the efforts to select the most suitable approach for the particular problems solved and (e)the computational cost of the Richardson Extrapolation is much less than that of the underlying numerical method when a prescribed accuracy has to be achieved.

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.