An Explicit–Implicit Upwind Difference Splitting Scheme in Directions for a Mixed Boundary Control Problem for a Two-Dimensional Symmetric t-Hyperbolic System

Abstract

In this paper, we introduce a numerical integration method for hyperbolic systems problems known as the splitting method, which serves as an effective tool for solving complex multidimensional problems in mathematical physics. The exponential stability of the upwind explicit–implicit difference scheme split into directions is established for the mixed problem of a linear two-dimensional symmetric t-hyperbolic system with variable coefficients and lower-order terms. It is noteworthy that there are control functions in the dissipative boundary conditions. A discrete quadratic Lyapunov function was devised to address this issue. A condition for the problem’s boundary data, ensuring the exponential stability of the difference scheme with directional splitting for the mixed problem in the l² norm, has been identified.

1. Introduction

In this paper, we examine a mixed problem for a two-dimensional linear hyperbolic system with variable coefficients, lower-order terms and dissipative boundary conditions. We suggest a numerical integration technique for hyperbolic systems known as the “splitting method”. This method stands out as an efficient tool for addressing complex multidimensional problems in mathematical physics. The essence of the method revolves around simplifying a complex initial-boundary differential problem by sequential solving simpler initial-boundary problems, thus enabling the creation of simple, flexible and cost-effective difference schemes.

It is well-acknowledged that simple approximation schemes, when transitioning from one temporal layer to another, must simultaneously satisfy stability and approximation the conditions. This simplifies the numerical expressions in the schemes. However, the scheme becomes less adaptable and has fewer control parameters. As a result, it might not fulfill all the demands imposed on such schemes.

In contrast, the splitting method segments the transition between layers into several intermediate stages. During each stage, the properties of the approximation of the original equation and stability are not mandatorily required. This gives the differential scheme an array of parameters, allowing the selection of a cost-effective scheme. The splitting method addresses a tangible need that has emerged in computational mathematics: the need to devise simple, cost-effective schemes for solving complex multidimensional natural science problems.

Currently, the splitting method continues to evolve. It not only aids in constructing optimal algorithms, but also serves as a tool for the theoretical analysis of difference and differential equations.

The primary objective of this study is to provide a theoretical foundation for the directional splitting method applied to hyperbolic systems.

It is well-understood that difference schemes, besides adhering to the requirements of approximation and correctness, must also fulfill other less strict, but practically essential requirements. Foremost among them is the demand for the scheme’s cost-effectiveness, which is measured using a certain conditional machine time. The efficiency of a difference scheme is not only a means to save the machine time, but in some cases, it is a practically mandatory condition for implementing the scheme as a program.

Hyperbolic equations are known to have a finite dependency domain. Therefore, it would seem that their natural approximation is an explicit scheme. However, as illustrated by the well-known Courant–Friedrichs–Lewy criterion, the stability requirements are determined by values at a given point, while the accuracy requirements are defined by gradients.

For flows with small gradients (river flows, atmospheric flows, etc.), the time step dictated by the accuracy demands significantly exceeds the one dictated by stability. Therefore, there is a need for implicit schemes for hyperbolic equation systems.

Our research is focused on the upwind explicit–implicit difference scheme with splitting designed for the numerical modeling of stable solutions for the aforementioned system. We developed a discrete counterpart of the Lyapunov function for the given problem and obtained a priori estimates for its solution. These findings affirm the exponential stability of the numerical solution, which in turn justifies the convergence of the difference scheme.

In another study [1], a multidimensional initial-boundary problem for hyperbolic systems can be found. The exponential stability of solutions for a one-dimensional mixed problem pertaining to hyperbolic systems has been explored in [2,3].

It is worth mentioning that numerous investigations [4,5,6] focus on the stability analysis of differential methods for hyperbolic systems. However, in all these studies, stability was evaluated using dissipative energy integrals. As a consequence, the findings from these studies do not imply the exponential stability of numerical solutions.

In references [7,8,9,10,11], the stability of initial-boundary problems for hyperbolic systems using the Lyapunov function method has been scrutinized.

Control problems and the stability theory are applied in studies of various phenomena in chemistry and biology [12,13], as well as in bifurcation analysis and the stability of neural network models [14,15].

