A study of a finite-difference method for solving gas dynamics equations in the Euler coordinates

Abstract

The paper investigates a two-level completely conservative finite-difference scheme for solving gas dynamics equations in the Euler coordinates. It is proved that the rate of convergence in the levelwise norm w£ h of the difference solution to the smooth solution of the original differential problem is of order τ + h under the conditions that the mesh steps are sufficiently small and τ ^ Ch. The problem of an additional error caused by the round off errors due to the algorithm implementation is analyzed. The paper considers the cases of one, two, and three space coordinates.Construction and analysis of nonlinear finite-difference schemes are of importance from both practical and theoretical points of view. Stability and convergence of nonlinear finite-difference schemes are mainly analyzed for particular formulations of differential problems [1,2,8], A general approach to the justification of convergence of nonlinear finite-difference schemes is proposed in [3], it is based on the presentation of a finite-difference scheme in the form of an operator equation and on the existence theorem for the solution of the operator equation in the vicinity of a known element. The main convergence condition is the uniform stability of the linearized scheme with respect to the equation coefficients. The application of this approach to the analysis of particular finite-difference schemes was demonstrated in [3,4].At the same time the straightforward investigation of some nonlinear finitedifference schemes using the energy method (for example, for gas dynamics equations [5,9,10] and equations describing flows of viscous gas [12][13][14]) yields that under some conditions on the original differential problem and the mesh steps the norm of the error of the numerical integration satisfies a quadratic recurrent inequality. In [12] a method for deriving a linear inequality from the quadratic one was suggested which requires no additional constraints on the parameters of the difference problem. Then to derive an error estimate of the numerical integration it remains only to apply the difference counterpart of the Gronwall lemma to this linear inequality. Note that the coefficients of the quadratic inequality should satisfy certain conditions. This paper investigates the completely conservative finite-difference scheme constructed in [7]. In the case of an isotermal gas flow we consider the Cauchy problem for the gas dynamics equations. Note that the results obtained remain valid for the equations describing a flow of an ideal gas even in the case of the complete system of equations. The equation for the temperature is not taken into account only for simplicity of manipulations. We assume that the differential problem has a unique solution periodic in the spatial variables in the classic sense which possesses a sufficient smoothness. Under the conditions that the mesh steps are sufficiently small and that τ < Ch we derive the estimate of accuracy of the difference solution of order τ 2 + h 2 in the W^h levelwise norm on the segment [Ο,Γ...