河网水力数值模拟中New ton-Raphson法收敛性的证明

Proof of convergence of Newton-Raphson iterative method in simulating unsteady flow in river networks

Newton- Raphson迭代法以其收敛速度快的优点而常被用于求解非线性代数方程组 ,但对各种不同的问题其局部收敛性条件的证明往往是十分困难的。本文针对用 Newton-Raphson迭代法求解河网数值模拟中所出现的非线性代数方程组的问题 ,证明了只要当时间步长取得足够小时 ,迭代法的局部收敛性条件就一定可以满足 ,从而给出了 Newton- Raphson迭代法在河网非恒定流计算中应用的一个理论基础。

The Newton-Raphson iterative method is commonly used to solve nonlinear algebraic equations due to its fast convergence speed. However it is very difficult to prove the conditions of local convergence of the method for different kinds of problems. For the nonlinear algebraic equations derived from a set of partial differential equations which describe the unsteady flow in river networks, this paper proves that the local convergence conditions of the Newton-Raphson iterative method are satisfied. This provides a theoretical foundations for the application of the Newton-Raphson iterative method in the computation of unsteady flow in the river networks.