Pub Date : 2009-11-24DOI: 10.1080/00411450903372084
A. Constantinescu, Dmitriy Anistratov
We study the convergence of the quasidiffusion (QD) method on one-dimensional spatially periodic heterogeneous problems. The QD method is a nonlinear projection-iterative method. A Fourier analysis of the linearized QD equations is performed. The convergence rates of the QD method in the vicinity of the solution are obtained. We also analyze the Second Moment (SM) method, which can be interpreted as a linear version of the QD method. The presented analysis gives a new insight on the convergence behavior of the QD method in a discretized form and reveals the differences in the convergence of the QD and SM methods. Numerical results are presented to confirm theoretical predictions.
{"title":"Stability Analysis of the Quasidiffusion Method on Periodic Heterogeneous 1D Transport Problems","authors":"A. Constantinescu, Dmitriy Anistratov","doi":"10.1080/00411450903372084","DOIUrl":"https://doi.org/10.1080/00411450903372084","url":null,"abstract":"We study the convergence of the quasidiffusion (QD) method on one-dimensional spatially periodic heterogeneous problems. The QD method is a nonlinear projection-iterative method. A Fourier analysis of the linearized QD equations is performed. The convergence rates of the QD method in the vicinity of the solution are obtained. We also analyze the Second Moment (SM) method, which can be interpreted as a linear version of the QD method. The presented analysis gives a new insight on the convergence behavior of the QD method in a discretized form and reveals the differences in the convergence of the QD and SM methods. Numerical results are presented to confirm theoretical predictions.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"38 1","pages":"295 - 316"},"PeriodicalIF":0.0,"publicationDate":"2009-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450903372084","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58913310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-10-28DOI: 10.1080/00411450.2011.604568
D. B'enisti, O. Morice, L. Gremillet, D. Strozzi
In this article, we provide a theoretical description and calculate the nonlinear frequency shift, group velocity, and collionless damping rate, ν, of a driven electron plasma wave (EPW). All these quantities, whose physical content will be discussed, are identified as terms of an envelope equation allowing one to predict how efficiently an EPW may be externally driven. This envelope equation is derived directly from Gauss’ law and from the investigation of the nonlinear electron motion, provided that the time and space rates of variation of the EPW amplitude, , are small compared to the plasma frequency or the inverse of the Debye length. ν arises within the EPW envelope equation as a more complicated operator than a plain damping rate and may only be viewed as such because [] remains nearly constant before abruptly dropping to zero. We provide a practical analytic formula for ν and show, without resorting to complex contour deformation, that in the limit 0, ν is nothing but the Landau damping rate. We then term ν the “nonlinear Landau damping rate” of the driven plasma wave. As for the nonlinear frequency shift of the driven EPW, it is also derived theoretically and found to assume values significantly different from previously published ones, which were obtained by assuming that the wave was freely propagating. Moreover, we find no limitation in , being the plasma wavenumber and the Debye length, for a solution to the dispertion relation to exist, and want to stress here the importance of specifying how an EPW is generated to discuss its properties. Our theoretical predictions are in excellent agreement with results inferred from Vlasov simulations of stimulated Raman scattering (SRS), and an application of our theory to the study of SRS is presented.
{"title":"Nonlinear Envelope Equation and Nonlinear Landau Damping Rate for a Driven Electron Plasma Wave","authors":"D. B'enisti, O. Morice, L. Gremillet, D. Strozzi","doi":"10.1080/00411450.2011.604568","DOIUrl":"https://doi.org/10.1080/00411450.2011.604568","url":null,"abstract":"In this article, we provide a theoretical description and calculate the nonlinear frequency shift, group velocity, and collionless damping rate, ν, of a driven electron plasma wave (EPW). All these quantities, whose physical content will be discussed, are identified as terms of an envelope equation allowing one to predict how efficiently an EPW may be externally driven. This envelope equation is derived directly from Gauss’ law and from the investigation of the nonlinear electron motion, provided that the time and space rates of variation of the EPW amplitude, , are small compared to the plasma frequency or the inverse of the Debye length. ν arises within the EPW envelope equation as a more complicated operator than a plain damping rate and may only be viewed as such because [] remains nearly constant before abruptly dropping to zero. We provide a practical analytic formula for ν and show, without resorting to complex contour deformation, that in the limit 0, ν is nothing but the Landau damping rate. We then term ν the “nonlinear Landau damping rate” of the driven plasma wave. As for the nonlinear frequency shift of the driven EPW, it is also derived theoretically and found to assume values significantly different from previously published ones, which were obtained by assuming that the wave was freely propagating. Moreover, we find no limitation in , being the plasma wavenumber and the Debye length, for a solution to the dispertion relation to exist, and want to stress here the importance of specifying how an EPW is generated to discuss its properties. Our theoretical predictions are in excellent agreement with results inferred from Vlasov simulations of stimulated Raman scattering (SRS), and an application of our theory to the study of SRS is presented.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"48 1","pages":"185 - 224"},"PeriodicalIF":0.0,"publicationDate":"2009-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2011.604568","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58909251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-10-27DOI: 10.1080/00411450.2011.654750
L. Marechal, Julien Perez
This article presents elements about the radial orbit instability, which occurs in spherical self-gravitating systems with a strong anisotropy in the radial velocity direction. It contains an overview on the history of radial orbit instability. We also present the symplectic method we use to explore stability of equilibrium states directly related to the dissipation induced instability mechanism well known in theoretical mechanics and plasma physics.