2. Proposed Methods

2.1. Problem Statement and Preliminary Information

Consider the case when the characteristic velocities $a_i(x), i=\overline{1,n}$ depend on spatial coordinates. Consider a linear symmetric t-hyperbolic system:

$$\frac{\partial v}{\partial t}+K\left(x\right)\frac{\partial v}{\partial x}+C\left(x\right)\frac{\partial v}{\partial y}+Q\left(x\right)v=0$$

(1)

with a diagonal matrix $K\left(x\right)\triangleq diag\left\{K^{+}\left(x\right),-K^{-}\left(x\right)\right\}$, such that

$$\begin{array}{l}{K}^{+}\left(x\right)=diag\left\{a_1\left(x\right),\cdots ,a_m\left(x\right)\right\}, K^{-}\left(x\right)=diag\left\{a_{m+1}\left(x\right),\cdots ,a_n\left(x\right)\right\},\\ a_i\left(x\right)\in C^1\left(\left[0,X\right]\right), a_i\left(x\right)>0 , i=\overline{1,n}, x\in \left[0,X\right]; \end{array}$$

Here, $C\left(x\right)$ is a given symmetric, positive definite matrix; $Q\left(x\right)$ is a given diagonal matrix;

$$Q\left(x\right)=diag\left(q_1\left(x\right),\cdots ,q_n\left(x\right)\right); q_i\left(x\right)\in C^0\left(\left[0,X\right]\right), i=\overline{1,n};$$

For $x=0$:

$$v^{\mathrm{I}}=s{v}^{\mathrm{II}}+{\psi }^{\mathrm{I}}, t\in \left[0,+\infty \right)$$

(2)

and for $x=X$:

$$v^{\mathrm{II}}=r{v}^{\mathrm{I}}+{\psi }^{\mathrm{II}}, t\in \left[0,+\infty \right)$$

(3)

with boundary conditions for $y=0$

$$v(x,0,t)=0, x\in \left[0,X\right], t\in \left[0,+\infty \right)$$

(4)

and with the initial data

$$v(x,y,0)=v_0(x,y), x\in \left(0,X\right), y\in \left(0,Y\right)$$

(5)

where $v(x,y,t)={(v^{\mathrm{I}}, v^{\mathrm{II}})}^{\mathrm{T}}$, $v^{\mathrm{I}}={(v_1,v_2,\cdots ,v_m)}^{\mathrm{T}}, v^{\mathrm{II}}={(v_{m+1},v_{m+2},\cdots ,v_n)}^{\mathrm{T}}$, $s=\Vert s_{pg}\Vert$ and $r=\Vert r_{gp}\Vert$ are rectangular matrices of dimensions $m\times \left(n-m\right)$ and $\left(n-m\right)\times m$, with elements $s_{pg}, p=\overline{1,m}, g=\overline{m+1,n}$ and $r_{gp}, p=\overline{m+1,n}, g=\overline{1,m}$, respectively.

$${\psi }^{\mathrm{I}}={({\psi }_1,{\psi }_2,\cdots ,{\psi }_m)}^{\mathrm{T}}, {\psi }^{\mathrm{II}}={({\psi }_{m+1},{\psi }_{m+2},\cdots ,{\psi }_n)}^{\mathrm{T}}, {\psi }_i\left(t\right)\in C^0\left(\left[0,+\infty \right)\right);$$

Suppose that the components of the initial data vector functions

$$v_0(x,y)=\phi (x,y)={({\phi }_1,{\phi }_2,\cdots ,{\phi }_n)}^T\in L^2((0,X)\times (0,Y),{\mathrm{R}}^n)$$

satisfy the compatibility condition:

$$\{\begin{array}{l}{\phi }^{\mathrm{I}}=s{\phi }^{\mathrm{II}}+{\psi }^{\mathrm{I}}, x=0, 0\le y\le Y,\\ {\phi }^{\mathrm{II}}=r{\phi }^{\mathrm{I}}+{\psi }^{\mathrm{II}}, x=X, 0\le y\le Y,\\ \phi \left(x,0\right)=0, 0\le x\le X.\end{array}$$

(6)

Here, ${\phi }^{\mathrm{I}}={({\phi }_1,{\phi }_2,\cdots ,{\phi }_m)}^{\mathrm{T}}, {\phi }^{\mathrm{II}}={({\phi }_{m+1},{\phi }_{m+2},\cdots ,{\phi }_n)}^{\mathrm{T}}, \phi ={({\phi }^{\mathrm{I}},{\phi }^{\mathrm{II}})}^{\mathrm{T}}$.

