Analysis of the implicit upwind finite volume scheme with rough coefficients

Abstract

We study the implicit upwind finite volume scheme for numerically approximating the linear continuity equation in the low regularity DiPerna–Lions setting. That is, we are concerned with advecting velocity fields that are spatially Sobolev regular and data that are merely integrable. We prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique distributional solution of the continuous model is at $\hbox {least}~{1}/{2}$. The numerical error is estimated in terms of logarithmic Kantorovich–Rubinstein distances and provides thus a bound on the rate of weak convergence.

Appendix A: The q-mean

We briefly describe some helpful estimates for the q-mean defined for $q>1$ by

$$\begin{aligned} \theta _q: \mathbf {R}_+ \times \mathbf {R}_+ \rightarrow \mathbf {R}_+ \qquad \text {with}\qquad \theta _q(a,b) := \frac{q-1}{q} \frac{a^q - b^q}{a^{q-1} -b^{q-1}}. \end{aligned}$$

• (i) The function $\theta _q$ has the following integral representation

  

• (ii) The function $\theta _q$ is 1-homogeneous: for any $c>0$ it holds $\theta _q(c\,a, c\,b) = c\,\theta _q(a,b)$.

• (iii) For any positive numbers $a\ne b$ is $q\mapsto \theta _q(a,b)$ strictly increasing.

• (iv) The function $(a,b)\mapsto \theta _q(a,b)$ is concave for $q\in (1,2)$ and convex for $q\in (2,\infty )$.

• (v) For any $a,b>0$ it holds

  

Proof

For the identity (i) let denote the integrand on the right hand side. Then, by a straightforward calculation it follows $\partial _s t_q(a,b;s) = \frac{b^{q-1}-a^{q-1}}{q-1} t(s)^{2-q}$ and hence $T_q(a,b;s) := \frac{q-1}{q} \frac{t(s)^q}{b^{q-1}-a^{q-1}}$ is its primitive from which (i) follows.

The 1-homogeneity as stated in (ii) follows immediately from the definition.

For proving (iii), we note that $t_q(a,b;s)$ is the $\ell ^{q-1}$ norm on the probability space for a function taking values $f(0)=a$ and $f(1)=b$. Hence, the statement is a consequence of the ordering of the $\ell ^p$ spaces, which extends to any value $p\in \mathbf {R}$.

The property (iv) follows by calculating the Hessian of $(a,b)\mapsto t_s(a,b)$:

Now, one immediately recovers that $(a,b)\mapsto t_q(a,b;s)$ is negative semidefinite for $q\in [0,2)$ and positive semidefinite for $q>2$.

For (v), we can assume by symmetry and 1-homogeneity that $a\in (0,1)$ and $b=1$. From (iv), we have that the mapping $(0,1) \ni a\mapsto \theta _q(a,1)$ is concave for $q\in (1,2)$ and convex for $q>2$. Let us first assume $q\in (1,2)$, then by convexity of $a\mapsto \theta _2(a,1)-\theta _q(a,1)$, we estimate using the secant inequality between the points 0 and 1 : 

For $q>2$, we apply the same argument to the convex function $a\mapsto \theta _q(a,1)-\theta _2(a,1)$.

$\square$