  We consider in this work the convergence of a split-step Euler type scheme
(SSM) for the numerical simulation of interacting particle Stochastic
Differential Equation (SDE) systems and McKean-Vlasov Stochastic Differential
Equations (MV-SDEs) with full super-linear growth in the spatial and the
interaction component in the drift, and non-constant Lipschitz diffusion
coefficient.
  The super-linear growth in the interaction (or measure) component stems from
convolution operations with super-linear growth functions allowing in
particular application to the granular media equation with multi-well confining
potentials. From a methodological point of view, we avoid altogether functional
inequality arguments (as we allow for non-constant non-bounded diffusion maps).
  The scheme attains, in stepsize, a near-optimal classical (path-space) root
mean-square error rate of $1/2-\varepsilon$ for $\varepsilon>0$ and an optimal
rate $1/2$ in the non-path-space mean-square error metric. Numerical examples
illustrate all findings. In particular, the testing raises doubts if taming is
a suitable methodology for this type of problem (with convolution terms and
non-constant diffusion coefficients).
