scholarly journals The Stability of the Solutions for a Porous Medium Equation with a Convection Term

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Huashui Zhan ◽  
Miao Ouyang
Keyword(s):  
Porous Medium ◽  
Boundary Value ◽  
Porous Medium Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Convection Term ◽  
Medium Equation ◽  
Special Cases ◽  
The Stability ◽  
Initial Boundary Value

This paper studies the initial-boundary value problem of a porous medium equation with a convection term. If the equation is degenerate on the boundary, then only a partial boundary condition is needed generally. The existence of the weak solution is proved by the monotone convergent method. Moreover, according to the different boundary value conditions, the stability of the solutions is studied. In some special cases, the stability can be proved without any boundary value condition.

scholarly journals On the Solutions of a Porous Medium Equation with Exponent Variable

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Huashui Zhan
Keyword(s):  
Porous Medium ◽  
Weak Solutions ◽  
Boundary Value ◽  
Porous Medium Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Medium Equation ◽  
Special Cases ◽  
The Stability ◽  
Initial Boundary Value

The paper studies the initial-boundary value problem of a porous medium equation with exponent variable. How to deal with nonlinear term with the exponent variable is the main dedication of this paper. The existence of the weak solution is proved by the monotone convergent method. Moreover, according to the different boundary value conditions, the stability of weak solutions is studied. In some special cases, the stability of weak solutions can be proved without any boundary value condition.

scholarly journals On a Partial Boundary Value Condition of a Porous Medium Equation with Exponent Variable

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Huashui Zhan
Keyword(s):  
Porous Medium ◽  
Variable Exponent ◽  
Boundary Value ◽  
Porous Medium Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Value Condition ◽  
Medium Equation ◽  
Partial Boundary Value Condition ◽  
Initial Boundary Value

The initial-boundary value problem of a porous medium equation with a variable exponent is considered. Both the diffusion coefficient ax,t and the variable exponent px,t depend on the time variable t, and this makes the partial boundary value condition not be expressed as the usual Dirichlet boundary value condition. In other words, the partial boundary value condition matching up with the equation is based on a submanifold of ∂Ω×0,T. By this innovation, the stability of weak solutions is proved.

scholarly journals Numerical Solotion of a Problem of Fluid Filtration in a Viscoelastic Porous Medium

2020 ◽  
pp. 72-76
Author(s):  
R.A. Virts ◽  
A.A. Papin ◽  
W.A. Weigant
Keyword(s):  
Porous Medium ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Order Equation ◽  
Second Order ◽  
Initial System ◽  
Effective Pressure ◽  
Initial Boundary Value

The paper considers a model for filtering a viscous incompressible fluid in a deformable porous medium. The filtration process can be described by a system consisting of mass conservation equations for liquid and solid phases, Darcy's law, rheological relation for a porous medium, and the law of conservation of balance of forces. This paper assumes that the poroelastic medium has both viscous and elastic properties. In the one-dimensional case, the transition to Lagrange variables allows us to reduce the initial system of governing equations to a system of two equations for effective pressure and porosity, respectively. The aim of the work is a numerical study of the emerging initial-boundary value problem. Paragraph 1 gives the statement of the problem and a brief review of the literature on works close to this topic. In paragraph 2, the initial system of equations is transformed, as a result of which a second-order equation for effective pressure and the first-order equation for porosity arise. Paragraph 3 proposes an algorithm to solve the initial-boundary value problem numerically. A difference scheme for the heat equation with the righthand side and a Runge–Kutta second-order approximation scheme are used for numerical implementation.

Linear stability analysis in the numerical solution of initial value problems

Acta Numerica ◽  
1993 ◽  
Vol 2 ◽  
pp. 199-237 ◽  
Author(s):  
J.L.M. van Dorsselaer ◽  
J.F.B.M. Kraaijevanger ◽  
M.N. Spijker
Keyword(s):  
Stability Analysis ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Upper Bounds ◽  
New Developments ◽  
Partial Linear ◽  
Linear Differential ◽  
The Stability ◽  
Initial Boundary Value

This article addresses the general problem of establishing upper bounds for the norms of the nth powers of square matrices. The focus is on upper bounds that grow only moderately (or stay constant) where n, or the order of the matrices, increases. The so-called resolvant condition, occuring in the famous Kreiss matrix theorem, is a classical tool for deriving such bounds.Recently the classical upper bounds known to be valid under Kreiss's resolvant condition have been improved. Moreover, generalizations of this resolvant condition have been considered so as to widen the range of applications. The main purpose of this article is to review and extend some of these new developments.The upper bounds for the powers of matrices discussed in this article are intimately connected with the stability analysis of numerical processes for solving initial(-boundary) value problems in ordinary and partial linear differential equations. The article highlights this connection.The article concludes with numerical illustrations in the solution of a simple initial-boundary value problem for a partial differential equation.

