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.
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