The approximation properties and the conditions of convergence of a computational scheme of the generalized method of Lie-algebraic discrete approximations for the solution of the Cauchy problem with a one-dimensional advection equation are proved. The reduction of the Cauchy problem with the advection equation to a system of linear algebraic equations ensures the factorial convergence in all variables of the equation.
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)
Abstract. We consider solving the Cauchy problem with an abstract linear evolution equation by means of the Generalized Method of Lie-algebraic discrete approximations. Discretization of the equation is performed by all variables in equation and leads to a factorial rate of convergence if Lagrange interpolation is used for building quasi representation of differential operator. The rank of a finite dimensional operator and approximation properties have been determined. Error estimations and the factorial rate of convergence have been proved.
Abstract. In the article the combined algorithm for finding conservation laws and implectic operators has been proposed. Using the Novikov-Bogoyavlensky method the finite dimensional reductions have been found. The structure of invariant submanifolds has been examined. Having analyzed phase portraits of Hamiltonian systems, partial periodical solutions have been found.
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