scholarly journals An iterative method for solving difference problems of gas dynamics in the mixed Euler-Lagrangian variables

2021 ◽  
Vol 2099 (1) ◽  
pp. 012013
Author(s):  
M Sablin
Keyword(s):  
Iterative Method ◽  
Gas Dynamics ◽  
Nonlinear Function ◽  
Initial Boundary ◽  
Single Equation ◽  
Iteration Scheme ◽  
Time Step ◽  
Lagrangian Variables ◽  
The Difference ◽  
Initial Boundary Value

Abstract The method proposed is intended to solve implicit conservative operator difference schemes for a grid initial-boundary value problems on a simplex grid for a system of equations of gas dynamics in the mixed Euler-Lagrangian variables. To find a solution to such a scheme at a time step, it is represented as a single equation for a nonlinear function of two arguments from space – the direct product of the grid spaces of gas-dynamic quantities. To solve such an equation, a combination of the generalized Gauss-Seidel iterative method (external iterations) and an implicit two-layer iteration scheme (internal iterations at each external iteration) is used. The feature of the method is that, the equation, which is solved by internal iterations, is obtained from the equation of the difference scheme using symmetrization – such a non-degenerate linear transformation that the function in this equation has a self-adjoint positive Frechet derivative.

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.

On the convergence of difference schemes for the generalized BBM–Burgers equation

2019 ◽  
Vol 26 (3) ◽  
pp. 341-349 ◽  
Author(s):  
Givi Berikelashvili ◽  
Manana Mirianashvili
Keyword(s):  
Exact Solution ◽  
Sobolev Space ◽  
Difference Scheme ◽  
Absolute Stability ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Algebraic Equations ◽  
The Difference ◽  
Initial Boundary Value

Abstract A three-level finite difference scheme is studied for the initial-boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation. The obtained algebraic equations are linear with respect to the values of the desired function for each new level. The unique solvability and absolute stability of the difference scheme are shown. It is proved that the scheme is convergent with the rate of order {k-1} when the exact solution belongs to the Sobolev space {W_{2}^{k}(Q)} , {1<k\leq 3} .

THE GREEN'S FUNCTION FOR THE BROADWELL MODEL WITH A TRANSONIC BOUNDARY

2008 ◽  
Vol 05 (02) ◽  
pp. 279-294 ◽  
Author(s):  
CHIU-YA LAN ◽  
HUEY-ER LIN ◽  
SHIH-HSIEN YU
Keyword(s):  
Boundary Value Problem ◽  
Green’S Function ◽  
Green's Function ◽  
Boundary Data ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Convergent Series ◽  
Iteration Scheme ◽  
Initial Boundary Value

We study an initial boundary value problem for the Broadwell model with a transonic physical boundary. The Green's function for the initial boundary value problem is obtained by combining the estimates of the full boundary data and the Green's function for the initial value problem. The full boundary data is constructed from the imposed boundary data through an iteration scheme. The iteration scheme is designed to separate the interaction between the boundary wave and the interior wave and leads to a convergent series in the iterative boundary estimates.

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.

scholarly journals On the existence of global solution of the system of equations of liquid movement in porous medium

2021 ◽  
Vol 234 ◽  
pp. 00095
Author(s):  
Margarita Tokareva ◽  
Alexander Papin
Keyword(s):  
Porous Medium ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Solvability ◽  
Compressible Liquid ◽  
Fluid Filtration ◽  
One Dimensional ◽  
Lagrangian Variables ◽  
Initial Boundary Value ◽  
Liquid Movement

The initial-boundary value problem for the system of one-dimensional isothermal motion of viscous liquid in deformable viscous porous medium is considered. Local theorem of existence and uniqueness of problem is proved in case of compressible liquid. In case of incompressible liquid the theorem of global solvability in time is proved in Holder classes. A feature of the model of fluid filtration in a porous medium considered in this paper is the inclusion of the mobility of the solid skeleton and its poroelastiс properties. The transition from Euler variables to Lagrangian variables is used in the proof of the theorems.

scholarly journals Some features of numerical diagnostics of instantaneous blow-up of the solution by the example of solving the equation of slow diffusion

2021 ◽  
pp. 77-86
Author(s):  
И.В. Пригорный ◽  
А.А. Панин ◽  
Д.В. Лукьяненко
Keyword(s):  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
A Posteriori ◽  
Order Of Accuracy ◽  
Ill Posed ◽  
The Difference ◽  
Initial Boundary Value ◽  
Richardson Extrapolation Method ◽  
Posteriori Estimation

В работе демонстрируется, как метод апостериорной оценки порядка точности разностной схемы по Ричардсону позволяет сделать вывод о некорректности постановки (в смысле отсутствия решения) решаемой численно начально-краевой задачи для уравнения в частных производных. Это актуально в ситуации, когда аналитическое доказательство некорректности постановки ещё не получено или принципиально невозможно. The paper demonstrates how the method of a posteriori estimation of the order of accuracy for the difference scheme according to the Richardson extrapolation method allows one to conclude that the formulation of the numerically solved initial-boundary value problem for a partial differential equation is ill-posed (in the sense of the absence of a solution). This is important in a situation when the ill-posedness of the formulation is not analytically proved yet or cannot be proved in principle.

scholarly journals Conservative Linear Difference Scheme for Rosenau-KdV Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jinsong Hu ◽  
Youcai Xu ◽  
Bing Hu
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Kdv Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
The Difference ◽  
Second Order Convergence ◽  
Initial Boundary Value ◽  
Theoretical Results

A conservative three-level linear finite difference scheme for the numerical solution of the initial-boundary value problem of Rosenau-KdV equation is proposed. The difference scheme simulates two conservative quantities of the problem well. The existence and uniqueness of the difference solution are proved. It is shown that the finite difference scheme is of second-order convergence and unconditionally stable. Numerical experiments verify the theoretical results.

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