Chernoff approximations are a flexible and powerful tool of functional analysis, which can be used, in particular, to find numerically approximate solutions of some differential equations with variable coefficients. For many classes of equations such approximations have already been constructed since pioneering papers of Prof. O.G.Somlyanov in 2000, however, the speed of their convergence to the exact solution has not been properly studied. We select the heat equation (because its exact solutions are already known) as a simple yet informative model example for the study of the rate of convergence of Chernoff approximations. Examples illustrating the rate of convergence of Chernoff approximations to the solution of the Cauchy problem for the heat equation are constructed in the paper. Numerically we show that for initial conditions that are smooth enough the order of approximation is equal to the order of Chernoff tangency of the Chernoff function used. We also consider not smooth enough initial conditions and show how H\"older class of initial condition is related to the rate of convergence. This method of study in the future can be applied to general second order parabolic equation with variable coefficients by a slight modification of our Python 3 code. This arXiv version of the text is a supplementary material for our journal article. Here we include all the written text from the article and additionally all illustrations (Appendix A) and full text of the Python 3 code (Appendix B).

翻译：Chernoff 近似是泛函分析中一种灵活且强大的工具，尤其可用于求解变系数微分方程的数值近似解。自 2000 年 O.G.Somlyanov 教授的奠基性论文以来，针对多类方程已构造出这类近似，然而其向精确解的收敛速度尚未得到充分研究。本文选取热方程（因其精确解已知）作为简单且具代表性的模型实例，用于研究 Chernoff 近似的收敛速度。文中构造了 Chernoff 近似向热方程柯西问题解收敛的算例。数值结果表明：对于足够光滑的初始条件，近似阶数等于所使用 Chernoff 函数的 Chernoff 相切阶数。本文还考虑非充分光滑的初始条件，揭示了初始条件的 Hölder 类与收敛速度之间的关系。通过略微修改我们的 Python 3 代码，该研究方法未来可推广至一般的二阶变系数抛物型方程。本文的 arXiv 版本是期刊论文的补充材料，此处包含论文全部正文，并额外附上所有图示（附录 A）及 Python 3 代码全文（附录 B）。