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Paul Chernoff in 1968 proposed his approach to approximations ofone-parameter operator semigroups while trying to give a rigorous mathematicalmeaning to Feynman's path integral formulation of quantum mechanics. In early2000's Oleg Smolyanov noticed that Chernoff's theorem may be used to obtainapproximations to solutions of initial-value problems for linear partialdifferential equations (LPDEs) of evolution type with variable coefficients,including parabolic equations, Schr\"odinger equation, and some other. Chernoffexpressions are explicit formulas containing variable coefficients of LPDE andthe initial condition, hence they can be used as a numerical method for solvingLPDEs. However, the speed of convergence of such approximations at the presenttime is understudied which makes it risky to employ this class of numericalmethods. In the present paper we take two equations with known solutions (heatequation and transport equation) and study both analytically and numericallythe speed of decay of the norm of the difference between Chernoffapproximations and exact solutions. We also provide graphical illustrations ofconvergence and its rate. These model examples, being relatively simple, allowto demonstrate general properties of Chernoff approximations. The observationsobtained build a base for the future employment of the approach based onChernoff's theorem to the problem of construction of new numerical methods forsolving initial-value problem for parabolic LPDEs with variable coefficients.

Speed of convergence of Chernoff approximations for two model examples: heat equation and transport equation