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Juergen Geiser

Publications and source records attributed to Juergen Geiser.

At least 19 recordsLinked to original sources
arXiv2023

Splitting Approach for Solving Multi-Component Transport Models with Maxwell-Stefan-Diffusion

In this paper, we present splitting algorithms to solve multicomponent transport models with Maxwell-Stefan-diffusion approaches. The multicomponent models are related to transport problems, while we consider plasma processes, in which the local thermodynamic equilibrium and weakly ionized plasma-mixture models are given. Such processes are used for medical and technical applications. These multi-component transport modelling equations are related to convection-diffusion-reactions equations, which are wel-known in transport processes. The multicomponent transport models can be derived from the microscopic multi-component Boltzmann equations with averaging quantities and leads into the macroscopic mass, momentum and energy equations, which are nearly Navier-Stokes-like equations. We discuss the benefits of the decomposition into the convection, diffusion and reaction parts, which allows to use fast numerical solvers for each part. Additional, we concentrate on the nonlinear parts of the multicomponent diffusion, which can be effectively solved with iterative splitting approaches In the numerical experiments, we see the benefit of combining iterative splitting methods with nonlinear solver methods, while these methods can relax the nonlinear terms. In the outview, we discuss the future investigation of the next steps in our multicomponent diffusion approaches.

math.NA↗
arXiv2019

Splitting Methods For Solving Multi-Component Transport Model: A Multicomponent Mixture for Hydrogen Plasma

In this paper, we present a splitting algorithm to solve multicomponent transport models. These models are related to plasma simulations, in which we consider the local thermodynamic equilibrium and weakly ionised plasma-mixture models that are used for medical and technical applications in etching processes. These multi-component transport-mixture models can be derived by approximately solving a linearised multi-component Boltzmann equation with an approximation of the collision terms in the mass, momentum and energy equations. The model-equations are nonlinear partial differential equations and they are known as Stefan-Maxwell equations. However, these partial differential equations are delicate to solve and we propose to use noniterative and iterative splitting methods. In the numerical experiments, we see the benefit of the iterative splitting methods, while these methods can relax the nonlinear terms.

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arXiv2019

Iterative Implicit Methods for Solving Hodgkin-Huxley Type Systems

We are motivated to approximate solutions of a Hodgkin-Huxley type model with implicit methods. As a representative we chose a psychiatric disease model containing stable as well as chaotic cycling behaviour. We analyze the bifurcation pattern and show that some implicit methods help to preserve the limit cycles of such systems. Further, we applied adaptive time stepping for the solvers to boost the accuracy, allowing us a preliminary zoom into the chaotic area of the system.

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arXiv2019

Comparison of Splitting methods for Gross-Pitaevskii Equation

In this paper, we discuss the different splitting approaches to solve the Gross-Pitaevskii equation numerically. We consider conservative finite-difference schemes and spectral methods for the spatial discretisation. Further, we apply implicit or explicit time-integrators and combine such schemes with different splitting approaches. The numerical solutions are compared based on the conservation of the $L_2$-norm with the analytical solutions. The advantages of the splitting methods for large time-domains are presented in several numerical examples of different solitons applications.

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arXiv2018

Modelling approach of a near-far-field model for bubble formation and transport

In this paper, we present a model based on a near-far-field bubble formation. We simulate the formation of a gas-bubble in a liquid, e.g., water and the transportation of such a gas-bubble in the liquid. The modelling approach is based on coupling the near-field model, which is done by the Young-Laplace equation, with the far-field model, which is done with a convection-diffusion equation. We decouple the small and large time- and space scales with respect to each adapted model. Such a decoupling allows to apply the optimal solvers for each near- or far-field model. We discuss the underlying solvers and present the numerical results for the near-far-field bubble formation and transport model.

