human
I am working on the file "Euler_approximate.ipynb".
The first task is:
Notice that in the exact solution, the right edge of the rarefaction travels to the right.  In the Roe solution, all waves travel to the left.  As in the case of the shallow water equations, here too this behavior can lead to unphysical solutions when this approximate solver is used in a numerical discretization.  In order to correct this, we can split the single wave into two when a transonic rarefaction is present, in a way similar to what is done in the shallow water equations.  We do not go into details here.
HLLE Solver
Recall that an HLL solver uses only two waves with a constant state between them.  The Euler equations are our first example for which the number of waves in the true solution is larger than the number of waves in the approximate solution.  As one might expect, this leads to noticeable inaccuracy in solutions produced by the solver.
Again following Einfeldt, the left-going wave speed is chosen to be the minimum of the Roe speed for the 1-wave and the characterstic speed $\lambda^1$ in the left state $q_\ell$.  The right-going wave speed is chosen to be the maximum of the Roe speed for the 3-wave and the characteristic speed $\lambda^3$ in the right state $q_r$.  Effectively, this means that
\begin{align}
    s_1 & = \min(u_\ell - c_\ell, \hat{u}-\hat{c}) \
    s_2 & = \max(u_r + c_r, \hat{u}+\hat{c})
\end{align}
Recall that once we have chosen these two wave speeds, conservation dictates the value of the intermediate state:
\begin{align}  \label{SWA:hll_middle_state}
q_m = \frac{f(q_r) - f(q_\ell) - s_2 q_r + s_1 q_\ell}{s_1 - s_2}.
\end{align}
Can you write Python program for it?
assistant
#Here is done in the shallow water equations.

def Euler_hlle(q_l, q_r, gamma=1.4):
    """HLLE approximate solver for the Euler equations."""
    
    rho_l = q_l[0]
    rhou_l = q_l[1]
    u_l = rhou_l/rho_l
    rho_r = q_r[0]
    rhou_r = q_r[1]
    u_r = rhou_r/rho_r
    E_r = q_r[2]
    E_l = q_l[2]
    
    u_hat, c_hat, H_hat = roe_averages(q_l, q_r, gamma)
    p_r = (gamma-1.) * (E_r - rho_r*u_r**2/2.)
    p_l = (gamma-1.) * (E_l - rho_l*u_l**2/2.)
    H_r = (E_r+p_r) / rho_r
    H_l = (E_l+p_l) / rho_l
    c_r = np.sqrt((gamma-1.)*(H_r-u_r**2/2.))
    c_l = np.sqrt((gamma-1.)*(H_l-u_l**2/2.))
    
    s1 = min(u_l-c_l,u_hat-c_hat)
    s2 = max(u_r+c_r,u_hat+c_hat)
    
    rho_m = (rhou_r - rhou_l - s2*rho_r + s1*rho_l)/(s1-s2)
    rhou_m = (rho_r*u_r**2 - rho_l*u_l**2 \
              + p_r - p_l - s2*rhou_r + s1*rhou_l)/(s1-s2)
    E_m = ( u_r*(E_r+p_r) - u_l*(E_l+p_l) - s2*E_r + s1*E_l)/(s1-s2)
    q_m = np.array([rho_m, rhou_m, E_m])
    
    states = np.column_stack([q_l,q_m,q_r])
    speeds = [s1, s2]
    wave_types = ['contact','contact']
    
    def reval(xi):
        rho  = (xi<s1)*rho_l + (s1<=xi)*(xi<=s2)*rho_m + (s2<xi)*rho_r
        mom  = (xi<s1)*rhou_l + (s1<=xi)*(xi<=s2)*rhou_m + (s2<xi)*rhou_r
        E  = (xi<s1)*E_l + (s1<=xi)*(xi<=s2)*E_m + (s2<xi)*E_r
        return rho, mom, E

    return states, speeds, reval, wave_types