Our goals are to construct a difference problem for solving a system with linear boundary conditions in canonical form, taking into account the above conditions, and to study its solution for stability.

2.2. Explicit–Implicit Upwind Difference Splitting Scheme

In $G=\{(x,y,t): 0\le x\le X, 0\le y\le Y,0\le t\le T\}$ build a difference grid with steps $\Delta x$ in the $x$ direction, with $\Delta y$ in the $y$ direction and $\Delta t$ in the $t$ direction. The nodal points of the differential grid (indicated by the intersections of the lines $x=x_j=j\Delta x, y=y_l=l\Delta y$ and $t=t^{\kappa }=\kappa \Delta t$) are represented as $(x_j,y_l,t^{\kappa })$. The collection of these nodal points is termed as $G_h$, where

$$G_h=\left\{\left(x_j,y_l,t^{\kappa }\right): j=0,\dots ,J; l=0,\dots ,L; \kappa =0,\dots ,{\rm K}\right\}.$$

The numerical solution values at the grid points are represented by

$${(v_i)}_{jl}^{\kappa }=v_i\left(x_j,y_l,t^{\kappa }\right), i=1,\dots ,n; j=0,\dots ,J; l=0,\dots ,L; \kappa =0,\dots ,{\rm K}.$$

Let us define the steps of the difference grid as $\Delta x=\frac{X}{J}$, $\Delta y=\frac{Y}{L}$ and $\Delta t=\frac{T}{K}$.

To find a numerical solution of mixed problems (1)–(6) over the difference grid $G_h$, we suggest the following difference splitting scheme in the directions.

Suppose, for simplicity of presentation, that the matrix C is a diagonal matrix

$$C_j=diag\left({\left(c_1\right)}_j,{\left(c_2\right)}_j,\cdots ,{\left(c_n\right)}_j\right), {\left(c_i\right)}_j>0, i=\overline{1,n}; j=\overline{1,J-1}.$$

The initial system (1) is approximated by an explicit–implicit difference scheme of splitting in different directions:

$$\begin{array}{l} {\left(w_i\right)}_{jl}^{\kappa }={\left(v_i\right)}_{jl}^{\kappa }-{\left(c_i\right)}_j\frac{\Delta t}{\Delta y}\left[{\left(v_i\right)}_{jl}^{\kappa }-{\left(v_i\right)}_{jl-1}^{\kappa }\right], \\ i=\overline{1,n}; j=\overline{1,J-1}; l=\overline{1,L-1}; \kappa =\overline{0,{\rm K}-1}.\end{array}$$

(7)

$$\begin{array}{c}\{\begin{array}{l}{\left(v_i\right)}_{jl}^{\kappa +1}={\left(w_i\right)}_{jl}^{\kappa }-{\left(a_i\right)}_j\frac{\Delta t}{\Delta x}\left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]-\Delta t{}_{i}q_j{\left(v_i\right)}_{jl}^{\kappa +1}, i=\overline{1,m}; \\ {\left(v_i\right)}_{jl}^{\kappa +1}={\left(w_i\right)}_{jl}^{\kappa }-{\left(a_i\right)}_j\frac{\Delta t}{\Delta x}\left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(v_i\right)}_{j+1l}^{\kappa +1}\right]-\Delta t{}_{i}q_j{\left(v_i\right)}_{jl}^{\kappa +1}, i=\overline{m+1,n};\end{array}\\ j=\overline{1,J-1}; l=\overline{1,L-1}; \kappa =\overline{0,{\rm K}-1}.\end{array}$$

(8)

And with the initial data, we take the exact value of the initial functions (5) at the nodal points of the initial layer in time:

$${\left(v_i\right)}_{jl}^0={\left({\phi }_i\right)}_{jl}, i=\overline{1,n}; j=\overline{0,J}; l=\overline{0,L}.$$

(9)

