Optimal Approximations of Transport Equations by Particle and Pseudoparticle Methods

Source abstract

The convergence rate of particle methods for solving linear transport equations is revisited. Denoting h the initial discretization parameter, we prove a quasi-optimal rate of convergence like $h^{s-\e}$ for all $\e>0$ for an initial data in the Sobolev space W s,p when choosing appropriate initial integration rules and general convolution. As it is well known, this suboptimality is due to the form and width of the convolution kernel. In particular, it can be fixed by computing an additional quantity, the cell deformation. Then one can restore the optimal rate of convergence, up to the first order (s=1), while keeping the built-in conservative aspect. To avoid these additional computations and move to higher order optimality, another strategy is introduced and analyzed. It is based on a discretization of the solution at initial time by local averages but differs from the usual particle methods: the local averages are viewed as point values of an approximation of the solution, and the regularization of the solution at time t >0 is performed by interpolation rather than convolution. This strategy allows us to recover optimal error estimates in L p or Sobolev norms (up to any prescribed order).