| This dissertation discusses higher order methods for solving partial differential equations. A common theme throughout this dissertation will be the use of spectral methods. One of the challenges of the spectral Fourier method is approximating smooth functions that are non-periodic. The rate of convergence deteriorates to O({dollar}{lcub}1{rcub}over{lcub}N{rcub}{dollar}) away from the boundaries and there are spurious oscillations at the boundary points, known as the Gibbs phenomenon.; Chapter 2 of this dissertation discusses an analytic but non-periodic function f(x) defined on ({dollar}-{dollar}1, 1), for which the first 2N + 1 Fourier coefficients are known, and focuses on two methods that successfully eliminate the Gibbs phenomenon based on the information obtained from the Fourier coefficients. A new method is suggested to improve upon the results that have been obtained thus far.; Chapter 3 discusses the Gibbs phenomenon for spherical harmonic spectral methods applied to functions that are piecewise analytic on spheres. We prove that knowledge of the first N spherical harmonic coefficients yield an exponentially convergent approximation to a spherical piecewise smooth function, hence completely overcoming the Gibbs phenomenon.; In Chapter 4 higher order numerical methods are applied to a specific one-dimensional hyperbolic system that describes the shallow water dynamics of a linearly sloping gulf. This problem is challenging due to the constantly changing location of where the water touches the shore. At this point the equation loses its strong hyperbolicity, and conventional numerical methods may not produce the correct solution. Also, any discontinuities that may arise as a result of the nonlinearity cannot be predicted, and this must be considered in selecting a numerical method. |