To approximate these boundary conditions (2), we take the values ${\left(v_i\right)}_{0l}^{\kappa +1}, i=\overline{m+1,n}; l=\overline{0,L-1}; k=\overline{0,K-1}$, which are equal to their values at the neighboring point ${\left(v_i\right)}_{1l}^{\kappa +1}, i=\overline{m+1,n}; l=\overline{0,L-1}; k=\overline{0,K-1}$, respectively. And for these boundary conditions (3), the values of the grid functions are exactly the same ${\left(v_i\right)}_{Jl}^{\kappa +1}, i=\overline{1,m}; l=\overline{0,L-1}; k=\overline{0,K-1}$, which we suppose are equal to ${\left(v_i\right)}_{J-1l}^{\kappa +1}, i=\overline{1,m};$ $l=\overline{0,L-1}; k=\overline{0,K-1}$, respectively. In this case, it is obvious that the approximation of the boundary conditions will be of the first order of accuracy:

$$\{\begin{array}{l}{\left(v^{\mathrm{I}}\right)}_{0l}^{\kappa +1}=s{\left(v^{\mathrm{II}}\right)}_{1l}^{\kappa +1}+{\left({\psi }^{\mathrm{I}}\right)}^{\kappa +1}, \\ {\left(v^{\mathrm{II}}\right)}_{Jl}^{\kappa +1}=r{\left(v^{\mathrm{I}}\right)}_{J-1l}^{\kappa +1}+{\left({\psi }^{\mathrm{II}}\right)}^{\kappa +1},\end{array} l=\overline{0,L-1}; \kappa =\overline{0,{\rm K}-1}.$$

(10)

The boundary conditions (4) are approximated as follows

$${\left(v_i\right)}_{j0}^{\kappa +1}=0, i=\overline{1,n}; j=\overline{0,J-1}; k=\overline{0,K-1}.$$

(11)

Let us investigate the exponential stability of the numerical solution to the difference problems presented in (7)–(11). This article introduces the definition of exponential stability in relation to the numerical solution of problems (7)–(11).

Definition 1. 

The solution of problems (7)–(11) is called the Lyapunov stable if there are such positive constants $\eta >0$ and $c>0, c_2>0, \xi >0$, such that for any initial vector function

$${\phi }_{jl}\in l^2\left({\mathsf{\Omega }}_h;{\mathrm{R}}^n\right), {\mathsf{\Omega }}_h=\left\{\left(x_j,y_l\right): j=1,2,\dots ,J-1; l=1,2,\dots ,L-1\right\}$$

the solution for initial-boundary difference problems (7)–(11) satisfies the inequality

$$\begin{array}{c}\Delta y\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}{\displaystyle \sum _{i=1}^n{\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2}}}\le c{e}^{-\eta t_{\kappa }}\Delta y\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}{\displaystyle \sum _{i=1}^n{\left[{\left({\phi }_i\right)}_{jl}\right]}^2}}}\\ + \frac{c_2}{\eta }\left(1+\frac{1}{\xi }\right)\underset{0\le s<k}{\sup }{\displaystyle \sum _{i=1}^n\left({\left|{\left({\psi }_i\right)}^s\right|}^2\right)}, \kappa =\overline{1,K}.\\ \end{array}$$

Here, $l^2\left({\mathsf{\Omega }}_h;{\mathrm{R}}^n\right)$ is a discrete space $L^2$ and is a family of grid vector functions sequences in the form $w_{jl}$, such that

$${\left\{\Delta y\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\left(w_{jl},w_{jl}\right)}}\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}<\infty .$$

We investigate the Lyapunov stability of difference problems (7)–(11) based on the approach of constructing a discrete quadratic Lyapunov function. As a discrete quadratic Lyapunov function, we propose the following function

$$L^{\kappa }={}_{1}L^{\kappa }+{}_{2}L^{\kappa }, W^{\kappa }= {}_{1}W^{\kappa }+{}_{2}W^{\kappa },$$

where

$$\begin{array}{l} L_i^{\kappa }\triangleq \{\begin{array}{c}\Delta y\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_l}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}{\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2}, i=\overline{1,m}; \\ \Delta y\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_l}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j+1}{\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2} , i=\overline{m+1,n};\end{array} \\ W_i^{\kappa }\triangleq \{\begin{array}{c}\Delta y\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_l}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}{\left[{\left(w_i\right)}_{jl}^{\kappa }\right]}^2}, i=\overline{1,m}; \\ \Delta y\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_l}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j+1}{\left[{\left(w_i\right)}_{jl}^{\kappa }\right]}^2} , i=\overline{m+1,n};\end{array}\end{array}$$

