NONLIENEAR CONSISTENT METHOD (NC-METHOD) TO ACCELERATE CONVERGENCE OF ITERATIONS FOR TRANSPORT EQUATION
R. M. Shagaliev, A. A. Busalov
VANT. Ser.: Mat. Mod. Fiz. Proc 2021. Вып.2. С. 3-10.
The paper is devoted to constructing a new convergence accelerating method, namely, the nonlinear consistent method (NC-method) for one-dimensional computations. The NC-method is derived as applied to the grid approximation of the transport equation using a difference scheme that provides positive grid solutions on structured grids. The simplest case of a 1D transport equation in Cartesian coordinates is considered. The method admits generalization to a multidimensional case and can be also used to solve the transport equation in curvilinear coordinates. This is a two-stage method. The first stage is finding an approximate solution to the transport equation with the simple iteration method using the sweep algorithm at all points of the phase space. The second stage is constructing a system of grid equations relative to the scalar flow function that associates its values at a given point of the space grid with the values in neighboring intervals. The way of constructing the second stage equations ensures the consistency of the NC-method of accelerating the convergence of simple iterations. Results of numerical studies are given and they demonstrate a high efficiency of the NC-method and its unconditional convergence. Keywords: the transport equation, iterative methods for solving grid equations, acceleration methods, the NC-method.
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