Iterative operator-splitting methods for linear problems

The operator-splitting methods base on splitting of the complex problem into the sequence of the 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 splitting the time interval and the traditional iterative operator splitting. We analyze the local splitting error of the method. Numerical examples are given in order to demonstrate the method

Serial and Parallel Iterative Splitting Methods: Algorithms and Applications to Fractional Convection-Diffusion Equations

[EN] The benefits and properties of iterative splitting methods, which are based on serial versions, have been studied in recent years, this work, we extend the iterative splitting methods to novel classes of parallel versions to solve nonlinear fractional convection-diffusion equations. For such interesting partial differential examples with higher dimensional, fractional, and nonlinear terms, we could apply the parallel iterative splitting methods, which allow for accelerating the solver methods and reduce the computational time. Here, we could apply the benefits of the higher accuracy of the iterative splitting methods. We present a novel parallel iterative splitting method, which is based on the multi-splitting methods, The flexibilisation with multisplitting methods allows for decomposing large scale operator equations. In combination with iterative splitting methods, which use characteristics of waveform-relaxation (WR) methods, we could embed the relaxation behavior and deal better with the nonlinearities of the operators. We consider the convergence results of the parallel iterative splitting methods, reformulating the underlying methods with a summation of the individual convergence results of the WR methods. We discuss the numerical convergence of the serial and parallel iterative splitting methods with respect to the synchronous and asynchronous treatments. Furthermore, we present different numerical applications of fluid and phase field problems in order to validate the benefit of the parallel versions.This research was partially supported by Ministerio de Economia y Competitividad, Spain, under grant PGC2018-095896-B-C21-C22 and German Academic Exchange Service grant number 91588469.Geiser, J.; Martínez Molada, E.; Hueso, JL. (2020). 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Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations

[EN] This article proposes adaptive iterative splitting methods to solve Multiphysics problems, which are related to convection-diffusion-reaction equations. The splitting techniques are based on iterative splitting approaches with adaptive ideas. Based on shifting the time-steps with additional adaptive time-ranges, we could embedded the adaptive techniques into the splitting approach. The numerical analysis of the adapted iterative splitting schemes is considered and we develop the underlying error estimates for the application of the adaptive schemes. The performance of the method with respect to the accuracy and the acceleration is evaluated in different numerical experiments. We test the benefits of the adaptive splitting approach on highly nonlinear Burgers' and Maxwell-Stefan diffusion equations.This research was funded by German Academic Exchange Service grant number 91588469. We acknowledge support by the DFG Open Access Publication Funds of the Ruhr-Universität of Bochum, Germany and by Ministerio de Economía y Competitividad, Spain, under grant PGC2018-095896-B-C21-C22.Geiser, J.; Hueso, JL.; Martínez Molada, E. (2020). Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations. Mathematics. 8(3):1-22. https://doi.org/10.3390/math8030302S12283Auzinger, W., & Herfort, W. (2014). Local error structures and order conditions in terms of Lie elements for exponential splitting schemes. Opuscula Mathematica, 34(2), 243. doi:10.7494/opmath.2014.34.2.243Auzinger, W., Koch, O., & Quell, M. (2016). Adaptive high-order splitting methods for systems of nonlinear evolution equations with periodic boundary conditions. Numerical Algorithms, 75(1), 261-283. doi:10.1007/s11075-016-0206-8Descombes, S., & Massot, M. (2004). 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