{"title":"Radial Orbit Instability: Review and Perspectives","authors":"L. Marechal, Julien Perez","doi":"10.1080/00411450.2011.654750","DOIUrl":"https://doi.org/10.1080/00411450.2011.654750","url":null,"abstract":"This article presents elements about the radial orbit instability, which occurs in spherical self-gravitating systems with a strong anisotropy in the radial velocity direction. It contains an overview on the history of radial orbit instability. We also present the symplectic method we use to explore stability of equilibrium states directly related to the dissipation induced instability mechanism well known in theoretical mechanics and plasma physics.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"40 1","pages":"425 - 439"},"PeriodicalIF":0.0,"publicationDate":"2009-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2011.654750","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58909117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-10-20DOI: 10.1080/00411450903238665
K. C. Assi, M. Laforest
We consider the spatial difference in L 1 between solutions to two different discrete velocity models in one space dimension. We assume that the second (fine) model is obtained by adding new velocities to the first (coarse) model, although the collision operators can be completely different. The 1-D discrete velocity models studied here include projections of n-D models, as described by Beale. This work adapts the nonlinear and decreasing interaction functional of Ha and Tzavaras for discrete velocity models in 1-D in order to measure the distance in L 1 . The resulting functional increases by a term proportional to the residual of the modeling error for the coarse model. The modeling error can therefore be computed a posteriori and can be used to determine which discrete velocity model within a hierarchy satisfies a prescribed accuracy.
{"title":"Modeling Error in L1 for a Hierarchy of 1-D Discrete Velocity Models","authors":"K. C. Assi, M. Laforest","doi":"10.1080/00411450903238665","DOIUrl":"https://doi.org/10.1080/00411450903238665","url":null,"abstract":"We consider the spatial difference in L 1 between solutions to two different discrete velocity models in one space dimension. We assume that the second (fine) model is obtained by adding new velocities to the first (coarse) model, although the collision operators can be completely different. The 1-D discrete velocity models studied here include projections of n-D models, as described by Beale. This work adapts the nonlinear and decreasing interaction functional of Ha and Tzavaras for discrete velocity models in 1-D in order to measure the distance in L 1 . The resulting functional increases by a term proportional to the residual of the modeling error for the coarse model. The modeling error can therefore be computed a posteriori and can be used to determine which discrete velocity model within a hierarchy satisfies a prescribed accuracy.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"38 1","pages":"245 - 272"},"PeriodicalIF":0.0,"publicationDate":"2009-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450903238665","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58913245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-10-20DOI: 10.1080/00411450903238707
A. Yilmazer, C. Kocar
This study is aimed at demonstrating the numerical convergency of spectral polynomial approximations to the radiative transfer equation in spherical media. To this end, T N method is employed as a representative of classical polynomial approximations to the corresponding pseudo-slab problem of spherical media radiative transfer equation. The method is used to calculate the albedo and density for isotropic scattering in a homogeneous spherical medium. Spherical harmonics or P N method is also applied to the same problem for comparison purposes. Benchmark results of both methods for an absorbing and scattering spherical medium transfer problem are reported. The results contribute remarkable improvements to the previously reported data in the literature. More importantly, on contrary to some literature works, numerical divergence of spherical harmonics method in spherical media transfer problems is illustrated to be unsound provided that arbitrary precision arithmetic is available when considering higher order approximations.
