Pub Date : 2010-03-31DOI: 10.1080/00411450.2010.542711
P. Choquard
The Monge problem (Monge 1781; Taton 1951), as reformulated by Kantorovich (2006a, 2006b) is that of the transportation at a minimum “cost” of a given mass distribution from an initial to a final position during a given time interval. It is an optimal transport problem (Villani, 2003, sects. 1, 2). Following the fluid mechanical solution provided by Benamou and Brenier (2000) for quadratic cost functions (Villani, 2003, sects. 5.4, 8.1), Lagrangian formulations are needed to solve this boundary value problem in time and to determine the Actions as time integral of Lagrangians that are measures of the “cost” of the transportations (Benamou and Brenier, 2000, prop. 1.1). Given canonical Hamilltonians of perfect and self-interacting systems expressed in function of mass densities and velocity potentials, four versions of explicit constructions of Lagrangians, with their corresponding generalized coordinates, are proposed: elimination of the velocity potentials as a function of the densities and their time derivatives by inversion of the continuity equations; elimination of the gradient of the velocity potentials from the continuity equations thanks to the introduction of vector fields such that their divergences give the mass densities; generalization in nD of Gelfand mass coordinate (1963) by the introduction of n-dimensional vector fields such that the determinant of their Jacobian matrices give the mass densities; and, last, introduction of the Lagrangian coordinates that describe the characteristics of the different models and are parametrized by the former auxiliary vector fields. Using this version, weak solutions of several models of Coulombian and Newtonian systems known in Plasma Physics and in Cosmology, with spherically symmetric boundary densities, are given as illustrations.
{"title":"On a Class of Mean Field Solutions of the Monge Problem for Perfect and Self-Interacting Systems","authors":"P. Choquard","doi":"10.1080/00411450.2010.542711","DOIUrl":"https://doi.org/10.1080/00411450.2010.542711","url":null,"abstract":"The Monge problem (Monge 1781; Taton 1951), as reformulated by Kantorovich (2006a, 2006b) is that of the transportation at a minimum “cost” of a given mass distribution from an initial to a final position during a given time interval. It is an optimal transport problem (Villani, 2003, sects. 1, 2). Following the fluid mechanical solution provided by Benamou and Brenier (2000) for quadratic cost functions (Villani, 2003, sects. 5.4, 8.1), Lagrangian formulations are needed to solve this boundary value problem in time and to determine the Actions as time integral of Lagrangians that are measures of the “cost” of the transportations (Benamou and Brenier, 2000, prop. 1.1). Given canonical Hamilltonians of perfect and self-interacting systems expressed in function of mass densities and velocity potentials, four versions of explicit constructions of Lagrangians, with their corresponding generalized coordinates, are proposed: elimination of the velocity potentials as a function of the densities and their time derivatives by inversion of the continuity equations; elimination of the gradient of the velocity potentials from the continuity equations thanks to the introduction of vector fields such that their divergences give the mass densities; generalization in nD of Gelfand mass coordinate (1963) by the introduction of n-dimensional vector fields such that the determinant of their Jacobian matrices give the mass densities; and, last, introduction of the Lagrangian coordinates that describe the characteristics of the different models and are parametrized by the former auxiliary vector fields. Using this version, weak solutions of several models of Coulombian and Newtonian systems known in Plasma Physics and in Cosmology, with spherically symmetric boundary densities, are given as illustrations.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"39 1","pages":"313 - 359"},"PeriodicalIF":0.0,"publicationDate":"2010-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2010.542711","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58908676","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 : 2010-03-31DOI: 10.1080/00411450.2011.563711
B. Eliasson
We present a review of recent developments of simulations of the Vlasov-Maxwell system of equations using a Fourier transform method in velocity space. In this method, the distribution functions for electrons and ions are Fourier transformed in velocity space, and the resulting set of equations are solved numerically. In the original Vlasov equation, phase mixing may lead to an oscillatory behavior and sharp gradients of the distribution function in velocity space, which is problematic in simulations where it can lead to unphysical electric fields and instabilities and to the recurrence effect where parts of the initial condition recur in the simulation. The particle distribution function is in general smoother in the Fourier-transformed velocity space, which is desirable for the numerical approximations. By designing outflow boundary conditions in the Fourier-transformed velocity space, the highest oscillating terms are allowed to propagate out through the boundary and are removed from the calculations, thereby strongly reducing the numerical recurrence effect. The outflow boundary conditions in higher dimensions including electromagnetic effects are discussed. The Fourier transform method is also suitable to solve the Fourier-transformed Wigner equation, which is the quantum mechanical analogue of the Vlasov equation for classical particles.
