In this paper, we present splitting methods that are based on iterative schemes for stochastic di... more In this paper, we present splitting methods that are based on iterative schemes for stochastic differential equations. The methods are applied to plasma simulations. The motivation arose from solving problems involving Coulomb collisions, which are modeled by nonlinear stochastic differential equations. We apply Langevin equations to model these collisions and we obtain coupled nonlinear stochastic differential equations, which are difficult to solve. We propose stochastic splitting schemes that generalise well-known deterministic splitting schemes. The benefit of decomposing the equations into different parts and solving each part individually is taken into account in the analysis of the new iterative splitting schemes. The increase in the convergence order of the iterative splitting scheme with the number of iteration steps is an important and valuable property. The numerical analysis and applications to various problems involving Coulomb collisions in plasma applications are pres...
We are motivated to solve differential algebraic equations with new multi-stage and multisplittin... more 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.
plasmas (CAPs). Such problems are related to the atmospheric pressure and room-temperature regime... more plasmas (CAPs). Such problems are related to the atmospheric pressure and room-temperature regimes. The plasmas are weakly ionized and have high relations of radical concentrations, e.g., oxygen, which are important for applications on surface-modifications. We derive a model based on multicomponent plasma regimes, while. each single species influences the flux-characteristics and the characteristic of the mixture, i.e., the diffusive effects of each specie are important. We assume that in the temperature-and pressure-regimes, the particles have small characteristic length instead of the length in the apparatus and we can derive and apply macroscopic equations. We extend the multicomponent systems (plasma-model) with the Stefan-Maxwell (SM) equation instead of a standard Fickian approach, while we assume to deal with non-dominant gaseous species. For solving such delicate nonlinear SM equations, we propose new iterative splitting methods based on the relaxation approaches. The novel solver methods are tested with multicomponent models in the literature.
We simulate the particle transport in a thin film deposition process made by PVD (physical vapor ... more We simulate the particle transport in a thin film deposition process made by PVD (physical vapor deposition) and present several models for projectile and tar-get collisions in order to compute the mean free path and the differential cross section (angular distribution of scattered projectiles) of the scattering process. A detailed description of collision models is of the highest importance in Monte Carlo simulations of high power impulse magnetron sputtering and DC sputtering. We derive an equation for the mean free path for arbitrary interactions (cross sections) that includes the relative velocity between the particles. We apply our results to two major interaction models: hard sphere interaction & screened Coulomb interaction. Both types of interaction separate DC sputtering from HIPIMS.
In this paper, we present a Picard's iterative method for the solution of nonlinear multicomponen... more In this paper, we present a Picard's iterative method for the solution of nonlinear multicomponent transport equations. The multicomponent transport equations are important for mixture models of the ionized and neutral particles in plasma simulations. Such mixtures deal with the so-called Stefan-Maxwell approaches for the multicomponent diffusion. The underlying nonlinearities are delicate and it is not necessary to be an analytical function of the dependent variables. The proposed solver method is based on Banach's contraction fix-point principle that allows to solve such nonlinearities without making any use to Lagrange multipliers and constrained variations. Such an improvement allows to solve delicate nonlinear problems and we test the application to model with multicomponent transport equations.
In this paper, we present splitting methods that are based on iterative schemes for stochastic di... more In this paper, we present splitting methods that are based on iterative schemes for stochastic differential equations. The methods are applied to plasma simulations. The motivation arose from solving problems involving Coulomb collisions, which are modeled by nonlinear stochastic differential equations. We apply Langevin equations to model these collisions and we obtain coupled nonlinear stochastic differential equations, which are difficult to solve. We propose stochastic splitting schemes that generalise well-known deterministic splitting schemes. The benefit of decomposing the equations into different parts and solving each part individually is taken into account in the analysis of the new iterative splitting schemes. The increase in the convergence order of the iterative splitting scheme with the number of iteration steps is an important and valuable property. The numerical analysis and applications to various problems involving Coulomb collisions in plasma applications are pres...
We are motivated to solve differential algebraic equations with new multi-stage and multisplittin... more 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.
plasmas (CAPs). Such problems are related to the atmospheric pressure and room-temperature regime... more plasmas (CAPs). Such problems are related to the atmospheric pressure and room-temperature regimes. The plasmas are weakly ionized and have high relations of radical concentrations, e.g., oxygen, which are important for applications on surface-modifications. We derive a model based on multicomponent plasma regimes, while. each single species influences the flux-characteristics and the characteristic of the mixture, i.e., the diffusive effects of each specie are important. We assume that in the temperature-and pressure-regimes, the particles have small characteristic length instead of the length in the apparatus and we can derive and apply macroscopic equations. We extend the multicomponent systems (plasma-model) with the Stefan-Maxwell (SM) equation instead of a standard Fickian approach, while we assume to deal with non-dominant gaseous species. For solving such delicate nonlinear SM equations, we propose new iterative splitting methods based on the relaxation approaches. The novel solver methods are tested with multicomponent models in the literature.
We simulate the particle transport in a thin film deposition process made by PVD (physical vapor ... more We simulate the particle transport in a thin film deposition process made by PVD (physical vapor deposition) and present several models for projectile and tar-get collisions in order to compute the mean free path and the differential cross section (angular distribution of scattered projectiles) of the scattering process. A detailed description of collision models is of the highest importance in Monte Carlo simulations of high power impulse magnetron sputtering and DC sputtering. We derive an equation for the mean free path for arbitrary interactions (cross sections) that includes the relative velocity between the particles. We apply our results to two major interaction models: hard sphere interaction & screened Coulomb interaction. Both types of interaction separate DC sputtering from HIPIMS.
In this paper, we present a Picard's iterative method for the solution of nonlinear multicomponen... more In this paper, we present a Picard's iterative method for the solution of nonlinear multicomponent transport equations. The multicomponent transport equations are important for mixture models of the ionized and neutral particles in plasma simulations. Such mixtures deal with the so-called Stefan-Maxwell approaches for the multicomponent diffusion. The underlying nonlinearities are delicate and it is not necessary to be an analytical function of the dependent variables. The proposed solver method is based on Banach's contraction fix-point principle that allows to solve such nonlinearities without making any use to Lagrange multipliers and constrained variations. Such an improvement allows to solve delicate nonlinear problems and we test the application to model with multicomponent transport equations.
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Papers by Juergen Geiser