Consider the scalar conservation law $u_t+f(u)_x=0,$ whose monotone conservative and consistent first order numerical scheme is given by \begin{equation}\label{1}u_i^{n+1}=u_i^n-\lambda\left(F(u_i^n,u_{i+1}^n)-F(u_{i-1}^n,u_{i}^n) \right) \end{equation}
Suppose we modify the above scheme by $$u_i^{n+1}=u_i^n-\lambda\left[F\left(u_{i+\frac{1}{2}}^{L,n},u_{i+\frac{1}{2}}^{R,n}\right)-F\left(u_{i-\frac{1}{2}}^{L,n},u_{i-\frac{1}{2}}^{R,n}\right) \right]$$ where $u_{i+\frac{1}{2}}^{L,n},u_{i+\frac{1}{2}}^{R,n}$ are the left and right limits at the point $x_{i+\frac{1}{2}}$ of the piecewise linear approximations of the piecewise constant functions given by $$v(x)=\sum\limits_{i \in \mathbb{Z}}\displaystyle{\chi_{C_i}u_i^n},$$ with slopes $p_i$ in interval $C_i.$
Suppose, $p_i, i\in \mathbb{Z}$ are chosen so that total variation of the piecewiese linear approximation is same as the total variation of the piecewise constant function (For example minmod limiter), then how to show that second order reconstruction as explained above is also TVD?.