Legendre polynomial expansions and Runge-Kutta methods for ordinary differential equations

In this thesis we obtain a numerical solution to the initial value problem for ordinary differential equations in two steps. First we expand the solution in a finite Legendre Polynomial series. This series solution is determined by solving a nonlinear system of algebraic equations for the coefficients in the series. The purpose of choosing Legendre series is to produce stable methods so that, for example, solving stiff equations presents no particular difficulty. Our study happens to treat ultra-spherical polynomials and the Legendre Polynomials are determined to be those that lead to the approximation of greatest accuracy. The second step discretizes the definite integrals that occur in the algebraic equations for the coefficients. By means Gauss-Legendre of quadrature we obtain implicit Runge-Kutta methods.;In 1964, J. C. Butcher derived n-stage, 2n order implicit Runge-Kutta methods by making assumptions to simplify highly non-linear equations. The solution of the resulting equations involved the explicit form for the inverse of order n matrices and hence was cumbersome. By applying Gauss-Legendre quadrature to the Legendre series expansion of the solution, we obtain in a straight forward manner, all n-stage, 2n order implicit Runge-Kutta schemes, including those found with more difficulty by Butcher. By using Radau and Lobatto type quadratures we are also led to the Runge-Kutta methods of Radau and Lobatto type.;Having reproduced the implicit Runge-Kutta methods for initial value problems we apply the Legendre Polynomial method and Gauss-Legendre quadrature to two point boundary value problem, obtaining implicit Runge-Kutta schemes there as well. When attempting the same procedure in the case of the heat equation, however, the Legendre method fails to give the degree of accuracy obtained in the former problems and appears not to lead easily to implicit Runge-Kutta methods.