{"title":"Some Benchmark Results in Spherical Media Radiative Transfer Problems","authors":"A. Yilmazer, C. Kocar","doi":"10.1080/00411450903238707","DOIUrl":"https://doi.org/10.1080/00411450903238707","url":null,"abstract":"This study is aimed at demonstrating the numerical convergency of spectral polynomial approximations to the radiative transfer equation in spherical media. To this end, T N method is employed as a representative of classical polynomial approximations to the corresponding pseudo-slab problem of spherical media radiative transfer equation. The method is used to calculate the albedo and density for isotropic scattering in a homogeneous spherical medium. Spherical harmonics or P N method is also applied to the same problem for comparison purposes. Benchmark results of both methods for an absorbing and scattering spherical medium transfer problem are reported. The results contribute remarkable improvements to the previously reported data in the literature. More importantly, on contrary to some literature works, numerical divergence of spherical harmonics method in spherical media transfer problems is illustrated to be unsound provided that arbitrary precision arithmetic is available when considering higher order approximations.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"13 1","pages":"273 - 292"},"PeriodicalIF":0.0,"publicationDate":"2009-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450903238707","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58913298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-09-16DOI: 10.1080/00411450903196228
M. Boulanouar
This article deals with new trace theorems for neutronic function spaces.
本文讨论了中子函数空间的新迹定理。
{"title":"New Trace Theorems for Neutronic Function Spaces","authors":"M. Boulanouar","doi":"10.1080/00411450903196228","DOIUrl":"https://doi.org/10.1080/00411450903196228","url":null,"abstract":"This article deals with new trace theorems for neutronic function spaces.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"38 1","pages":"228 - 242"},"PeriodicalIF":0.0,"publicationDate":"2009-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450903196228","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58912570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-09-16DOI: 10.1080/00411450903192938
L. Bourhrara
In this article, we propose new approximations of the neutron transport equation based on the transport variational formulations presented in Bourhrara (2004). These approximations are solutions of associated variational problems. We give an error estimate between the angular flux, exact solution of the transport equation, and the approximate solutions. We also derive variational formulations for even- and odd- parity fluxes as approximations of the neutron transport equation.
{"title":"W N Approximations Of Neutron Transport Equation","authors":"L. Bourhrara","doi":"10.1080/00411450903192938","DOIUrl":"https://doi.org/10.1080/00411450903192938","url":null,"abstract":"In this article, we propose new approximations of the neutron transport equation based on the transport variational formulations presented in Bourhrara (2004). These approximations are solutions of associated variational problems. We give an error estimate between the angular flux, exact solution of the transport equation, and the approximate solutions. We also derive variational formulations for even- and odd- parity fluxes as approximations of the neutron transport equation.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"1 1","pages":"195 - 227"},"PeriodicalIF":0.0,"publicationDate":"2009-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450903192938","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58913001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-09-11DOI: 10.1080/00411450.2011.651055
G. Belmont, T. Chust, F. Mottez, S. Hess
The linear Landau effect is revisited by the means of numerical simulations and analytical calculations. The existence of non-Landau solutions to the Vlasov-Poisson system is emphasized and the consistency of these solutions with respect to the arguments based on energy is investigated. The present article briefly summarizes the content of two articles already published on the subject and introduces a discussion based on the exchanges that occurred at Marseille during the Vlasovia meeting.
{"title":"Landau and Non-Landau Linear Damping: Physics of the Dissipation","authors":"G. Belmont, T. Chust, F. Mottez, S. Hess","doi":"10.1080/00411450.2011.651055","DOIUrl":"https://doi.org/10.1080/00411450.2011.651055","url":null,"abstract":"The linear Landau effect is revisited by the means of numerical simulations and analytical calculations. The existence of non-Landau solutions to the Vlasov-Poisson system is emphasized and the consistency of these solutions with respect to the arguments based on energy is investigated. The present article briefly summarizes the content of two articles already published on the subject and introduces a discussion based on the exchanges that occurred at Marseille during the Vlasovia meeting.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"40 1","pages":"419 - 424"},"PeriodicalIF":0.0,"publicationDate":"2009-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2011.651055","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58909106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-09-04DOI: 10.1080/00411450903187722
Haoyang Gao, Hongkai Zhao
In this article, we develop an efficient forward solver for steady-state or frequency-domain radiative transfer equation on 2D and 3D structured and unstructured meshes with vacuum boundary condition or reflection boundary condition. Although we emphasize applications in optical imaging, our method may be applied to more general radiative transfer equations, for example, other types of scattering function. Efficiency and accuracy of our algorithm is demonstrated by comparison with both analytical solutions and Monte Carlo solutions and with various numerical tests in optical imaging.