{"title":"Numerical Simulations of the Fourier-Transformed Vlasov-Maxwell System in Higher Dimensions—Theory and Applications","authors":"B. Eliasson","doi":"10.1080/00411450.2011.563711","DOIUrl":"https://doi.org/10.1080/00411450.2011.563711","url":null,"abstract":"We present a review of recent developments of simulations of the Vlasov-Maxwell system of equations using a Fourier transform method in velocity space. In this method, the distribution functions for electrons and ions are Fourier transformed in velocity space, and the resulting set of equations are solved numerically. In the original Vlasov equation, phase mixing may lead to an oscillatory behavior and sharp gradients of the distribution function in velocity space, which is problematic in simulations where it can lead to unphysical electric fields and instabilities and to the recurrence effect where parts of the initial condition recur in the simulation. The particle distribution function is in general smoother in the Fourier-transformed velocity space, which is desirable for the numerical approximations. By designing outflow boundary conditions in the Fourier-transformed velocity space, the highest oscillating terms are allowed to propagate out through the boundary and are removed from the calculations, thereby strongly reducing the numerical recurrence effect. The outflow boundary conditions in higher dimensions including electromagnetic effects are discussed. The Fourier transform method is also suitable to solve the Fourier-transformed Wigner equation, which is the quantum mechanical analogue of the Vlasov equation for classical particles.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"39 1","pages":"387 - 465"},"PeriodicalIF":0.0,"publicationDate":"2010-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2011.563711","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58908737","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 : 2010-03-31DOI: 10.1080/00411450.2011.563707
M. Firpo
When starting from specific initial conditions, the ferromagnetic-like XY Hamiltonian mean field (HMF) model evolves toward quasistationary states, with lifetimes diverging with the number N of degrees of freedom that violate equilibrium statistical mechanics predictions. Phase transitions have been reported between low-energy magnetized quasistationary states and large energy unexpected, antiferromagnetic-like ones with low, but nonvanishing, magnetization. This issue is addressed here in the Vlasov N→∞ limit. It is argued that the time asymptotic states emerging in the Vlasov limit can be related to simple generic time asymptotic forms for the force field. The proposed picture unveils the nature of the out-of-equilibrium phase transitions reported for the ferromagnetic HMF in the second order regime: This is a bifurcation point connecting an effective integrable Vlasov one-particle time-asymptotic dynamic to a partly ergodic one, which means an abrupt open-up of the Vlasov one-particle phase space. This is proposed as a mechanism for second-order phase transitions compatible with nonvanishing time-asymptotic values of the order parameter in mean-field long-range systems.
