scholarly journals Stability and convergence of the difference schemes for equations of isentropic gas dynamics in Lagrangian coordinates

2012 ◽  
Vol 91 (105) ◽  
pp. 137-153
Author(s):  
Piotr Matus ◽  
Dmitry Polyakov
Keyword(s):  
Gas Dynamics ◽  
A Priori Estimates ◽  
Initial Boundary ◽  
Stability And Convergence ◽  
Lagrangian Coordinates ◽  
Smooth Functions ◽  
Differential Problem ◽  
Isentropic Gas Dynamics ◽  
Isentropic Gas ◽  
The Difference

For the initial-boundary value problem (IBVP) if isentropic gas dynamics written in Lagrangian coordinates written in terms of Riemann invariants we show how to obtain necessary conditions for existence of global smooth solution using the Lax technique. Under these conditions we formulate the existence theorem in the class of piecewise-smooth functions. A priori estimates with respect to the input data for the difference scheme approximating this problem are obtained. The estimates of stability are proved using only restrictions on the initial and boundary conditions corresponding the differential problem. In the general case the estimates have been obtained only for the finite instant of time t < t0. The monotonicity has been proved in the both cases. The uniqueness and convergence of the difference solution are also considered. The results of the numerical experiment illustrating theoretical statements are given.

Initial boundary value problems for isentropic gas dynamics

1992 ◽  
Vol 120 (1-2) ◽  
pp. 1-23 ◽  
Author(s):  
Shigeharu Takeno
Keyword(s):  
Difference Scheme ◽  
Boundary Value Problems ◽  
Gas Dynamics ◽  
Initial Boundary ◽  
Compensated Compactness ◽  
Initial Boundary Value Problems ◽  
Isentropic Gas Dynamics ◽  
Isentropic Gas ◽  
The Difference ◽  
Initial Boundary Value

SynopsisFor piston problems for a system of isentropic gas dynamics, convergence theorems of a difference scheme are obtained by compensated compactness theory and by analysis of the difference scheme.

Nonlinear Stability Of The Difference Schemes For Equations Of Isentropic Gas Dynamics

2008 ◽  
Vol 8 (2) ◽  
pp. 155-170 ◽  
Author(s):  
P. MATUS ◽  
A. KOLODYNSKA
Keyword(s):  
Initial Data ◽  
Numerical Experiment ◽  
Difference Scheme ◽  
Nonlinear Stability ◽  
Gas Dynamics ◽  
Differential Problem ◽  
Isentropic Gas Dynamics ◽  
Isentropic Gas ◽  
The Difference ◽  
Theoretical Results

AbstractFor the difference scheme approximating the gas dynamics problem in Riemann invariants a priory estimates with respect to the initial data have been obtained. These estimates are proved without any assumptions about the solution of the differential problem using only limitations for the initial and boundary conditions. Estimates of stability in the general case have been obtained only for the finite instant of time. The uniqueness and convergence of the difference solution are also considered. The results of the numerical experiment confirming theoretical results are given.

scholarly journals Study of a Numerical Method for Solving a Boundary Value Problem for a Differential Equation with a Fractional Time Derivative

2020 ◽  
pp. 64-69
Author(s):  
N.B Alimbekova ◽  
D.R. Baigereyev ◽  
M.N. Madiyarov
Keyword(s):  
Boundary Value Problem ◽  
Difference Scheme ◽  
A Priori Estimates ◽  
Boundary Value ◽  
Time Derivative ◽  
Smooth Functions ◽  
Differential Problem ◽  
Difference Problem ◽  
The Difference ◽  
Fractional Time Derivative

Recently, there has been an increased interest in the problem of numerical implementation of multiphase filtration models due to its enormous economic importance in the oil industry, hydrology, and nuclear waste management. In contrast to the classical models of filtration, filtration models in highly porous fractured formations with the fractal geometry of wells are not fully understood. The solution to this problem reduces to solving a system of differential equations with fractional derivatives. In the paper, a finite-difference scheme is constructed for solving the initial-boundary value problem for the convection-diffusion equation with a fractional time derivative in the sense of Caputo-Fabrizio. A priori estimates are obtained for solving a difference problem under the assumption that there is a solution to the problem in the class of sufficiently smooth functions that prove the uniqueness of the solution and the stability of the difference scheme. The convergence of the solution of the difference problem to the solution of the original differential problem with the second order in time and space variables is shown. The results of computational experiments confirming the reliability of theoretical analysis are presented.

LARGE TIME DECAY OF SOLUTIONS TO ISENTROPIC GAS DYNAMICS WITH SPHERICAL SYMMETRY

2009 ◽  
Vol 06 (02) ◽  
pp. 371-387
Author(s):  
NAOKI TSUGE
Keyword(s):  
Spherical Symmetry ◽  
Gas Dynamics ◽  
Large Time ◽  
Approximate Solutions ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Time Decay ◽  
Decay Of Solutions ◽  
Isentropic Gas Dynamics ◽  
Isentropic Gas

We consider the large time behavior of solutions to isentropic gas dynamics with spherical symmetry. In the present paper, we show the decay of the pressure in particular. To do this, we investigate approximate solutions constructed by a difference scheme.

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