91Ó°ÊÓ

To develop a fixed-point iteration scheme to solve for \(H\), we first need to rewrite the Manning equation in terms of \(H\). The Manning equation is given as: \( Q = \frac{\sqrt{S}(BH)^{5/3}}{n(B+2H)^{2/3}} \) Let's express \(H\) in terms of other variables. \(H = \frac{nQ(B + 2H)^{2/3}}{\sqrt{S}(B)^{5/3}} \)

Fixed-point iteration is a method of finding the root of a function by iteratively applying a specific function that will eventually lead to the fixed point (root) solution. In this case, we want to find the value of \(H\) that makes the function equal to \(H\) itself. Let's define a function, \(g(H)\), in terms of \(H\) using the equation we found in Step 1. \( g(H) = \frac{nQ(B + 2H)^{2/3}}{\sqrt{S}(B)^{5/3}} \) So, our fixed-point iteration scheme can be defined as follows: \( H_{k+1} = g(H_k) \)

We are asked to prove that the scheme converges for all initial guesses greater than or equal to zero. Let's start with the initial guess, \(H_0 = 0\). Furthermore, we have the following values for the other variables: - \(Q = 5 \, \mathrm{m^3/s}\) - \(S = 0.0002 \, \mathrm{m/m}\) - \(B = 20 \, \mathrm{m}\) - \(n = 0.03\)

To iterate using the scheme defined above and to show the convergence of the method, we will use the following formula to calculate the successive values of \(H\): \( H_{k+1} = g(H_k) = \frac{nQ(B + 2H_k)^{2/3}}{\sqrt{S}(B)^{5/3}} \) As we continue to iterate more and more, the values of \(H_{k+1}\) and \(H_{k}\) will eventually become close enough that we can conclude that the method converges as the iteration continues.

To prove convergence for all initial guesses greater than or equal to zero, we can use the fact that the Manning equation is a monotonic function of \(H\) when \(H\) is non-negative. This means that, as we move along the \(H\) axis, the function only increases or decreases, and never changes direction. Since the function is monotonic, we can conclude that the fixed-point iteration will converge starting from any non-negative initial guess, as long as the underlying slopes and coefficients are also non-negative. In our given problem, the slope \(S\) and the Manning roughness coefficient \(n\) are both positive, ensuring the convergence using the fixed-point iteration method. This completes the proof of convergence for fixed-point iteration using the Manning equation for all initial guesses greater than or equal to zero.