{"title":"Out-of-Equilibrium Phase Transitions and Time-Asymptotic One-Particle Dynamics in the Vlasov Limit of The Hamiltonian Mean Field Model","authors":"M. Firpo","doi":"10.1080/00411450.2011.563707","DOIUrl":"https://doi.org/10.1080/00411450.2011.563707","url":null,"abstract":"When starting from specific initial conditions, the ferromagnetic-like XY Hamiltonian mean field (HMF) model evolves toward quasistationary states, with lifetimes diverging with the number N of degrees of freedom that violate equilibrium statistical mechanics predictions. Phase transitions have been reported between low-energy magnetized quasistationary states and large energy unexpected, antiferromagnetic-like ones with low, but nonvanishing, magnetization. This issue is addressed here in the Vlasov N→∞ limit. It is argued that the time asymptotic states emerging in the Vlasov limit can be related to simple generic time asymptotic forms for the force field. The proposed picture unveils the nature of the out-of-equilibrium phase transitions reported for the ferromagnetic HMF in the second order regime: This is a bifurcation point connecting an effective integrable Vlasov one-particle time-asymptotic dynamic to a partly ergodic one, which means an abrupt open-up of the Vlasov one-particle phase space. This is proposed as a mechanism for second-order phase transitions compatible with nonvanishing time-asymptotic values of the order parameter in mean-field long-range systems.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"39 1","pages":"370 - 386"},"PeriodicalIF":0.0,"publicationDate":"2010-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2011.563707","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58908727","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 : 2010-03-01DOI: 10.1080/00411450.2010.535088
R. McClarren
In celebration of the 50th anniversary of the simplified Pn equations (SPn), this work reviews the theory underpinning the SPnequations for neutral particle transport. We recount the derivation of these equations by Gelbard’s formal procedure and by more recent asymptotic and variational analyses. The relation between the SPn equations and several other low order approximations is discussed. Also, the conditions under which the SPn equations are equivalent to the full Pn equations of the same order are discussed as well as the accuracy of the SPnequations. Several open problems in the theory of the SPn equations are posed.
{"title":"Theoretical Aspects of the Simplified Pn Equations","authors":"R. McClarren","doi":"10.1080/00411450.2010.535088","DOIUrl":"https://doi.org/10.1080/00411450.2010.535088","url":null,"abstract":"In celebration of the 50th anniversary of the simplified Pn equations (SPn), this work reviews the theory underpinning the SPnequations for neutral particle transport. We recount the derivation of these equations by Gelbard’s formal procedure and by more recent asymptotic and variational analyses. The relation between the SPn equations and several other low order approximations is discussed. Also, the conditions under which the SPn equations are equivalent to the full Pn equations of the same order are discussed as well as the accuracy of the SPnequations. Several open problems in the theory of the SPn equations are posed.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"39 1","pages":"109 - 73"},"PeriodicalIF":0.0,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2010.535088","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58908629","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 : 2010-03-01DOI: 10.1080/00411450.2010.531879
R. McClarren
In 1968, Ely Gelbard quipped in reference to the spherical harmonics (Pn) method, “Now we live in a period when whole fields of technology are born, mature, and, sometimes, die in a decade. In times when physics changes so quickly a 40-year-old method must be regarded as very old indeed,” (Gelbard, 1968). In this special issue we celebrate a very old method conceived by Gelbard as a simplification of the spherical harmonics method: the simplified Pn or SPn method. In a September 1960 review of nuclear reactor technology at the Bettis Atomic Power Laboratory (Gelbard, 1960), Gelbard presented a simplification to the full spherical harmonics method that greatly reduced the number of unknowns, and also provided an analysis of when the method was equivalent to the full Pn equations. A new method was born and researchers have been applying and studying it for the past 50 years. I was motivated to organize this special issue in 2009 while trying to extend the equivalence of the SPn and Pn equations to more general cases. In reading the literature I realized that the next year was the 50th anniversary of SPn. Like many in the field of transport I was initially intrigued by the simplicity of SPn, later amazed that in many cases SPn is equivalent to the more complicated Pn equations, and finally disappointed that in some problems it can give worse answers than diffusion.1 To encapsulate all of these feelings and to highlight the successes in using SPn, I decided that a special issue commemorating SPn was in order. I hope that this issue both takes a snapshot of the current uses and understanding of