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arXiv2018

Simulations of Multiscale Schroedinger Equations with Multiscale Splitting Approaches: Theory and Application

In this paper we present a novel multiscale splitting approach to solve multiscale Schroedinger equation, which have large different time-scales. The energy potential is based on highly oscillating functions, which are magnitudes faster than the transport term. We obtain a multiscale problem and a highly stiff problem, while standard solvers need to small time-steps. We propose multiscale solvers, which are based on operator splitting methods and we decouple the diffusion and reaction part of the Schroedinger equation. Such a decomposition allows to apply a large time step for the implicit time-discretization of the diffusion part and small time steps for the explicit and highly oscillating reaction part. With extrapolation steps, we could reduce the computational time in the highly-oscillating time-scale, while we relax into the slow time-scale. We present the numerical analysis of the extrapolated operator splitting method. First numerical experiments verified the benefit of the extrapolated splitting approaches.

math.NA↗
arXiv2017

Iterative Splitting Methods for Coulomb Collisions in Plasma Simulations

In this paper, we present splitting methods that are based on iterative schemes and applied to plasma simulations. The motivation arose of solving the Coulomb collisions, which are modeled by nonlinear stochastic differential equations. We apply Langevin equations to model the characteristics of the collisions and we obtain coupled nonlinear stochastic differential equations, which are delicate to solve. We propose well-known deterministic splitting schemes that can be extended to stochastic splitting schemes, by taking into account the stochastic behavior. The benefit decomposing the different equation parts and solve such parts individual is taken into account in the analysis of the new iterative splitting schemes. Numerical analysis and application to various Coulomb collisions in plasma applications are presented.

math.NA↗
arXiv2016

Multi-stage waveform Relaxation and Multisplitting Methods for Differential Algebraic Systems

We are motivated to solve differential algebraic equations with new multi-stage and multisplitting methods. The multi-stage strategy of the waveform relaxation (WR) methods are given with outer and inner iterations. While the outer iterations decouple the initial value problem of differential algebraic equations (DAEs) in the form of $A \frac{d y(t)}{dt} + B y(t) = f(t)$ to $M_A \frac{d y^{k+1}(t)}{dt} + M_1 y^{k+1}(t) = N_1 y^k(t) + N_A \frac{d y^{k}(t)} + f(t)$, where $A = M_A - N_A$, $B = M_1 - N_1$. The inner iterations decouple further $M_1 = M_2 - N_2$ and $M_2 = M_3 - N_3$ with additional iterative processes, such that we result to invert simpler matrices and accelerate the solver process. The multisplitting method use additional a decomposition of the outer iterative process with parallel algorithms, based on the partition of unity, such that we could improve the solver method. We discuss the different algorithms and present a first experiment based on a DAE system.

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arXiv2015

Numerical Methods of the Maxwell-Stefan Diffusion Equations and Applications in Plasma and Particle Transport

In this paper, we present a model based on a local thermodynamic equilibrium, weakly ionized plasma-mixture model used for medical and technical applications in etching processes. We consider a simplified model based on the Maxwell-Stefan model, which describe multicomponent diffusive fluxes in the gas mixture. Based on additional conditions to the fluxes, we obtain an irreducible and quasi-positive diffusion matrix. Such problems results into nonlinear diffusion equations, which are more delicate to solve as standard diffusion equations with Fickian's approach. We propose explicit time-discretisation methods embedded to iterative solvers for the nonlinearities. Such a combination allows to solve the delicate nonlinear differential equations more effective. We present some first ternary component gaseous mixtures and discuss the numerical methods.

math.NA↗
arXiv2014

Iterative Splitting Methods: Almost Asymptotic Symplectic Integrator for Stochastic Nonlinear Schr\"odinger Equation

In this paper we present splitting methods which are based on iterative schemes and applied to stochastic nonlinear Schroedinger equation. We will design stochastic integrators which almost conserve the symplectic structure. The idea is based on rewriting an iterative splitting approach as a successive approximation method based on a contraction mapping principle and that we have an almost symplectic scheme. We apply a stochastic differential equation, that we can decouple into a deterministic and stochatic part, while each part can be solved analytically. Such decompositions allow accelerating the methods and preserving, under suitable conditions, the symplecticity of the schemes. A numerical analysis and application to the stochastic Schroedinger equation are presented.

math.NA↗
arXiv2014

Effective Simulation Methods for Structures with Local Nonlinearity: Magnus integrator and Successive Approximations

In the following, we discuss nonlinear simulations of nonlinear dynamical systems, which are applied in technical and biological models. We deal with different ideas to overcome the treatment of the nonlinearities and discuss a novel splitting approach. While Magnus expansion has been intensely studied and widely applied for solving explicitly time-dependent problems, it can also be extended to nonlinear problems. By the way it is delicate to extend, while an exponential character have to be computed. Alternative methods, like successive approximation methods, might be an attractive tool, which take into account the temporally in-homogeneous equation (method of Tanabe and Sobolevski). In this work, we consider nonlinear stability analysis with numerical experiments and compare standard integrators to our novel approaches.

