  We present a comprehensive discretization scheme for linear and nonlinear
stochastic differential equations (SDEs) driven by either Brownian motions or
$\alpha$-stable processes. Our approach utilizes compound Poisson particle
approximations, allowing for simultaneous discretization of both the time and
space variables in McKean-Vlasov SDEs. Notably, the approximation processes can
be represented as a Markov chain with values on a lattice. Importantly, we
demonstrate the propagation of chaos under relatively mild assumptions on the
coefficients, including those with polynomial growth. This result establishes
the convergence of the particle approximations towards the true solutions of
the McKean-Vlasov SDEs. By only imposing moment conditions on the intensity
measure of compound Poisson processes, our approximation exhibits universality.
In the case of ordinary differential equations (ODEs), we investigate scenarios
where the drift term satisfies the one-sided Lipschitz assumption. We prove the
optimal convergence rate for Filippov solutions in this setting. Additionally,
we establish a functional central limit theorem (CLT) for the approximation of
ODEs and show the convergence of invariant measures for linear SDEs. As a
practical application, we construct a compound Poisson approximation for
2D-Navier Stokes equations on the torus and demonstrate the optimal convergence
rate.
