New method for solving the Navier-Stokes equations with artificial relations between variations of quantities, applied at nearest nodes

With the use of the Newton method, a new numerical method previously published [1] for solving the three-dimensional Navier-Stokes equations, is theoretically proved for the most simple case of one-dimensional acoustic equations. The convergence of iteration scheme is proved. In this paper, we also recall some theoretical and numerical results presented earlier in [1]. The gradient of internal energy (see [1]) has to be redefined. This yielded in [1] that, along with descending temperature of internal walls, some small variations of balance of mass arose within the flow of a gas heated from its motion along tube walls. The author succeeded [1] in achieving the maximal time step Dtmax=h/uflow (h is the average size of cell, uflow stands for the flow velocity) along with the condition that, on every step, the required computation time exceeds approximately 6 times the time necessary for computation via an explicit scheme. Every step requires a number of arithmetic operations of order of N; here N is the number of nodes and cells. The stability and velocity of convergence were estimated in a numerical experience. Satisfactory correlation is obtained between the analytic and computed balances of mass in a tube for a given wall temperature dependence. Next, briefly, the idea of the method includes an artificial binding of unknowns' corrections at neighbouring nodes or cells; the respective corrections are determined not via solving bounded system of equations, but in a way directly based on the residual of equation for the corresponding unknown at either a node or a cell. A staggered arrangement of variables is used, this means that the pressure, density, and internal energy are located at the usual cell centres, whereas the velocity vectors are positioned at the displaced cell centres which are the vertices of usual cells. The three-dimensional Navier-Stokes equations are solved via the Newton iteration procedure. The initial guesses are taken for the time t + Dt as known values for time t, and the time step Dt is chosen with the requirement to provide the convergence within an approximately given number of iterations; then the divergence will be avoided due that restriction of time step. The introduction of artificial relations between the variations of quantities at the nearest nodes or cells and the use of approximate equality with opposite signs of vectors relating the geometric coefficients of both displaced and usual cells, make it possible to obtain formulas for correct rates of change of the residuals of equations