$$ \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\halfi}{{1/2}} \newcommand{\xpoint}{\boldsymbol{x}} \newcommand{\normalvec}{\boldsymbol{n}} \newcommand{\Oof}[1]{\mathcal{O}(#1)} \newcommand{\Ix}{\mathcal{I}_x} \newcommand{\Iy}{\mathcal{I}_y} \newcommand{\It}{\mathcal{I}_t} \newcommand{\setb}[1]{#1^0} % set begin \newcommand{\sete}[1]{#1^{-1}} % set end \newcommand{\setl}[1]{#1^-} \newcommand{\setr}[1]{#1^+} \newcommand{\seti}[1]{#1^i} \newcommand{\Real}{\mathbb{R}} $$

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Implementation of variable coefficients

Assume c[i] holds \( c_i \) the spatial mesh points 

for i in range(1, Nx):
    u[i] = - u_2[i] + 2*u_1[i] + \ 
           C2*(0.5*(q[i] + q[i+1])*(u_1[i+1] - u_1[i])  - \ 
               0.5*(q[i] + q[i-1])*(u_1[i] - u_1[i-1])) + \ 
           dt2*f(x[i], t[n])

Here: C2=(dt/dx)**2

Vectorized version: 

u[1:-1] = - u_2[1:-1] + 2*u_1[1:-1] + \ 
          C2*(0.5*(q[1:-1] + q[2:])*(u_1[2:] - u_1[1:-1]) -
              0.5*(q[1:-1] + q[:-2])*(u_1[1:-1] - u_1[:-2])) + \ 
          dt2*f(x[1:-1], t[n])

Neumann condition \( u_x=0 \): same ideas as in 1D (modified stencil or ghost cells).

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