Discrete entropy inequality for scalar conservation laws

Question

Consider a scalar conservation law $u_t+f(u)_x=0.$

A three point monotone scheme given by, \begin{eqnarray} u_i^{n+1}=u_i^{n}-\lambda (F(u_i^n,u_{i+1}^n)-F(u_{i-1}^n,u_i^n)) \end{eqnarray} where $F(u,u)=f(u).$

For a general entropy flux pair $(\eta,q)$ the discrete entropy condition is given by \begin{eqnarray} \eta(u_i^{n+1})-\eta(u_i^{n})+\lambda \left( Q(u_i^n,u_{i+1}^n)-Q(u_{i-1}^n,u_i^n)\right) \leq 0, \end{eqnarray} where $Q$ is the numerical entropy flux for the entropy flux pair $(\eta,q)$ which satisfies for every constant $k \in \mathbb{R}$ $$Q(u,u)=q(u)=\int\limits^u_k\eta'(s)f'(s)ds=\eta'(u)f(u)-\int\limits^u_k\eta''(s)f(s)ds.$$

If $\eta(u)=|u-k|$ then $q(u)=\operatorname{sgn}(u-k)(f(u)-f(k))$ and hence $Q(a,b)=f(\max(a,k))-f(\min(b,k))$ satisfies the discrete entropy inequality (see Theorem 4.2, Hyperbolic systems of conservation laws by Godlewski and Raviart).

I have the following questions.

1. For a general $\eta$ does the numerical entropy flux exist?
2. How to prove such a $Q$ satisfies the discrete entropy condition?

Answer 1

The definition of the numerical flux $Q(u_i, u_{i+1})$ I am aware of is the one given by Crandall & Majda. In equation (4.3) they state (I adapted their original version to your notation in 1D) $$Q(u_i, u_{i+1}) \colon = F\Big( \max \{u_i, k \}, \max \{u_{i+1},k \} \Big) - F\Big( \min \{u_i, k \}, \min \{u_{i+1},k \} \Big) $$

Thus, to answer question 1: The numerical flux function $Q$ depends only on your numerical flux $F$, not on the entropy $\eta$ forming a pair with the "analytical" entropy flux $q$.

As a side note: You recover your formula for $Q(u,u) = q(u, k) = \mathrm{sign} (u - k) \big( f(u) - f(k) \big)$ for a consistent numerical flux $\Big(F(u, u) = f(u) \Big)$: $$ \begin{align} Q(u, u) &= F\Big( \max \{u, k \}, \max \{u,k \} \Big) - F\Big( \min \{u, k \}, \min \{u,k \} \Big) \\ & = f\Big( \max \{u, k \} \Big) - f\Big( \min \{u, k \} \Big) \\ &= \mathrm{sign} (u - k) \big( f(u) - f(k) \big)\end{align} $$

2. $u$ has to satisfy the entropy condition for every convex $\eta$. As stated in Dafermos: Hyperbolic Conservation Laws in Continuum Physics:

To verify [the entropy condition] for all convex $\eta$, it would suffice to test it just for some family of convex $\eta$ with the property that the set of linear combinations of its members, with nonnegative coefficients, spans the entire set of convex functions. [...] one may use the celebrated family of entropy-entropy flux pairs of Kruzkov.

The same strategy is (slightly generalized) employed in this paper, Lemma 1. Thus, if you adapt your discrete entropy condition with $\eta = \eta_\text{Kruzkov} = \vert u - k \vert $ you can follow this proof (Lemma 4.10) showing that entropy fluxes $Q$ based on a conservative, consistent and monotone numerical flux $F$ satisfy the discrete entropy inequality.