Three numerical schemes for solving nonlinear partial differential equations

This dissertation is concerned with numerical solutions to steady state 2-dimensional nonlinear elliptical partial differential equations with Dirichlet boundary conditions in a rectangular region. Three numerical schemes discussed in this dissertation are the iterative method, Newton's method, and the shooting method. In general the shooting method is only applied in ordinary differential equations, not in partial differential equations. In order to apply the shooting method and the Newton's method in a nonlinear partial differential equation, the alternating direction procedure is applied. Details of how to apply these three schemes are discussed in this dissertation as well as the implementation and numerical experiments.; Numerical results show that the alternating direction shooting method and the alternating direction Newton's method performed better than the commonly used linearized iterative method. Moreover, the greatest advantage of using the alternating direction procedure on the shooting method and Newton's method is that the computation is totally parallel. Therefore, if a parallel computing were to be used, it could achieve greater speedup. However the shooting method suffers stability problems as in ordinary differential equations. One possible way in which instability may be conquered is the consideration of multiple shooting methods which is not the objective of this dissertation.