(12)

$${}_{1}L^{\kappa }={\displaystyle \sum _{i=1}^m{L}_i^{\kappa }}, {}_{2}L^{\kappa }={\displaystyle \sum _{i=m+1}^n{L}_i^{\kappa }}, {}_{1}W^k={\displaystyle \sum _{i=1}^m{W}_i^{\kappa }}, {}_{2}W^k={\displaystyle \sum _{i=m+1}^n{W}_i^{\kappa }} .$$

$${}_{i}e_j=\{\begin{array}{c}\exp \left(-\Delta x{\displaystyle \sum _{q=1}^j\frac{\nu }{{\left(a_i\right)}_q}}\right), i=\overline{1,m};\\ \exp \left(\Delta x{\displaystyle \sum _{q=1}^j\frac{\nu }{{\left(a_i\right)}_q}}\right), i=\overline{m+1,n};\end{array} \nu >0.$$

$${}_{i}e_l=\exp \left(-\frac{y_l\nu }{{\left(c_i\right)}_j}\right), \nu >0.$$

Here, $\nu ; {\mu }_{x1},{\mu }_{x2},\cdots ,{\mu }_{xn}; {\mu }_{y1},{\mu }_{y2},\cdots ,{\mu }_{yn}$ are the positive constants to be determined.

3. Results

Using $M_{n,n}(ℝ)$, we denote the set of real matrices in dimension $n\times n$. In order to formulate the stability theorem, we first introduce function ${\rho }_2(M)$, which is similar to that in [2] (see page 86, Formula (3.6)); it is defined by

$${\rho }_2(M)= \left(1+\gamma \right)\inf \left\{{\Vert \Delta M{\Delta }^{-1}\Vert }_2, \Delta \in D_n^{+}\right\},$$

where $D_n^{+}$ denotes the set of diagonal $n\times n$ real matrices with strictly positive diagonal elements, $\gamma$ is strictly positive number and

• for $\zeta ={({\zeta }_1,\dots ,{\zeta }_n)}^{\mathrm{T}}\in {\mathrm{R}}^n, {\Vert \zeta \Vert }_2={\left[{\displaystyle \sum _{i=1}^n{\left|{\zeta }_i\right|}^2}\right]}^{\frac{1}{2}},$
• for $M\in M_{n,n}(ℝ), {\Vert M\Vert }_{2,\max }=\underset{{\Vert \zeta \Vert }_2=1}{\max } {\Vert M\zeta \Vert }_2$.

Also the following composite matrix consisting of boundary matrices $r, s$ is denoted by $R$:

$$R=\left(\begin{array}{cc}0& s\\ r& 0\end{array}\right).$$

And now, we formulate the following basic stability theorem.

Theorem 1. 

Let $T>0$ and the discrete Lyapunov function is defined using Formula (12). If matrices $r$ and $s$ of the parameters of the boundary conditions (10) obey the inequality ${\rho }_2(R)<1$, (dissipative conditions of boundary conditions), then numerical solution $v_{jl}^{\kappa }$ for difference boundary value problems (7)–(11) is exponentially stable in the $l^2$ norm (in the sense of the definition).

To prove the theorem, we prove some auxiliary lemmas.

Consider differential Equation (7) in conjunction with the boundary condition (11). Presume that the intervals of the differential grid adhere to the CFL condition:

$$\underset{1\le j\le J-1}{\max }{\left(c_i\right)}_j\frac{\Delta t}{\Delta y}\le 1, i\in \overline{1,n}.$$

(13)

Lemma 1. 

Let conditions (13) be fulfilled. Then, for the solution of difference scheme (7), the following inequality is valid:

$$\frac{W_i^k-L_i^k}{\Delta t}\le {\mu }_{yi}\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{jl-1}^{\kappa }\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2\right\}}, i=\overline{1,m}.$$

(14)

$$\frac{W_i^k-L_i^k}{\Delta t}\le {\mu }_{yi}\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j+1}\left\{{\left[{\left(v_i\right)}_{jl-1}^{\kappa }\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2\right\}} , i=\overline{m+1,n}.$$

(15)

Proof. 

