我們考慮噴嘴流體問題， 先利用有限差分法把問題離散得到一個大型非線性系統， 對於這個大型非線性系統我們藉由牛頓法來解數值解。 在這類的問題中，流體急速減速過程中所產生的震波對於牛頓法迭代收歛性造成影響。 本論文考慮並測試一個新的方法叫做非線性塊狀高斯消去牛頓法。在這個方法中，我們定義一個 局部問題並利用局部問題找到震波正確的位置，以改進牛頓法的收歛性與收斂速度。數值實驗結果證明， 在所需電腦運算時間上之比較，我們這個新的方法優於傳統的牛頓法。 Newton type method is one of most popular methods for solving a large nonlinear algebraic system of equations arising from the discretization of partial differential equations with applications in science and engineering. Due to the presence of normal shock wave the convergence rate of Newton type methods for solving the discrete nozzle flow problem becomes very slow. In this thesis, we proposed and tested some right nonlinear preconditioned iterative algorithm to enhance robustness of Newton's method and to improve it's convergence rate. In this method, we define a local problem, which is governed by the same differential equation as the global problem we try to solve while the boundary conditions are imposed to satisfy the current global approximation at these grid points. Such solution of the local problem is able to quickly detect the exact location of shock wave. Finally, we show numerically that our approach is better than some traditional Newton's method in terms of total CPU time.
