<mods:mods xmlns:mods="http://www.loc.gov/mods/v3" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-4.xsd"><mods:titleInfo><mods:title>Numerical Methods for Hyperbolic Equations: Generalized Definition of Local Conservation and Discontinuous Galerkin Methods for Maxwell's equations in Drude Metamaterials</mods:title></mods:titleInfo><mods:name type="personal"><mods:namePart>Shi, Cengke</mods:namePart><mods:role><mods:roleTerm type="text">creator</mods:roleTerm></mods:role></mods:name><mods:name type="personal"><mods:namePart>Shu, Chi-Wang</mods:namePart><mods:role><mods:roleTerm type="text">Advisor</mods:roleTerm></mods:role></mods:name><mods:name type="personal"><mods:namePart>Guzman, Johnny</mods:namePart><mods:role><mods:roleTerm type="text">Reader</mods:roleTerm></mods:role></mods:name><mods:name type="personal"><mods:namePart>Darbon, Jerome</mods:namePart><mods:role><mods:roleTerm type="text">Reader</mods:roleTerm></mods:role></mods:name><mods:name type="corporate"><mods:namePart>Brown University. Department of Applied Mathematics</mods:namePart><mods:role><mods:roleTerm type="text">sponsor</mods:roleTerm></mods:role></mods:name><mods:originInfo><mods:copyrightDate>2018</mods:copyrightDate></mods:originInfo><mods:physicalDescription><mods:extent>xi, 101 p.</mods:extent><mods:digitalOrigin>born digital</mods:digitalOrigin></mods:physicalDescription><mods:note type="thesis">Thesis (Ph. D.)--Brown University, 2018</mods:note><mods:genre authority="aat">theses</mods:genre><mods:abstract>This dissertation presents two topics on numerical solutions solving hyperbolic equations from both theoretical and practical points of view. &#13;
&#13;
In the first part, we introduce a definition of the local conservation property for numerical methods &#13;
solving time dependent conservation laws, which generalizes the classical local conservation&#13;
definition.  The motivation of our definition is the Lax-Wendroff theorem, and thus we prove it &#13;
for locally conservative numerical schemes per our definition in one and two space dimensions. &#13;
Several numerical methods, including continuous Galerkin methods and compact schemes,&#13;
which do not fit the classical local conservation definition,&#13;
are given as examples of locally conservative methods under our generalized definition. &#13;
&#13;
In the second part, we develop and analyze non-dissipative&#13;
discontinuous Galerkin (DG) methods for solving the Maxwell's equations &#13;
in Drude metamaterials.  &#13;
Our method achieves provable non-dissipative stability and optimal error&#13;
estimates simultaneously on rectangular meshes.&#13;
However, on triangular meshes, the DG schemes only have suboptimal convergence rate. &#13;
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.</mods:abstract><mods:subject><mods:topic>Discontinuous Galerkin Methods</mods:topic></mods:subject><mods:subject authority="fast" authorityURI="http://id.worldcat.org/fast" valueURI="http://id.worldcat.org/fast/01041273"><mods:topic>Numerical analysis</mods:topic></mods:subject><mods:subject><mods:topic>hyperbolic conservation law</mods:topic></mods:subject><mods:language><mods:languageTerm authority="iso639-2b">English</mods:languageTerm></mods:language><mods:recordInfo><mods:recordContentSource authority="marcorg">RPB</mods:recordContentSource><mods:recordCreationDate encoding="iso8601">20180615</mods:recordCreationDate></mods:recordInfo><mods:identifier type="doi">10.26300/nxfc-nw57</mods:identifier><mods:accessCondition type="rights statement" xlink:href="http://rightsstatements.org/vocab/InC/1.0/">In Copyright</mods:accessCondition><mods:accessCondition type="restriction on access">Collection is open for research.</mods:accessCondition><mods:typeOfResource authority="primo">dissertations</mods:typeOfResource></mods:mods>