scholarly journals Numerical Solution of One Problem of Carbon Dioxide Injection into the Rock

2021 ◽  
pp. 81-85
Author(s):  
R.A . Virts
Keyword(s):  
Carbon Dioxide ◽  
Porous Medium ◽  
Boundary Value Problem ◽  
Numerical Study ◽  
Boundary Value ◽  
Original System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Carbon Dioxide Injection ◽  
Initial Boundary Value

The paper considers a two-dimensional mathematical model of filtration of a viscous incompressible liquid or gas in a porous medium. A unique feature of the model under consideration is the incorporation of poroelastic properties of the solid skeleton. From a mathematical point of view, the equations of mass conservation for liquid / gaseous and solid phases, Darcy's law, the rheological ratio for a porous medium, and the conservation law of the balance of forces are considered. The work is aimed at numerical study of the model initial-boundary value problem of carbon dioxide injection into the rock with minimum initial porosity. Also, it is necessary to find out the parameters at which the porosity will increase in the upper layers of the rock and, as a result, the gas will come to the surface. Section 1 contains a statement of the problem and a brief review of scientific papers related to this topic. In Section 2, the original system of constitutive equations is transformed. In the case of slow flows, when the convective term can be neglected, a system arises that consists of a second-order parabolic equation for the effective pressure of the medium and a first-order equation for porosity. Section 3 presents the results and conclusions of a numerical study of the initial-boundary value problem.

scholarly journals Countable stability of a weak solution of a parabolic differential-difference system with distributed parameters on the graph

2020 ◽  
Vol 16 (4) ◽  
pp. 402-414
Author(s):  
Vyacheslav V. Provotorov ◽  
◽  
Sergey M. Sergeev ◽  
Van Nguyen Hoang ◽  
◽  
...  
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Optimal Control Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Control Problems ◽  
Distributed Parameters ◽  
The Stability ◽  
Initial Boundary Value

The article proposes an analog of E. Rothe’s method (semi-discretization with respect to the time variable) for construction convergent different schemes when analyzing the countable stability of a weak solution of an initial boundary value problem of the parabolic type with distributed parameters on a graph in the class of summable functions. The proposed method leads to the study of the input initial boundary value problem to analyze the boundary value problem in a weak setting for elliptical type equations with distributed parameters on the graph. By virtue of the specifics of this method, the stability of a weak solution is understood in terms of the spectral criterion of stability (Neumann’s countable stability), which establishes the stability of the solution with respect to each harmonic of the generalized Fourier series of a weak solution or a segment of this series. Thus, there is another possibility indicated, in addition to the Faedo—Galerkin method, for constructing approaches to the desired solution of the initial boundary value problem, to analyze its stability and the way to prove the theorem of the existence of a weak solution to the input problem. The approach is applied to finding sufficient conditions for the countable stability of weak solutions to other initial boundary value problems with more general boundary conditions — in which elliptical equations are considered with the boundary conditions of the second or third type. Further analysis is possible to find the conditions under which Lyapunov stability is established. The approach can be used to analyze the optimal control problems, as well as the problems of stabilization and stability of differential systems with delay. Presented method of finite difference opens new ways for approximating the states of a parabolic system, analyzing their stability in the numerical implementation and algorithmization of optimal control problems.

scholarly journals Numerical Solution of a Two-Dimensional Problem of Fluid Filtration in a Deformable Porous Medium

2021 ◽  
pp. 88-92
Author(s):  
R.A. Virts
Keyword(s):  
Porous Medium ◽  
Boundary Value Problem ◽  
Numerical Solution ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Deformable Porous Medium ◽  
Balance Of Forces ◽  
Initial Boundary Value

The paper considers a two-dimensional mathematical model of filtration of a viscous incompressible fluid in a deformable porous medium. The model is based on the equations of conservation of mass for liquid and solid phases, Darcy’s law, the rheological relationship for a porous medium, and the law of conservation of the balance of forces. In this article, the equation of the balance of forces is taken in full form, i.e. the viscous and elastic properties of the medium are taken into account. The aim of the work is a numerical study of a model initial-boundary value problem. Section 1 gives a statement of the problem and a brief review of the literature on works related to this topic. In item 2, the original system of equations is transformed. In the case of slow flows, when the convective term can be neglected, a system arises that consists of a second-order parabolic equation for the effective pressure of the medium and the first-order equation for porosity. Section 3 proposes an algorithm for the numerical solution of the resulting initial-boundary value problem. For the numerical implementation, a variable direction scheme for the heat equation with variable coefficients is used, as well as the Runge — Kutta scheme of the fourth order of approximation.

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