UC San Diego

In this thesis, we consider the discretization of two different PDE which govern physical phenomenon. First, we consider the diffusive diffusive logistic equation and develop several new results on weak solutions and on their approximation by Galerkintype methods. Our goal is to establish a rate of convergence for Galerkin approximations to solutions of this problem, and thus we first consider the continuous model, and briefly review the literature on the known solution theory. We then state and prove a new result on existence and uniqueness of weak solutions. Moreover, we provide numerical results to provide evidence of our theoretical results. Second, we consider the nonlinear systems of conservation laws which propagate shock waves, rarefactions, and contact discontinuities, and introduce what we call the C-method. We shall focus our attention on the compressible Euler equations in one space dimension. The novel feature of our approach involves the coupling of a linear scalar reaction -diffusion equation to our system of conservation laws, whose solution C(x, t) is the coefficient to an additional (and artificial) term added to the flux, which determines the location, localization, and strength of the artificial viscosity. Near shock discontinuities, C(x, t) is large and localized, and transitions smoothly in space-time to zero away from discontinuities. Our approach is a provably convergent, spacetime-regularized variant of the original idea of Richtmeyer and Von Neumann, and is provided at the level of the PDE, thus allowing a host of numerical discretization schemes to be employed. We demonstrate the effectiveness of the C-method with three different numerical implementations and apply these to a collection of classical problems. All three schemes yield higher- order discretization strategies, which provide sharp shock resolution with minimal overshoot and noise, and compare well with higher-order WENO schemes that employ approximate Riemann solvers, outperforming them for the difficult Leblanc shock tube experiment