# Preprint 2010-021

# Finite volume schemes for the approximation via characteristics of linear convection equations with irregular data

## F. Bouchut, R. Eymard and A. Prignet

**Abstract:**
We consider the approximation
by multidimensional finite volume schemes
of the transport of an initial measure by a Lipschitz flow.
We first consider a scheme defined via characteristics,
and we prove the convergence to the continuous solution,
as the time-step and the ratio of the space step
to the time-step tend to zero.
We then consider a second finite volume scheme,
obtained from the first one by addition
of some uniform numerical viscosity.
We prove that this scheme converges to the continuous solution,
as the space step tends to zero whereas
the ratio of the space step to the time-step
remains bounded by below and by above,
and under assumption of uniform regularity of the mesh.
This is obtained via an improved discrete Sobolev inequality
and a sharp weak BV estimate,
under some additional assumptions on the transport flow.
Examples show the optimality of these assumptions.