Grid stabilization for the one-dimensional advection equation using biased finite differnces of odd orders and orders higher than twenty-two

Abstract

This work utilizes finite differences to approximate the first derivative of non-periodic smooth functions. Math literature indicates that stabilizing Partial Differential Equation solvers based on high order finite difference approximations of spatial derivatives of a non-periodic function becomes problematic near a boundary. Hagstrom and Hagstrom have discovered a method of introducing additional grid points near a boundary, which has proven to be effective in stabilizing Partial Differential Equation solvers. Hagstrom and Hagstrom demonstrated their method for the case of the one-dimensional advection equation using spatial derivative approximations of even orders up to twenty-second order. In this dissertation, we explore the efficacy of the Hagstrom and Hagstrom method for the same Partial Differential Equation with spatial derivative approximations of odd orders and orders higher than twenty-two and report the number and locations of additional grid points required for stability in each case.