1968年，Ely Gelbard在提到球谐波(Pn)方法时打趣道:“现在我们生活在一个整个技术领域诞生、成熟，有时在十年内消亡的时期。在物理学变化如此之快的时代，一个有40年历史的方法必须被认为是非常古老的，”(Gelbard, 1968)。在这期特刊中，我们颂扬一个非常古老的方法，这个方法是由Gelbard提出的，它是球面谐波方法的简化:简化Pn或SPn方法。1960年9月，在贝蒂斯原子能实验室(Bettis Atomic Power Laboratory)对核反应堆技术的回顾中(Gelbard, 1960)， Gelbard提出了对全球面谐波法的简化，大大减少了未知数的数量，并分析了该方法何时与全Pn方程等效。一种新的方法诞生了，研究人员在过去的50年里一直在应用和研究它。我在2009年试图将SPn和Pn方程的等价性扩展到更一般的情况时，受到了组织这期特刊的激励。在阅读文献时，我意识到明年是SPn成立50周年。像许多输运领域的人一样，我最初对SPn的简单性很感兴趣，后来惊讶于在许多情况下SPn等同于更复杂的Pn方程，最后失望的是，在某些问题上它可以给出比扩散更差的答案为了概括所有这些感受并突出使用SPn的成功，我决定出版一期纪念SPn的专刊。我希望这个问题既能让大家对当前的使用和理解有一个大致的了解
{"title":"Fifty Years of the Simplified Pn Method","authors":"R. McClarren","doi":"10.1080/00411450.2010.531879","DOIUrl":"https://doi.org/10.1080/00411450.2010.531879","url":null,"abstract":"In 1968, Ely Gelbard quipped in reference to the spherical harmonics (Pn) method, “Now we live in a period when whole fields of technology are born, mature, and, sometimes, die in a decade. In times when physics changes so quickly a 40-year-old method must be regarded as very old indeed,” (Gelbard, 1968). In this special issue we celebrate a very old method conceived by Gelbard as a simplification of the spherical harmonics method: the simplified Pn or SPn method. In a September 1960 review of nuclear reactor technology at the Bettis Atomic Power Laboratory (Gelbard, 1960), Gelbard presented a simplification to the full spherical harmonics method that greatly reduced the number of unknowns, and also provided an analysis of when the method was equivalent to the full Pn equations. A new method was born and researchers have been applying and studying it for the past 50 years. I was motivated to organize this special issue in 2009 while trying to extend the equivalence of the SPn and Pn equations to more general cases. In reading the literature I realized that the next year was the 50th anniversary of SPn. Like many in the field of transport I was initially intrigued by the simplicity of SPn, later amazed that in many cases SPn is equivalent to the more complicated Pn equations, and finally disappointed that in some problems it can give worse answers than diffusion.1 To encapsulate all of these feelings and to highlight the successes in using SPn, I decided that a special issue commemorating SPn was in order. I hope that this issue both takes a snapshot of the current uses and understanding of","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"39 1","pages":"71 - 72"},"PeriodicalIF":0.0,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2010.531879","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58908985","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 : 2010-03-01DOI: 10.1080/00411450.2010.533742
J. Densmore, R. McClarren
We extend moment analysis, a technique developed for investigating the accuracy of discrete-ordinates spatial discretization schemes, to time-dependent radiation transport and apply it to several angular approximation methods. Specifically, we examine the diffusion approximation, the P 1/3 approximation, and three time-dependent generalizations of the simplified PNapproximation: the SP 2 , SP 3 , and SSP 3 approximations. We show that all of the these methods preserve the correct flux-weighted average of x but not the correct flux-weighted average of (x-xa)2, where x is the spatial variable and xais an arbitrary point. We also demonstrate that, for general cross sections and large elapsed time, the error in the flux-weighted average of (x-xa)2 is smallest in magnitude for the SP 2 and approximations. In addition, we present a simple improvement to the SP 2 approximation that allows this method to produce the correct flux-weighted average of (x-xa)2. We present numerical results that test this analysis. From these results, we find that the angular approximation methods with the most accurate solutions also have the most accurate values for the flux-weighted average of (x-xa)2. In particular, the SP 2 and SP 3 approximations are two of the most accurate methods at large elapsed times, while the improved SP 2 approximation is one of the most accurate methods at all times. We also observe, however, that an accurate value for the flux-weighted average of (x-xa)2 is not always accompanied by an accurate solution. Consequently, we conclude that an accurate flux-weighted average of (x-xa)2 is a necessary rather than sufficient condition for an overall accurate angular approximation method.
