  We study multiscale integrator numerical schemes for a class of stiff
stochastic differential equations (SDEs). We consider multiscale SDEs with
potentially multiple attractors that behave as diffusions on graphs as the
stiffness parameter goes to its limit. Classical numerical discretization
schemes, such as the Euler-Maruyama scheme, become unstable as the stiffness
parameter converges to its limit and appropriate multiscale integrators can
correct for this. We rigorously establish the convergence of the numerical
method to the related diffusion on graph, identifying the appropriate choice of
discretization parameters. Theoretical results are supplemented by numerical
studies on the problem of the recently developing area of introducing
irreversibility in Langevin samplers in order to accelerate convergence to
equilibrium.
