Variable piecewise interpolation solution of the transport equation

In this paper, we construct a piecewise interpolation method of approximate solution of the transport equation based on the Newton interpolation polynomial of two variables. We transform the polynomial to the algebraic form with numerical coefficients; this leads us to a sequence of iterations, which improves the accuracy of the approximation. The method is implemented in software and numerical experiments ares performed. The possibility of generalizations to systems of partial differential equations and integro-differential equations is discussed.