math.NA↗
arXiv2014

Comparison of Integrators for Electromagnetic Particle in Cell Methods: Algorithms and Applications

In this paper, we present different types of integrators for electro-magnetic particle-in-cell (PIC) methods. While the integrator is an important tool of the PIC methods, it is necessary to characterize the different conservation approaches of the integrators, e.g. symplecticity, energy- or charge-conservation. We discuss the different principles, e.g. composition, filtering, explicit and implicit ideas. While, particle in cell methods are well-studied, the combination between the different parts, i.e. pusher, solver and approximations are hardly to analyze. we concentrate on choosing the optimal pusher component, with respect to conservation and convergence behavior. We discuss oscillations of the pusher component, strong external magnetic fields and optimal conservation of energy and momentum. The algorithmic ideas are discussed and numerical experiments compare the exactness of the different schemes. An outlook to overcome the different error components is discussed in the future works.

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arXiv2012

Simulation of a Heat Transfer in Porous Media

We are motivated to model a heat transfer to a multiple layer regime and their optimization for heat energy resources. Such a problem can be modeled by a porous media with different phases (liquid and solid). The idea arose of a geothermal energy reservoir which can be used by cities, e.g. Berlin. While hot ground areas are covered to most high populated cites, the energy resources are important and a shift to use such resources are enormous. We design a model of the heat transport via the flow of water through the heterogeneous layer of the underlying earth sediments. We discuss a multiple layer model, based on mobile and immobile zones. Such numerical simulations help to economize on expensive physical experiments and obtain control mechanisms for the delicate heating process.

math-ph↗
arXiv2012

Multiple Iterative Splitting method for Higher order and Integro-differental equations

In this paper we present an extension of standard iterative splitting schemes to multiple splitting schemes for solving higher order differential equations. We are motivated by dynamical systems, which occur in dynamics of the electrons in the plasma using a simplified Boltzmann equation. Oscillation problems in spectroscopy problems using wave-equations. The motivation arose to simulate active plasma resonance spectroscopy which is used for plasma diagnostic techniques.

math.NA↗
arXiv2012

Embedded Zassenhaus Expansion to Operator Splitting Schemes: Theory and Application in Fluid Dynamics

In this paper, we contribute operator-splitting methods improved by the Zassenhaus product for the numerical solution of linear partial differential equations. We address iterative splitting methods, that can be improved by means of the Zassenhaus product formula, which is a sequnential splitting scheme. The coupling of iterative and sequential splitting techniques are discussed and can be combined with respect to their compuational time. While the iterative splitting schemes are cheap to compute, the Zassenhaus product formula is more expensive, based on the commutators but achieves higher order accuracy. Iterative splitting schemes and also Zassenhaus products are applied in physics and physical chemistry are important and are predestinated to their combinations of each benefits. Here we consider phase models in CFD (computational fluid dynamics). We present an underlying analysis for obtaining higher order operator-splitting methods based on the Zassenhaus product. Computational benefits are given with sparse matrices, which arose of spatial discretization of the underlying partial differential equations. While Zassenhaus formula allows higher accuracy, due to the fact that we obtain higher order commutators, we combine such an improved initialization process to cheap computable to linear convergent iterative splitting schemes. Theoretical discussion about convergence and application examples are discussed with CFD problems.

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arXiv2012

Multiscale methods for Levitron Problems: Theory and Applications

In this paper, we describe a multiscale model based on magneto-static traps of neutral atoms or ion traps. The idea is to levitate a magnetic spinning top in the air repelled by a base magnet. For such a problem, we have to deal with different time and spatial scales and we propose a novel splitting method for solving the levitron problem. We focus on the multiscale problem, which we obtain by coupling the kinetic T and the potential U part of our equation. The kinetic and potential parts, can be seen as generators of flows. The main problem is based on the accurate computation of the Hamiltonian equation and we propose a novel higher order splitting scheme to obtain stable states near the relative equilibrium. To improve the splitting scheme we apply a novel method so called MPE (multiproduct expansion method), which include higher order extrapolation schemes. In numerical studies, we discuss the stability near this relative equilibrium with our improved time-integrators. Best results are obtained by iterative and extrapolated Verlet schemes in comparison to higher order explicit Runge-Kutta schemes. Experiments are applied to a magnetic top in an axisymmetric magnetic field (i.e. the Levitron) and we discuss the future applications to quantum computations.

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