  We propose numerical schemes that enable the application of particle methods
for advection problems in general bounded domains. These schemes combine
particle fields with Cartesian tensor product splines and a fictitious domain
approach. Their implementation only requires a fitted mesh of the domain's
boundary, and not the domain itself, where an unfitted Cartesian grid is used.
We establish the stability and consistency of these schemes in $W^{s,p}$-norms,
$s\in\mathbb{R}$, $1<p\leq\infty$.
