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).

翻译：本文研究了一种分裂步欧拉型格式（SSM）在相互作用粒子随机微分方程系统和McKean-Vlasov随机微分方程数值模拟中的收敛性，其中漂移项在空间和相互作用分量中具有完全超线性增长，且扩散系数为非常数Lipschitz函数。相互作用（或测度）分量中的超线性增长源于与超线性增长函数的卷积运算，尤其适用于具有多阱约束势的颗粒介质方程。从方法论角度看，我们完全避免了泛函不等式论证（因允许非常数无界扩散映射）。该格式在步长下达到近最优经典（路径空间）均方根误差率$1/2-\varepsilon$（$\varepsilon>0$），并在非路径空间均方误差度量下达到最优率$1/2$。数值算例验证了所有结论。特别地，实验测试对驯服方法是否适用于此类问题（含卷积项和非常数扩散系数）提出了质疑。