Comparison of Numerical Analysis Methods for Solving One-Dimensional Elliptic Differential Equations

摘要

Four numerical methods are used to solve a specific set of problems and then the methods are compared for accuracy and efficiency. The methods are: standard finite differences, collocation, Galerkin, and least squares. The latter three methods are finite element methods which use either Lagrange linear, Hermite cubic, or Hermite septic piecewise polynominals as interpolation functions. The problem set consists of second- and fourth-order, linear and nonlinear, differential equations with constant and variable coefficients. The linear equations govern elementary structural members and the nonlinear equation is a one-dimensional analog for transonic flow past an airfoil. The three major conclusions are:(1) the least squares method with Hermite cubic polynomials was the method of choice for the second-order linear equations, (2) the collocation method was chosen over Galerkin and the finite difference methods for the fourth-order equations, and (3) Galerkin method was chosen over the collocation method for the nonlinear problem. (Author)