Direct Method of Lie-Algebraic Discrete Approximations for Advection Equation

Abstract

We propose the direct method of Lie-algebraic discrete approximation for numerical solving the Cauchy problem for advection equation in this paper. Discretization of the equation is performed by means of the Lie-algebraic quasi representations for space variables of the equation and by means of Taylor series expansion and small parameter method for the time variable. Such combination of approaches leads to a factorial rate of convergence with respect to all variables in the equation if the quasi representations for differential operator are built by means of Lagrange interpolation. The approximation properties and error estimations for the proposed scheme are investigated. The factorial rate of convergence for the proposed numerical scheme has been proven.Key words: direct method of Lie-algebraic discrete approximations, advection equation, finite dimensional quasi representation, Lagrange polynomial, small parameter method, factorial convergence. t q t q (1.1)