Numerical Methods for Hyperbolic Equations: Generalized Definition of Local Conservation and Discontinuous Galerkin Methods for Maxwell's equations in Drude MetamaterialsShi, CengkecreatorShu, Chi-WangAdvisorGuzman, JohnnyReaderDarbon, JeromeReaderBrown University. Department of Applied Mathematicssponsor2018xi, 101 p.born digitalThesis (Ph. D.)--Brown University, 2018thesesThis 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.Discontinuous Galerkin MethodsNumerical analysishyperbolic conservation lawEnglishRPB2018061510.26300/nxfc-nw57In CopyrightCollection is open for research.dissertations