Denote the Courant number by ${}_i\rho _j$

$${}_i\rho _j={\left(c_i\right)}_j\frac{\Delta t}{\Delta y}$$

(16)

Taking into account notation (16), difference scheme (7) can be written as:

$${\left(w_i\right)}_{jl}^{\kappa }={\left(v_i\right)}_{jl}^{\kappa }-{}_i\rho _j\left[{\left(v_i\right)}_{jl}^{\kappa }-{\left(v_i\right)}_{jl-1}^{\kappa }\right]$$

(17)

Taking into account the form of record (17) of difference scheme (7), we obtain the following expression for $\frac{W_i^k-L_i^k}{\Delta t}$:

$$\begin{array}{c}\frac{W_i^k-L_i^k}{\Delta t}=\frac{\Delta y}{\Delta t}\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_l}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(w_i\right)}_j^{\kappa }\right]}^2-{\left[{\left(v_i\right)}_j^{\kappa }\right]}^2\right\}}\\ =\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{}_i\rho _j}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(w_i\right)}_j^{\kappa }\right]}^2-{\left[{\left(v_i\right)}_j^{\kappa }\right]}^2\right\}}, i=\overline{1,m}.\end{array}$$

In the right part of the expression for $\frac{W_i^k-L_i^k}{\Delta t}$, by substituting values ${\left(w_i\right)}_j^{\kappa }$ according to Formula (17), we have

$$\begin{array}{c}\frac{W_i^k-L_i^k}{\Delta t}= \Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{}_i\rho _j}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}\left({\left\{{\left(v_i\right)}_{jl}^{\kappa }-{}_i\rho _j\left[{\left(v_i\right)}_{jl}^{\kappa }-{\left(v_i\right)}_{jl-1}^{\kappa }\right]\right\}}^2-{\left[{\left(v_i\right)}_j^{\kappa }\right]}^2\right)}\\ =\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{}_i\rho _j}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}}{}_{i}e_{l-1}{}_{i}e_{j-1}\\ \times \left(\left[{{}_i\rho _j}^2-2{}_i\rho _j\right]{\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2+2{}_i\rho _j\left(1-{}_i\rho _j\right){\left(v_i\right)}_{jl}^{\kappa }{\left(v_i\right)}_{jl-1}^{\kappa }+{{}_i\rho _j}^2{\left[{\left(v_i\right)}_{jl-1}^{\kappa }\right]}^2\right), i=\overline{1,m}.\end{array}$$

From the last part, taking into account (16), we have ${}_i\rho _j>0, 1-{}_i\rho _j>0$. Hence, ${}_i\rho _j\left(1-{}_i\rho _j\right)>0$. Using an obvious inequality

$$2{\left(v_i\right)}_{jl}^{\kappa }{\left(v_i\right)}_{jl-1}^{\kappa }\le {\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2+{\left[{\left(v_i\right)}_{jl-1}^{\kappa }\right]}^2$$

we obtain (14). Similarly, inequality (15) of Lemma 1 is proved. □

Lemma 2. 

Let the boundary condition (11) be satisfied. Then, the equality is valid.

$$\begin{array}{c}{\mu }_{yi}\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{jl-1}^{\kappa }\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2\right\}}\\ ={\mu }_{yi}\Delta x{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_0{}_{i}e_{j-1}{\left[{\left(v_i\right)}_{j0}^{\kappa }\right]}^2-{\mu }_{yi}\Delta x{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{L-1}{}_{i}e_{j-1}{\left[{\left(v_i\right)}_{jL-1}^{\kappa }\right]}^2\\ -\nu \Delta x\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_j}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{jl-1}^{\kappa }\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2\right\}}.\end{array}$$

(18)

Proof. 

Using the difference differentiation formula, we obtain

$$\begin{array}{c}{\displaystyle \sum _{l=1}^{L-1}{}_{i}e_{l-1}\left\{{\left[{\left(v_i\right)}_{jl-1}^{\kappa }\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2\right\}}\\ ={\displaystyle \sum _{l=1}^{L-1}\left\{{}_{i}e_{l-1}{\left[{\left(v_i\right)}_{jl-1}^{\kappa }\right]}^2-{}_{i}e_l{\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2\right\}} -{\displaystyle \sum _{l=1}^{L-1}\left({}_{i}e_{l-1}-{}_{i}e_l\right){\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2.}\end{array}$$

(19)

