Piecewise interpolation solution of the Cauchy problem for the transport equation with iterative refinement

Abstract

This article deals with the method of approximate solution of the Cauchy problem for the transport equation based on the Newton interpolation polynomial of two variables with iterative refinement. In each element of the division of a rectangular domain into subdomains, a polynomial approximation of the partial time derivative is constructed. The interpolation polynomial is transformed into the form of an algebraic polynomial with numerical coefficients. In the integrated form, the polynomial is substituted for the dependent variable in the right side of the expression of the partial time derivative. Iterative renewal of the process at a fixed polynomial degree leads to the analogue of Picard’s method of successive approximations.