{"title":"A Fast-Forward Solver of Radiative Transfer Equation","authors":"Haoyang Gao, Hongkai Zhao","doi":"10.1080/00411450903187722","DOIUrl":"https://doi.org/10.1080/00411450903187722","url":null,"abstract":"In this article, we develop an efficient forward solver for steady-state or frequency-domain radiative transfer equation on 2D and 3D structured and unstructured meshes with vacuum boundary condition or reflection boundary condition. Although we emphasize applications in optical imaging, our method may be applied to more general radiative transfer equations, for example, other types of scattering function. Efficiency and accuracy of our algorithm is demonstrated by comparison with both analytical solutions and Monte Carlo solutions and with various numerical tests in optical imaging.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"38 1","pages":"149 - 192"},"PeriodicalIF":0.0,"publicationDate":"2009-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450903187722","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58912988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-08-20DOI: 10.1080/00411450903081313
C. Lancellotti, J. Dorning
In this article, we report the mathematical details of a systematic analysis of the nonlinear Landau damping of longitudinal electrostatic waves propagating in a collisionless plasma. Many of the main results have been reported previously; unfortunately, major parts of the essential mathematical developments had to be omitted for various reasons, making it almost impossible for even the most well-prepared reader to follow the analysis. Sufficient details are provided here to remedy this situation. Some important results that have not been reported previously are also are included here. Most notably among these is the distinction between strong branches and weak branches of nonzero time-asymptotic electric field amplitudes that bifurcate from the zero-amplitude solution for the time-asymptotic electric field, and the results on the weak branches that had to be omitted in the earlier report of this research. Based on the decomposition of the electric field E into a transient part T and a time-asymptotic part A, we show that A is given by a finite superposition of wave modes, whose frequencies obey a Vlasov dispersion relation and whose amplitudes satisfy a set of nonlinear algebraic equations. These time-asymptotic mode amplitudes are calculated explicitly based on approximate solutions for the particle distribution functions obtained by linearizing only the term that contains T in the Vlasov equation for each particle species and then integrating the resulting equation along the nonlinear characteristics associated with A, which are obtained via Hamiltonian perturbation theory. For “linearly stable” initial Vlasov equilibria, we obtain a critical initial amplitude, separating the initial conditions that Landau damp to zero from those that lead to nonzero multiple-traveling-wave time-asymptotic states via nonlinear particle trapping. These theoretical results explain why in some cases experiments and large-scale numerical simulations have resulted in zero-field final states; whereas in other cases they have yielded nonzero multiple-traveling-wave final states because the theoretical results establish the existence of a “threshold” in the initial electric field below which the field damps to zero and above which it evolves to a finite-amplitude multiple-traveling-wave final state.
{"title":"Nonlinear Landau Damping","authors":"C. Lancellotti, J. Dorning","doi":"10.1080/00411450903081313","DOIUrl":"https://doi.org/10.1080/00411450903081313","url":null,"abstract":"In this article, we report the mathematical details of a systematic analysis of the nonlinear Landau damping of longitudinal electrostatic waves propagating in a collisionless plasma. Many of the main results have been reported previously; unfortunately, major parts of the essential mathematical developments had to be omitted for various reasons, making it almost impossible for even the most well-prepared reader to follow the analysis. Sufficient details are provided here to remedy this situation. Some important results that have not been reported previously are also are included here. Most notably among these is the distinction between strong branches and weak branches of nonzero time-asymptotic electric field amplitudes that bifurcate from the zero-amplitude solution for the time-asymptotic electric field, and the results on the weak branches that had to be omitted in the earlier report of this research. Based on the decomposition of the electric field E into a transient part T and a time-asymptotic part A, we show that A is given by a finite superposition of wave modes, whose frequencies obey a Vlasov dispersion relation and whose amplitudes satisfy a set of nonlinear algebraic equations. These time-asymptotic mode amplitudes are calculated explicitly based on approximate solutions for the particle distribution functions obtained by linearizing only the term that contains T in the Vlasov equation for each particle species and then integrating the resulting equation along the nonlinear characteristics associated with A, which are obtained via Hamiltonian perturbation theory. For “linearly stable” initial Vlasov equilibria, we obtain a critical initial amplitude, separating the initial conditions that Landau damp to zero from those that lead to nonzero multiple-traveling-wave time-asymptotic states via nonlinear particle trapping. These theoretical results explain why in some cases experiments and large-scale numerical simulations have resulted in zero-field final states; whereas in other cases they have yielded nonzero multiple-traveling-wave final states because the theoretical results establish the existence of a “threshold” in the initial electric field below which the field damps to zero and above which it evolves to a finite-amplitude multiple-traveling-wave final state.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"9 1","pages":"1 - 146"},"PeriodicalIF":0.0,"publicationDate":"2009-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450903081313","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58912943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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