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You searched for +publisher:"Brown University" +contributor:("Darbon, Jerome"). Showing records 1 – 2 of 2 total matches.

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1. Shi, Cengke. Numerical Methods for Hyperbolic Equations: Generalized Definition of Local Conservation and Discontinuous Galerkin Methods for Maxwell's equations in Drude Metamaterials.

Degree: Department of Applied Mathematics, 2018, Brown University

This dissertation presents two topics on numerical solutions solving hyperbolic equations from both theoretical and practical points of view. In the first part, we introduce a definition of the local conservation property for numerical methods solving time dependent conservation laws, which generalizes the classical local conservation definition. The motivation of our definition is the Lax-Wendroff theorem, and thus we prove it for locally conservative numerical schemes per our definition in one and two space dimensions. Several numerical methods, including continuous Galerkin methods and compact schemes, which do not fit the classical local conservation definition, are given as examples of locally conservative methods under our generalized definition. In the second part, we develop and analyze non-dissipative discontinuous Galerkin (DG) methods for solving the Maxwell's equations in Drude metamaterials. Our method achieves provable non-dissipative stability and optimal error estimates simultaneously on rectangular meshes. However, on triangular meshes, the DG schemes only have suboptimal convergence rate. We present extensive numerical results with convergence consistent of our error estimate, and simulations of wave propagation in Drude metamaterials to demonstrate the flexibility of triangular meshes. Advisors/Committee Members: Shu, Chi-Wang (Advisor), Guzman, Johnny (Reader), Darbon, Jerome (Reader).

Subjects/Keywords: Discontinuous Galerkin Methods

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

APA (6th Edition):

Shi, C. (2018). Numerical Methods for Hyperbolic Equations: Generalized Definition of Local Conservation and Discontinuous Galerkin Methods for Maxwell's equations in Drude Metamaterials. (Thesis). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:792729/

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):

Shi, Cengke. “Numerical Methods for Hyperbolic Equations: Generalized Definition of Local Conservation and Discontinuous Galerkin Methods for Maxwell's equations in Drude Metamaterials.” 2018. Thesis, Brown University. Accessed January 18, 2021. https://repository.library.brown.edu/studio/item/bdr:792729/.

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

MLA Handbook (7th Edition):

Shi, Cengke. “Numerical Methods for Hyperbolic Equations: Generalized Definition of Local Conservation and Discontinuous Galerkin Methods for Maxwell's equations in Drude Metamaterials.” 2018. Web. 18 Jan 2021.

Vancouver:

Shi C. Numerical Methods for Hyperbolic Equations: Generalized Definition of Local Conservation and Discontinuous Galerkin Methods for Maxwell's equations in Drude Metamaterials. [Internet] [Thesis]. Brown University; 2018. [cited 2021 Jan 18]. Available from: https://repository.library.brown.edu/studio/item/bdr:792729/.

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

Council of Science Editors:

Shi C. Numerical Methods for Hyperbolic Equations: Generalized Definition of Local Conservation and Discontinuous Galerkin Methods for Maxwell's equations in Drude Metamaterials. [Thesis]. Brown University; 2018. Available from: https://repository.library.brown.edu/studio/item/bdr:792729/

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

2. Qin, Tong. Positivity-Preserving High-Order Discontinuous Galerkin Methods: Implicit Time Stepping and Applications to Relativistic Hydrodynamics.

Degree: Department of Applied Mathematics, 2017, Brown University

The positivity-preserving property is a highly desirable property when designing high order numerical methods for hyperbolic conservation laws, since negative values sometimes cause ill-posedness of the problem and blow-ups of the algorithms. The general framework for constructing positivity-preserving schemes for solving hyperbolic conservation laws have been proposed in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), pp.~3091 – 3120) and (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), pp.~8918 – 8934). In this dissertation, we extend this framework to DG methods with implicit discretizations and to DG methods for solving the relativistic hydrodynamics (RHD). Due to the the Courant-Friedrichs-Levis (CFL) number constraint, explicit DG methods are impractical for problems involving unstructured and extremely varying meshes or long-time simulations. Instead, implicit DG schemes are often popular in practice, especially in the computational fluid dynamics (CFD) community. In the first part of this dissertation, we develop a high-order positivity-preserving DG method with the backward Euler time discretization for solving conservation laws, basing on a generalization of the Zhang-Shu positivity-preserving limiter. Both the analysis and numerical experiments indicate that a lower bound for the CFL number is required to obtain the positivity-preserving property for the numerical schemes. The proposed method not only preserves the positivity of the numerical approximation without compromising the designed high-order accuracy, but also helps accelerate the convergence towards the steady-state solution and add robustness to the nonlinear solver. For RHD systems, the density and the pressure are positive physical quantities and the velocity is bounded by the speed of light. The violation of these bounds will result in ill-posedness of the problem and blow-up of the code, especially in extreme relativistic cases. It is usually hard to maintain these physical bounds without sacrificing the numerical accuracy. In the second part, we develop a bound-preserving DG method to solve RHD systems by extending the bound-preserving limiter for the non-relativistic hydrodynamics. The proposed method has the following features. It can theoretically guarantee to preserve the physical bounds for the numerical approximation and maintain its designed high order accuracy. Moreover, it renders L1-stability to the numerical scheme. The robustness of the scheme is tested on various extreme relativistic cases, including relativistic jets. Advisors/Committee Members: Shu, Chi-Wang (Advisor), Guzman, Johnny (Reader), Darbon, Jerome (Reader).

Subjects/Keywords: Numerical Anlaysis

Record DetailsSimilar RecordsGoogle PlusoneFacebookTwitterCiteULikeMendeleyreddit

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

APA (6th Edition):

Qin, T. (2017). Positivity-Preserving High-Order Discontinuous Galerkin Methods: Implicit Time Stepping and Applications to Relativistic Hydrodynamics. (Thesis). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:733482/

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):

Qin, Tong. “Positivity-Preserving High-Order Discontinuous Galerkin Methods: Implicit Time Stepping and Applications to Relativistic Hydrodynamics.” 2017. Thesis, Brown University. Accessed January 18, 2021. https://repository.library.brown.edu/studio/item/bdr:733482/.

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

MLA Handbook (7th Edition):

Qin, Tong. “Positivity-Preserving High-Order Discontinuous Galerkin Methods: Implicit Time Stepping and Applications to Relativistic Hydrodynamics.” 2017. Web. 18 Jan 2021.

Vancouver:

Qin T. Positivity-Preserving High-Order Discontinuous Galerkin Methods: Implicit Time Stepping and Applications to Relativistic Hydrodynamics. [Internet] [Thesis]. Brown University; 2017. [cited 2021 Jan 18]. Available from: https://repository.library.brown.edu/studio/item/bdr:733482/.

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

Council of Science Editors:

Qin T. Positivity-Preserving High-Order Discontinuous Galerkin Methods: Implicit Time Stepping and Applications to Relativistic Hydrodynamics. [Thesis]. Brown University; 2017. Available from: https://repository.library.brown.edu/studio/item/bdr:733482/

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

.