Second order conservative scheme with piecewise linear approximations

Question

Consider the scalar conservation law $u_t+f(u)_x=0,$ whose 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 $C_i.$ Then clearly the modified scheme is conservative and consistent and suppose it converges it goes to the weak solution. I have the following doubts

1. How to show that it is second order accurate?

2. What are the additional assumptions on slopes $p_i$ required to show that scheme converges to the entropy solution. How to prove it?

Though there are many books which deal with this problem I could not find books which precisely answer the above question.

Answer 1

Let's consider a simple example, where $f(u) = a u$, $a > 0$, and where the numerical flux $$ F(u^L, u^R) = a u^L $$ is derived from the Godunov (upwind) method. The linearly extrapolated data with slopes $p_i$ are $$ u_{i+1/2}^{L,n} = u_i^n + \tfrac{\Delta x}{2} p_i , \qquad u_{i+1/2}^{R,n} = u_{i+1}^n - \tfrac{\Delta x}{2} p_{i+1} \, . $$ The first-order upwind method is recovered if the slopes $p_i$ are zero. One may consider the family of slopes $$p_i = \tfrac12 (1+\omega) \tfrac{u_i^n - u_{i-1}^n}{\Delta x} + \tfrac12 (1-\omega) \tfrac{u_{i+1}^n - u_i^n}{\Delta x}$$ see e.g. the book by Toro doi:10.1007/b79761. The choice $\omega = 1$ is a downwind slope, $\omega = {-1}$ is an upwind slope, and $\omega = 0$ is a centered slope. The proposed updating formula is conservative, i.e. $u_i^{n+1} = u_i^n - \lambda \big( F_{i+1/2} - F_{i-1/2}\big)$ with the numerical flux $$ F_{i+1/2} = a u_{i+1/2}^{L,n} = -a\tfrac{1+\omega}{4} u_{i-1}^n + a (1 + \tfrac\omega{2}) u_i^n + a\tfrac{1-\omega}{4} u_{i+1}^n . $$ Using Taylor series for an exact solution, we have (see this post using $\lambda = \Delta t/\Delta x$) $$ u_i^{n+1} = u_i^n - a \lambda (u_x)_i^n \Delta x + \tfrac12 a^2 \lambda^2 (u_{xx})_i^n \Delta x^2 + O(\Delta x^3) $$ and the scheme's updating formula gives $$ u_i^{n+1} = u_i^n - a \lambda (u_x)_i^n \Delta x + O(\Delta x^3) $$ up to possible algebra mistakes. Thus, contrary to the second-order REA algorithm (Sec. 6.4 p. 106-107 of doi:10.1017/CBO9780511791253), this scheme is never exactly second-order accurate at fixed Courant number $a\lambda$. Indeed, the Taylor series coefficients above match up to order $\Delta x$ only. Now, we can still look for the regions of monotonicity/TV-stability of the scheme.