Iterative operator-splitting methods for nonlinear differential equations and applications

Geiser, Jürgen

doi:10.1002/num.20568

Cited by 17 publications

(11 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The iterative splitting method defined in [13] and extensively studied for ordinary and partial differential equations in [3,10,15,19,20] are alternative operator splitting methods, which are based on iterative techniques.…”

Section: Iterative Splitting Methodsmentioning

confidence: 99%

Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations

2020

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Iterative Splitting Methodsmentioning

confidence: 99%

Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations

2020

Self Cite

View full text Add to dashboard Cite

show abstract

“…We assume that the boundary conditions are embedded into the spatial discretized matrices and that we deal with ordinary differential equations, see [26].…”

Section: Splitting Methodsmentioning

confidence: 99%

Engineering Applications

Geiser

2015

Multicomponent and Multiscale Systems

View full text Add to dashboard Cite

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 ...

show abstract

“…We first deal with the one-dimensional case, For the iterative operator-splitting as fixed point scheme, we have the following results, see Tables III and V. The result for the iterative operator-splitting method plus Newton's method as linearization technique, see [26], is given in Table IV. Figure 4 presents the profile of the 1D momentum equation.…”

Section: Test Example 3: Momentum Equation (Molecular Flow)mentioning

confidence: 96%

Consistency of iterative operator‐splitting methods: Theory and applications

Geiser

2009

Numerical Methods Partial

View full text Add to dashboard Cite

In this article, we describe a different operator-splitting method for decoupling complex equations with multidimensional and multiphysical processes for applications for porous media and phase-transitions. We introduce different operator-splitting methods with respect to their usability and applicability in computer codes. The error-analysis for the iterative operator-splitting methods is discussed. Numerical examples are presented.

show abstract

Iterative operator-splitting methods for nonlinear differential equations and applications

Cited by 17 publications

References 20 publications

Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations

Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations

Engineering Applications

Consistency of iterative operator‐splitting methods: Theory and applications

Contact Info

Product

Resources

About