Monotone numerical scheme and entropy solution to conservation laws

Question

A monotone numerical scheme is when I have two sequences $v=(v_j)_j$, $w=(w_j)_j$, and $v^{(n)} \ge w^{(n)}$ , then $v^{(n+1)} \ge w^{(n+1)}$ , where $ v_j^{(n)}$ is an appoximation of $u(x_j,t_n)$. Or in other words if I have a $2k+1$ Explicit difference scheme, then the derivatives at time $t_n$ are positive.

Why is a monotone numerical scheme a good approximation for an entropy solution? Is it because it ensures the existence of TVD schemes, which prevent oscillation, or does it have a direct relationship to the Entropy solution?

Answer 1

There is a theorem that links conservative monotonous schemes for conservation laws to entropy solutions:

Theorem. If a numerical method for the conservation law $u_t + f(u)_x = 0$ is conservative, consistent and monotonous, then

• The scheme is $L^\infty$ stable, and $\|u_h\|_\infty \leq \|u_0\|_\infty$;
• The scheme is entropic;
• The scheme is convergent (towards the entropy-satisfying weak solution).

Such a scheme is of order 1, at most.