human
I am working on the file "Shallow_water_approximate.ipynb".
The first task is:
Because the exact and approximate solutions are the same, the exact solution is not visible in these plots.  The only exception is the 1-wave in the first plot.
the speed of the 1-wave appears different for the Roe solution and the exact solution, but this is unimportant since the 1-wave has zero strength.
Entropy fix
As discussed previously, the use of an entropy-violating discontinuity in place of a rarefaction can be problematic in the case of a transonic rarefaction.  For the Roe solver, we can again apply an entropy fix, splitting the single discontinuity into two waves, to avoid this problem.  Several different entropy fixes have been proposed.
here we focus on one that is quite general and effective in practice.
Suppose that $\lambda_1(q_\ell) < 0$ while $\lambda_1(q_m)>0$.  In this case, it appears that the 1-wave should be a transonic rarefaction.  We take the 1-wave ${\mathcal W}1$ of the Roe solver, with speed $s_1$, and split it into two waves, ${\mathcal W}{1\ell} = \alpha_{1\ell} r_1$ and ${\mathcal W}{1r} = \alpha{1r} r_1$, with speeds $\lambda_1(q_\ell), \lambda_1(q_m)$.  Two requirements determine the strength of these waves.  First, the total jump across them should be equal to the jump across the original wave:
$${\mathcal W}{1\ell} + {\mathcal W}{1r} = {\mathcal W}_1.$$
Second, we must maintain conservation:
$$\lambda_1(q_\ell) {\mathcal W}{1\ell} + \lambda_1(q_m) {\mathcal W}{1r} = s_1 {\mathcal W}_1.$$
This leads to
\begin{align}
{\mathcal W}{1\ell} & = \frac{\lambda_1(q_m)-s_1}{\lambda_1(q_m)-\lambda_1(q\ell)}{\mathcal W}1 \
{\mathcal W}{1r} & = \frac{s_1 - \lambda_1(q_\ell)}{\lambda_1(q_m)-\lambda_1(q_\ell)}{\mathcal W}_1.
\end{align}
We can check in a similar way for a transonic 2-rarefaction, and modify the 2-wave accordingly.  Note that only one of the two characteristic fields can be transonic in the Riemann solution for the shallow water equations.
This entropy fix is implemented in the solver below.
Can you write Python code for it?
assistant
#Here is Python code to determine the strength of these waves.

def shallow_water_roe_with_efix(q_l, q_r, grav=1., efix=True):
    """
    Approximate Roe solver for the shallow water equations,
    with entropy fix.
    """
    
    h_l = q_l[0]
    hu_l = q_l[1]
    u_l = hu_l/h_l
    h_r = q_r[0]
    hu_r = q_r[1]
    u_r = hu_r/h_r
    
    delta = q_r - q_l
    
    # Roe averages
    h_hat = (h_r + h_l)/2.
    u_hat = (np.sqrt(h_r)*u_r + np.sqrt(h_l)*u_l) /  \
            (np.sqrt(h_r) + np.sqrt(h_l))
    c_hat = np.sqrt(grav*h_hat)
    
    s1 = u_hat - c_hat
    s2 = u_hat + c_hat
    
    alpha1 = ( (u_hat+c_hat)*delta[0] - delta[1])/(2*c_hat)
    #alpha2 = (-(u_hat-c_hat)*delta[0] + delta[1])/(2*c_hat)
    
    h_m = q_l[0] + alpha1
    hu_m = q_l[1] + alpha1 * (u_hat - c_hat)
    u_m = hu_m/h_m
    q_m = np.array([h_m, hu_m])
        
    transonic = None
    if efix:
        # Check if either wave appears transonic
        lambda_1_l = u_l - np.sqrt(grav*h_l)
        lambda_1_m = u_m - np.sqrt(grav*h_m)
        lambda_2_m = u_m + np.sqrt(grav*h_m)
        lambda_2_r = u_r + np.sqrt(grav*h_r)
        
        if lambda_1_l < 0 < lambda_1_m: # 1-wave appears transonic
            transonic = 1
        elif lambda_2_m < 0 < lambda_2_r: # 2-wave appears transonic
            transonic = 2
    
    if transonic == 1:
        beta = (lambda_1_m - s1)/(lambda_1_m - lambda_1_l)
        alpha1_l = beta*alpha1
        #alpha_1_r = (1-beta)*alpha1
        h_m_new = q_l[0] + alpha1_l
        hu_m_new = q_l[1] + alpha1_l * (u_hat-c_hat)
        q_m_new = np.array([h_m_new, hu_m_new])
        states = np.column_stack([q_l,q_m_new,q_m,q_r])
        speeds = [lambda_1_l, lambda_1_m, s2]
        wave_types = ['contact']*3
        
        def reval(xi):
            h_out  = (xi<speeds[0])*states[0,0] + \
                     (speeds[0]<=xi)*(xi<speeds[1])*states[0,1] + \
                     (speeds[1]<=xi)*(xi<speeds[2])*states[0,2] + \
                     (speeds[2]<=xi)*states[0,3]
            hu_out = (xi<speeds[0])*states[1,0] + \
                    ...