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dc.contributor.advisorJokipii, Jack R.en
dc.contributor.authorMcEvoy, Erica L.
dc.creatorMcEvoy, Erica L.en
dc.date.accessioned2017-09-27T21:51:14Z
dc.date.available2017-09-27T21:51:14Z
dc.date.issued2017
dc.identifier.urihttp://hdl.handle.net/10150/625664
dc.description.abstractStochastic differential equations are becoming a popular tool for modeling the transport and acceleration of cosmic rays in the heliosphere. In diffusive shock acceleration, cosmic rays diffuse across a region of discontinuity where the up- stream diffusion coefficient abruptly changes to the downstream value. Because the method of stochastic integration has not yet been developed to handle these types of discontinuities, I utilize methods and ideas from probability theory to develop a conceptual framework for the treatment of such discontinuities. Using this framework, I then produce some simple numerical algorithms that allow one to incorporate and simulate a variety of discontinuities (or boundary conditions) using stochastic integration. These algorithms were then modified to create a new algorithm which incorporates the discontinuous change in diffusion coefficient found in shock acceleration (known as Skew Brownian Motion). The originality of this algorithm lies in the fact that it is the first of its kind to be statistically exact, so that one obtains accuracy without the use of approximations (other than the machine precision error). I then apply this algorithm to model the problem of diffusive shock acceleration, modifying it to incorporate the additional effect of the discontinuous flow speed profile found at the shock. A steady-state solution is obtained that accurately simulates this phenomenon. This result represents a significant improvement over previous approximation algorithms, and will be useful for the simulation of discontinuous diffusion processes in other fields, such as biology and finance.
dc.language.isoen_USen
dc.publisherThe University of Arizona.en
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en
dc.subjectBoundary Conditionsen
dc.subjectDiffusive Shock Accelerationen
dc.subjectFermi Accelerationen
dc.subjectShock Wavesen
dc.subjectSkew Brownian Motionen
dc.subjectStochastic Differential Equationsen
dc.titleA Numerical Method for the Simulation of Skew Brownian Motion and its Application to Diffusive Shock Acceleration of Charged Particlesen_US
dc.typetexten
dc.typeElectronic Dissertationen
thesis.degree.grantorUniversity of Arizonaen
thesis.degree.leveldoctoralen
dc.contributor.committeememberJokipii, Jack R.en
dc.contributor.committeememberGiacalone, Joeen
dc.contributor.committeememberWatkins, Joeen
thesis.degree.disciplineGraduate Collegeen
thesis.degree.disciplineApplied Mathematicsen
thesis.degree.namePh.D.en
refterms.dateFOA2018-05-28T12:37:14Z
html.description.abstractStochastic differential equations are becoming a popular tool for modeling the transport and acceleration of cosmic rays in the heliosphere. In diffusive shock acceleration, cosmic rays diffuse across a region of discontinuity where the up- stream diffusion coefficient abruptly changes to the downstream value. Because the method of stochastic integration has not yet been developed to handle these types of discontinuities, I utilize methods and ideas from probability theory to develop a conceptual framework for the treatment of such discontinuities. Using this framework, I then produce some simple numerical algorithms that allow one to incorporate and simulate a variety of discontinuities (or boundary conditions) using stochastic integration. These algorithms were then modified to create a new algorithm which incorporates the discontinuous change in diffusion coefficient found in shock acceleration (known as Skew Brownian Motion). The originality of this algorithm lies in the fact that it is the first of its kind to be statistically exact, so that one obtains accuracy without the use of approximations (other than the machine precision error). I then apply this algorithm to model the problem of diffusive shock acceleration, modifying it to incorporate the additional effect of the discontinuous flow speed profile found at the shock. A steady-state solution is obtained that accurately simulates this phenomenon. This result represents a significant improvement over previous approximation algorithms, and will be useful for the simulation of discontinuous diffusion processes in other fields, such as biology and finance.


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