Advanced search options

Advanced Search Options 🞨

Browse by author name (“Author name starts with…”).

Find ETDs with:

in
/  
in
/  
in
/  
in

Written in Published in Earliest date Latest date

Sorted by

Results per page:

You searched for +publisher:"University of North Carolina" +contributor:("Chen, Dangxing"). One record found.

Search Limiters

Last 2 Years | English Only

No search limiters apply to these results.

▼ Search Limiters


University of North Carolina

1. Chen, Dangxing. Spatio-temporal integral equation methods with applications.

Degree: Mathematics, 2017, University of North Carolina

Electromagnetic interactions are vital in many applications including physics, chemistry, material sciences and so on. Thus, a central problem in physical modeling is the electromagnetic analysis of materials. Here, we consider the numerical solution of the Maxwell equation for the evolution of the electromagnetic field given the charges, and the Newton or Schrödinger equation for the evolution of particles. By combining integral equation techniques with new spectral deferred correction algorithms in time and hierarchical methods in space, we develop fast solvers for the calculation of electromagnetism with relaxations of the model in different scenarios. The dissertation consists of two parts, aiming to resolve the challenges in the temporal and spatial direction, respectively. In the first part, we study a new class of time stepping methods for time-dependent differential equations. The core algorithm uses the pseudo-spectral collocation formulation to discretize the Picard type integral equation reformulation, producing a highly accurate and stable representation, which is then solved via the deferred correction technique. By exploiting the mathematical properties of the formulation and the convergence procedure, we develop some new preconditioning techniques from different perspectives that are accurate, robust, and can be much more efficient than existing methods. As is typical of spectral methods, the solution to the discretization is spectral accurate and the time step-size is optimal, though the cost of solving the system can be high. Thus, the solver is particularly suited to problems where very accurate solutions are sought or large time-step is required, e.g., chaotic systems or long-time simulation. In the second part, we study the hierarchical methods with emphasis on the spatial integral equations. In the first application, we implement a parallel version of the adaptive recursive solver for two-point boundary value problem by Cilk multithreaded runtime system based on the integral equation formulation. In the second application, we apply the hierarchical method to two-layered media Helmholtz equations in the acoustic and electromagnetic scattering problems. With the method of images and integral representations, the spatially heterogeneous translation operators are derived with rigorous error analysis, and the information is then compressed and spread in a fashion similar to fast multipole methods. The preliminary results suggest that our approach can be faster than existing algorithms with several orders of magnitude. We demonstrate our solver on a number of examples and discuss various useful extensions. Preliminary results are favorable and show the viability of our techniques for integral equations. Such integral equation methods could well have a broad impact on many areas of computational science and engineering. We describe further applications in biology, chemistry, and physics, and outline some directions for future work. Advisors/Committee Members: Chen, Dangxing, Huang, Jingfang, Lu, Jianfeng, Marzuola, Jeremy, Newhall, Katherine, Kanai, Yosuke, University of North Carolina at Chapel Hill.

Subjects/Keywords: College of Arts and Sciences; Department of Mathematics

Record DetailsSimilar RecordsGoogle PlusoneFacebookTwitterCiteULikeMendeleyreddit

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Chen, D. (2017). Spatio-temporal integral equation methods with applications. (Thesis). University of North Carolina. Retrieved from https://cdr.lib.unc.edu/record/uuid:b34f5edd-7a20-4ba7-9b39-32914513d073

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Chen, Dangxing. “Spatio-temporal integral equation methods with applications.” 2017. Thesis, University of North Carolina. Accessed November 29, 2020. https://cdr.lib.unc.edu/record/uuid:b34f5edd-7a20-4ba7-9b39-32914513d073.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Chen, Dangxing. “Spatio-temporal integral equation methods with applications.” 2017. Web. 29 Nov 2020.

Vancouver:

Chen D. Spatio-temporal integral equation methods with applications. [Internet] [Thesis]. University of North Carolina; 2017. [cited 2020 Nov 29]. Available from: https://cdr.lib.unc.edu/record/uuid:b34f5edd-7a20-4ba7-9b39-32914513d073.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Chen D. Spatio-temporal integral equation methods with applications. [Thesis]. University of North Carolina; 2017. Available from: https://cdr.lib.unc.edu/record/uuid:b34f5edd-7a20-4ba7-9b39-32914513d073

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

.