Inviscid linear transport

We now consider the inviscid case for a linear transport equation. The BFN equations are:

Theorem 3.5 We consider the non viscous one-step BFN (3.32), with observations $u_{obs}$ satisfying (3.32-F) with . We denote

$\begin{displaymath}\begin{array}{rcl} w(t) &=& u(t) - u_{obs}(t), \\ \widetilde{w}(t) &=& \widetilde{u}(t)-u_{obs}(t). \end{array}\end{displaymath}$

(3.33)

We denote by

$\displaystyle (s,\psi(s,x))$

(3.34)

the characteristic curve of equation (3.32-F) with , with foot in time , i.e. such that

$\displaystyle (s,\psi(s,x))\vert _{s=0} = (0,x).$

(3.35)

We assume that the final time is such that the characteristics are well defined and do not intersect over . Then:

If , then we have, for all $t\in[0,T]$ ,

$\displaystyle \widetilde{w}(t) = w(t) e^{(-K-K^\prime)(T-t)}.$ (3.36)
If $K(t,x)=K \mathbbm{1}_{[t_{1},t_{2}]}(t)$ with $0\leq t_{1} < t_{2}\leq T$ , then we have

$\displaystyle \widetilde{w}(0) = w(0) e^{(-K-K^\prime)(t_{2}-t_{1})}.$ (3.37)
If , then we have, for all $t\in[0,T]$ ,

$\displaystyle \widetilde{w}(t,\psi(t,x)) = w(t,\psi(t,x)) \, \exp \left(-\int_{t}^T K(\psi(s,x))+K^\prime(\psi(s,x))\, ds \right).$ (3.38)

From this result, we deduce that the BFN algorithm applied to inviscid linear transport equation does converge if all the domain is observed (first two cases of theorem 3.5). Moreover, if the support of

does not cover all the domain (third case of theorem 3.5, e.g. when the system is not fully observed), the algorithm converges as soon as all the characteristics intersect the support of

. This constraint is satisfied as soon as the system is observable (see remarks below proposition 3.2).