{"title":"Moment Analysis of Angular APproximation Methods for Time-Dependent Radiation Transport","authors":"J. Densmore, R. McClarren","doi":"10.1080/00411450.2010.533742","DOIUrl":"https://doi.org/10.1080/00411450.2010.533742","url":null,"abstract":"We extend moment analysis, a technique developed for investigating the accuracy of discrete-ordinates spatial discretization schemes, to time-dependent radiation transport and apply it to several angular approximation methods. Specifically, we examine the diffusion approximation, the P 1/3 approximation, and three time-dependent generalizations of the simplified PNapproximation: the SP 2 , SP 3 , and SSP 3 approximations. We show that all of the these methods preserve the correct flux-weighted average of x but not the correct flux-weighted average of (x-xa)2, where x is the spatial variable and xais an arbitrary point. We also demonstrate that, for general cross sections and large elapsed time, the error in the flux-weighted average of (x-xa)2 is smallest in magnitude for the SP 2 and approximations. In addition, we present a simple improvement to the SP 2 approximation that allows this method to produce the correct flux-weighted average of (x-xa)2. We present numerical results that test this analysis. From these results, we find that the angular approximation methods with the most accurate solutions also have the most accurate values for the flux-weighted average of (x-xa)2. In particular, the SP 2 and SP 3 approximations are two of the most accurate methods at large elapsed times, while the improved SP 2 approximation is one of the most accurate methods at all times. We also observe, however, that an accurate value for the flux-weighted average of (x-xa)2 is not always accompanied by an accurate solution. Consequently, we conclude that an accurate flux-weighted average of (x-xa)2 is a necessary rather than sufficient condition for an overall accurate angular approximation method.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"39 1","pages":"192 - 233"},"PeriodicalIF":0.0,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2010.533742","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58909060","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 : 2010-03-01DOI: 10.1080/00411450.2010.531877
G. Coppa, V. Giusti, B. Montagnini, P. Ravetto
The purpose of the paper is, first, to recall the proof that the AN method and, therefore, the SP2N−1 method (of which AN was shown to be a variant) are equivalent to the odd order P2N−1, at least for a particular class of multi-region problems; namely the problems for which the total cross section has the same value for all the regions and the scattering is supposed to be isotropic. By virtue of the introduction of quadrature formulas representing first collision probabilities, this class is then enlarged in order to encompass the systems in which the regions may have different total cross sections. Some examples are reported to numerically validate the procedure.
{"title":"On the Relation Between Spherical Harmonics and Simplified Spherical Harmonics Methods","authors":"G. Coppa, V. Giusti, B. Montagnini, P. Ravetto","doi":"10.1080/00411450.2010.531877","DOIUrl":"https://doi.org/10.1080/00411450.2010.531877","url":null,"abstract":"The purpose of the paper is, first, to recall the proof that the AN method and, therefore, the SP2N−1 method (of which AN was shown to be a variant) are equivalent to the odd order P2N−1, at least for a particular class of multi-region problems; namely the problems for which the total cross section has the same value for all the regions and the scattering is supposed to be isotropic. By virtue of the introduction of quadrature formulas representing first collision probabilities, this class is then enlarged in order to encompass the systems in which the regions may have different total cross sections. Some examples are reported to numerically validate the procedure.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"39 1","pages":"164 - 191"},"PeriodicalIF":0.0,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2010.531877","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58908957","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 : 2010-03-01DOI: 10.1080/00411450.2010.531878
E. Larsen
The classic diffusion approximation to the linear Boltzmann equation is known to be accurate if the underlying physical system is (i) optically thick and (ii) scattering-dominated. This approximation has been mathematically justified by an asymptotic analysis having a scaling that is consistent with these two conditions. Also, the Simplified P N (SP N ) equations have been shown to be higher-order asymptotic corrections to the diffusion equation for the same class of physical problems. In this paper, we alter the asymptotic scaling that yields the standard diffusion and SP N approximations and obtain modified diffusion and SP N approximations that can be significantly more accurate for deep penetration problems, which are not scattering-dominated.
