An explicit Euler method for Sobolev vector fields with applications to the continuity equation on non cartesian grids

We prove a novel stability estimate in $L^\infty _t (L^p_x)$ between the regular Lagrangian flow of a Sobolev vector field and a piecewise affine approximation of such flow. This approximation of the flow is obtained by a (sort of) explicit Euler method, and it is the crucial tool to prove approximation results for the solution of the continuity equation by using the representation of the solution as the push-forward via the regular Lagrangian flow of the initial datum. We approximate the solution in two ways, using different approximations for both the flow and the initial datum. In the first case we give an estimate, which however holds only in probability, of the Wasserstein distance between the solution of the continuity equation and a discrete approximation of such solution. The approximate solution is defined as the push-forward of weighted Dirac deltas (whose centers are chosen in a probabilistic way). In the second case we give a deterministic estimate of the Wasserstein distance using a slightly different approximation of the regular Lagrangian flow and requiring more regularity on the velocity field $u$ than in the previous case. An advantage of both approximations is that they provide an algorithm which is easily parallelizable and does not rely on any particular structure of the mesh with which we discretize (only in space) the domain.