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

Euler simulation of interacting particle systems and McKean-Vlasov SDEs with fully superlinear growth drifts in space and interaction