Numerical Study of the Rate of Convergence of Chernoff Approximations to Solutions of the Heat Equation

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, however, the speed of their convergence to the exact solution has not been properly studied. We developed a program in Python 3 that allows to model a wide class of Chernoff approximations to a wide class of evolution equations on the real line. After that we select the heat equation (with already known exact solutions) 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 conduction 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 can be applied to general second order parabolic equation with variable coefficients by a slight modification of our Python 3 code, the full text of it is provided in the appendix to the paper.