Iterative Splitting Methods for Differential Equations

Geiser, Jürgen

doi:10.1201/b10947

Cited by 56 publications

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“…One of the main motivations to apply iterative splitting method is to reach higher accuracy based on the iterated solutions, see [30]. Further, such scheme can relax nonlinearities based on the smoothing behaviour of fix-point approaches or successive approximation schemes, see [31,32].…”

Section: Iterative Splitting Methodsmentioning

confidence: 99%

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General Principles

Geiser¹

2015

Multicomponent and Multiscale Systems

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In the general principle, we give an overview of the recently used methods and schemes to solve multicomponent and multiscale systems. While multicomponent systems are evolution equations based on each single species, which are coupled with the other species, e.g. with reaction-, diffusion-processes, multiscale systems are evolution equations based on different scales for each species, e.g. macroscopicor microscopic scale. We give the general criteria for practically performing the different splitting and multiscale methods, such that a modification to practical applications of the splitting schemes to a real-life problem can be done. Multicomponent SystemsIn the following, we deal with multicomponent systems. Multicomponent systems concentrate on disparate components in the underlying multiscale models, while they can be coupled by linear or nonlinear functions or differential systems, e.g. reactions or transport phenomena. Often, also different scales are important to resolve to understand the interactions of the different components. Here, we have to apply multicomponent schemes that are also related to disparate spatial and timescales to resolve such complexities, see algorithmic ideas of multicomponent problems in [1].In the following sections, we concentrate on modelling or algorithmical aspects of multicomponent systems for the following applications:• Multicomponent flows, see [2,3].• Multicomponent transport, see [1,4].We simulate the flow and transport systems based on their interactions with the different components. Hence, we allow to study weakly or strongly coupled components in the underlying modelling equation systems. Multicomponent FlowsThe class of multicomponent flows can be defined as a mixture of different chemical species on a molecular level, which are flown with the same velocity and temperature, The modelling of such behaviours is important in engineering, e.g. reactor design (Chemical vapour deposition reactors, see [6]) in chemical engineering. Such processes are very complex, while different physical processes occur, e.g. injection, heating, mixturing, homogeneous and heterogeneous chemistry and further, see [7].In the following, we present some typical problems in multicomponent flow problems:• Ionized Species, e.g. plasma problems, see [4].• Combustion of oil, coal or natural gas, see [8].• Chemical reaction processes in chemical engineering, see [6,9,10].• Atmospheric pollution, see [11].Remark 1.1 In the different applications, the term multiphase flow is often used. Here, we define multiphase flows where the phases are immiscible and not chemically related, see [2,12]. So each phase has a separately defined volume fraction and velocity field. Therefore, also the conservation equations for the flow of each species and their interchange between the phases are different from the multicomponent flow. Here, one is taken into account to define a common pressure field, while each phase is related to the gradient of this field and its volume fraction, see [13]. The applications a...

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Section: Iterative Splitting Methodsmentioning

confidence: 99%

“…In the following sections, we concentrate on a basic so-called iterative splitting method, which is discussed in [30].…”

Section: Iterative Splitting Methodsmentioning

confidence: 99%

General Principles

Geiser¹

2015

Multicomponent and Multiscale Systems

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“…Nonetheless, the accuracy of the solution in time can be improved up to the second order by using the bicyclic splitting [10,18,20,28]. …”

Section: Operator Splitting and Two Coordinate Mapsmentioning

confidence: 99%

On an efficient splitting‐based method for solving the diffusion equation on a sphere

Skiba

Filatov

2011

Numerical Methods Partial

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A novel numerical approach for solving the diffusion problem on a sphere is suggested. By using operator splitting, we develop a new method that allows constructing finite difference schemes of the second and fourth approximation orders in the spatial variables. Both schemes properly ensure the balance of mass and the energy dissipation in the L 2 -norm. The schemes are very cheap from the computational standpoint. Numerical results demonstrate the skillfulness of the approach in describing the diffusion dynamics on a sphere. It is shown the method can directly be extended to nonlinear diffusion problems.

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“…5.3 Velocity v of a particle and 3D presentation of the velocity components for one underlying particle, see [3] The SDE system of the Coulomb scattering test particle problem is given in the following form: Remark 5.1 The SDE system is strongly coupled and also nonlinear, therefore, we have taken into account linearization techniques, e.g. fixpoint or Newton's schemes, see also [14,15], or derive higher order methods, e.g. Mitstein schemes, as discussed in [13].…”

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confidence: 99%

Engineering Applications

Geiser

2015

Multicomponent and Multiscale Systems

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In this chapter, we discuss the different engineering applications related to multicomponent and multiscale models, that occur in different categories (microscopic, mesoscopic and macroscopic). As we have described in the Introduction on p. xxv. That such models can be used as basic model to couple to more complicate models, describing materials, interfaces, etc., see Rosso and de Baas (Review of materials modelling: what makes a material function? Let me compute the ways, 2014, [1]).We deal with the following engineering applications with the two classified multiscaling approaches, see also the Introduction on p. xxv and in Fig. 5.1:• Different time-or spatial scales of same model (mono model or basic model).• Linking of different models (different models or multimodel).One of the main contributions to link the different scales and models together are the coupling techniques. In our motivation, such coupling of different time-and spatial scales or coupling of different models, need the suggested methods, e.g. multiscale methods, multicomponent methods, that allow a data transfer between the different scales and models. Such methods, we have introduced in the previous sections and now, we will close the gap between the theoretical discussions of methods and their applications to engineering problems. In such a stage, we have to adapt the numerical schemes with respect to the real-life properties and we obtain truly working multiscale approaches, that we solve the engineering complexities, see [1].Based on the real-life applications, we could study such helpful coupling of different scales or different models. Therefore, we overcome the gap of the recent problem in linking different scales of complicated engineering models in the industrial applications. Multiscale Methods for Langevin-Like EquationsAbstract In this section, we discuss multiscale methods, that solve Langevin-like equations, see [2]. The underlying ideas are to split into a deterministic and stochastic part of the Langevin equation, see [3]. The splitting methods are based on additive and iterative schemes, which are discussed with respect of their benefits and drawbacks, see [4]. We are motivated to reduce computational time for Coulomb collisions in plasma done in particle simulations. We extend splitting schemes, which are well-known in deterministic applications, to stochastic applications and modify the methods with respect to the stochastic terms. Such an idea allows to solve the multiscale behaviour of the coarse deterministic and fine stochastic timescales in an adequate computational time. Introduction of the ProblemIn the underlying problem, we are motivated to develop fast algorithms to solve the Coulomb collisions in plasma simulations, see [5].Recently in the literature, we can find two main ideas to deal with the Coulomb collisions in particle simulations. Here we have the following two ideas:• Binary algorithm: Particles in a finite cell selected into binary pairs. The collision is computed by the scattered velocities by an ...

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Iterative Splitting Methods for Differential Equations

Cited by 56 publications

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General Principles

General Principles

On an efficient splitting‐based method for solving the diffusion equation on a sphere

Engineering Applications

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