{"title":"Asymptotic Diffusion and Simplified PN Approximations for Diffusive and Deep Penetration Problems. Part 1: Theory","authors":"E. Larsen","doi":"10.1080/00411450.2010.531878","DOIUrl":"https://doi.org/10.1080/00411450.2010.531878","url":null,"abstract":"The classic diffusion approximation to the linear Boltzmann equation is known to be accurate if the underlying physical system is (i) optically thick and (ii) scattering-dominated. This approximation has been mathematically justified by an asymptotic analysis having a scaling that is consistent with these two conditions. Also, the Simplified P N (SP N ) equations have been shown to be higher-order asymptotic corrections to the diffusion equation for the same class of physical problems. In this paper, we alter the asymptotic scaling that yields the standard diffusion and SP N approximations and obtain modified diffusion and SP N approximations that can be significantly more accurate for deep penetration problems, which are not scattering-dominated.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"39 1","pages":"110 - 163"},"PeriodicalIF":0.0,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2010.531878","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58908972","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 : 2010-03-01DOI: 10.1080/00411450.2010.533740
M. Frank, A. Klar, R. Pinnau
This paper summarizes work that the authors were involved in on the optimal control of cooling processes in industrial glass manufacturing. The problem is formulated as the minimization of a functional constrained by equations for the temperature and the radiation field. In an optimization algorithm, this system of constraints has to be solved tens of times. The problem becomes numerically tractable if we substitute the radiative transfer equation for the photons by its SPN approximation. We derive steady and time-dependent SPN approximations for radiative transfer in glass using an asymptotic analysis. An optimization algorithm using adjoint information from the forward problem is designed using an abstract Newton's method. We show the results of one sample cooling problem.
{"title":"Optimal Control of Glass Cooling Using Simplified PN Theory","authors":"M. Frank, A. Klar, R. Pinnau","doi":"10.1080/00411450.2010.533740","DOIUrl":"https://doi.org/10.1080/00411450.2010.533740","url":null,"abstract":"This paper summarizes work that the authors were involved in on the optimal control of cooling processes in industrial glass manufacturing. The problem is formulated as the minimization of a functional constrained by equations for the temperature and the radiation field. In an optimization algorithm, this system of constraints has to be solved tens of times. The problem becomes numerically tractable if we substitute the radiative transfer equation for the photons by its SPN approximation. We derive steady and time-dependent SPN approximations for radiative transfer in glass using an asymptotic analysis. An optimization algorithm using adjoint information from the forward problem is designed using an abstract Newton's method. We show the results of one sample cooling problem.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"39 1","pages":"282 - 311"},"PeriodicalIF":0.0,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2010.533740","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58908998","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 : 2010-03-01DOI: 10.1080/00411450.2010.533743
J. Ragusa
This paper is devoted to the application of isotropic and anisotropic mesh adaptivity techniques to the SP 3 equations. We compare uniform refinements, h-, p-, and hp-adaptivity techniques. h-versions only refine the mesh cells whereas p-versions increase locally the polynomial order of the approximation. hp-adaptivity combines both techniques but requires more knowledge regarding the shape of the error. The hp-finite element method (hp-FEM) generates a sequence of adapted meshes by either performing h- or p-refinements, based upon which choice is locally more efficient. At each adaptivity step, a pair of coarse/fine adapted meshes is employed to compute the local error and to determine the next adapted mesh that best minimizes the projection-based interpolation error. The implementation of the various hp-strategies analyzed is performed with the hp-FEM Hermes2D code and numerical results are presented for the SP 3 equations.
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