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

Abstract

We consider the approximation by multidimensional nite volume schemes of the transport of an initial measure by a Lipschitz ow. We rst consider a scheme dened 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 nite volume scheme, obtained from the rst 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 ow. Examples show the optimality of these assumptions.

Numerical methods for piecewise deterministic Markov processes with boundary

“…An explicit scheme is introduced in [2] for Lipschitz flows. The present scheme uses the value τ > 0 in the same way as the time step is used in [2] (where the convergence of the scheme is proved for general Lipschitz flow φ in the case δt → 0 and h/δt → 0).…”

“…In all application areas, most of interest quantities depend on the distribution of the process at each time, which means that the approximation of the marginal distributions is requested. For this purpose, Monte Carlo methods are widely used (see for instance [8,14,17,18]), but it has been shown in [2,4,6,9,10,11,15] that Finite volume schemes could also provide an efficient approximation of these marginal distributions. These methods consist in solving numerically equations which are fulfilled by the marginal distributions, namely generalized Kolmogorov equations.…”

Numerical methods for piecewise deterministic Markov processes with boundary

“…But in this scheme the considered flow is much more regular (it is assumed to be the solution of an EDO). (3) An explicit scheme is introduced in [2] for Lipschitz flows. The present scheme uses the value τ > 0 in the same way as the time step is used in [2] (where the convergence of the scheme is proved for general Lipschitz flow φ in the case δt → 0 and h/δt → 0).…”

“…(3) An explicit scheme is introduced in [2] for Lipschitz flows. The present scheme uses the value τ > 0 in the same way as the time step is used in [2] (where the convergence of the scheme is proved for general Lipschitz flow φ in the case δt → 0 and h/δt → 0). (4) In [10], the asymptotic states at large times have been obtained letting δt → ∞ in an implicit scheme.…”

“…The characterization of the marginal distributions by these equations is studied in [5] for a PDMP without boundary. Finite volume schemes are proposed in [2,4,6,[10][11][12]14].…”