We study each sum on the right side of equality (19) separately. Taking into account (11), it is not difficult to establish the validity of the following equalities:

$${\displaystyle \sum _{l=1}^{L-1}\left\{{}_{i}e_{l-1}{\left[{\left(v_i\right)}_{jl-1}^{\kappa }\right]}^2-{}_{i}e_l{\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2\right\}}={}_{i}e_0{\left[{\left(v_i\right)}_{j0}^{\kappa }\right]}^2-{}_{i}e_{L-1}{\left[{\left(v_i\right)}_{jL-1}^{\kappa }\right]}^2=-{}_{i}e_{L-1}{\left[{\left(v_i\right)}_{jL-1}^{\kappa }\right]}^2;$$

$${\displaystyle \sum _{l=1}^{L-1}\left({}_{i}e_l-{}_{i}e_{l-1}\right){\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2}=-\Delta y\frac{\nu }{{\left(c_i\right)}_j}{\displaystyle \sum _{l=1}^{L-1}{}_{i}e_{l-1}{\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2}.$$

Indeed, the proof of the first equality is followed by direct summation, taking into account boundary condition (11). To prove the second equality, we use the following chain of equalities up to $O\left({\left(\Delta y\right)}^2\right)$:

$${}_{i}e_l-{}_{i}e_{l-1}={}_{i}e_{l-1}\left\{\exp \left(-\Delta y\frac{\nu }{{\left(c_i\right)}_j}\right)-1\right\}=-\Delta y\frac{\nu }{{\left(c_i\right)}_j}{}_{i}e_{l-1}$$

Lemma 2 is proved. □

From Lemma 2, taking into account equality (19), we obtain

$$\begin{array}{c}\frac{W_i^k-L_i^k}{\Delta t}\le {\mu }_{yi}\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{jl-1}^{\kappa }\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2\right\}}\\ \le -{\mu }_{yi}\Delta x{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{L-1}{}_{i}e_{j-1}{\left[{\left(v_i\right)}_{jL-1}^{\kappa }\right]}^2\\ -\nu \Delta x\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_j}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{jl-1}^{\kappa }\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa }\right]}^2\right\}}\\ =-{\mu }_{yi}\Delta x{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}}{}_{i}e_{L-1}{}_{i}e_{j-1}{\left[{\left(v_i\right)}_{jL-1}^{\kappa }\right]}^2-\nu L_i^k\le -\nu L_i^k; i=\overline{1,m}\end{array}$$

or

$$\frac{W_i^k-L_i^k}{\Delta t}\le -\nu L_i^k; i=\overline{1,m}; \forall \kappa =\overline{0,N-1}.$$

(20)

By doing the same again, we can obtain the following inequalities in the cases $i=\overline{m+1,n}$:

$$\frac{W_i^k-L_i^k}{\Delta t}\le -\nu L_i^k; i=\overline{m+1,n}; \forall \kappa =\overline{0,N-1}.$$

(21)

Summing up the left and right sides of inequalities (20) and (21), respectively, we obtain

$$\frac{W^k-L^k}{\Delta t}\le -\nu L^k; \forall \kappa =\overline{0,N-1}.$$

Lemma 3. 

For any solution of difference Equation (8), the following inequality holds

$$\begin{array}{l}\frac{L_i^{\kappa +1}-W_i^{\kappa }}{\Delta t}\le \\ \{\begin{array}{c}{\mu }_{xi}\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2\right\}, {}_{i}q_j\ge 0, \forall i,j; }\\ {\mu }_{xi}\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2\right\}}+2\overline{q}{L}_i^{\kappa +1}, {}_{i}q_j<0, \forall i,j;\end{array}\\ i=\overline{1,m}.\end{array}$$

(22)

Here, $\overline{q}=\underset{\begin{array}{l}1\le j\le J-1\\ 1\le i\le m\end{array}}{\max }\left|{}_{i}q_j\right|, q_{\max }=\underset{\begin{array}{l}1\le j\le J-1\\ 1\le i\le m\end{array}}{\max }\left({}_{i}q_j\right), q_{\min }=\underset{\begin{array}{l}1\le j\le J-1\\ 1\le i\le m\end{array}}{\min }\left({}_{i}q_j\right).$

Proof. 

Denote the Courant number ${}_i\rho _{aj}={\left(a_i\right)}_j\frac{\Delta t}{\Delta x}, i=\overline{1,m}; j=\overline{1,J-1}$ by ${}_i\rho _{aj}$ for the first m difference equations of scheme (7). Then, the first m difference equations of scheme (7) will take the following form:

$${\left(v_i\right)}_{jl}^{\kappa +1}={\left(w_i\right)}_{jl}^{\kappa }-{}_i\rho _{aj}\left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]-\Delta t{}_{i}q_j{\left(v_i\right)}_{jl}^{\kappa +1}, i=\overline{1,m}.$$

(23)

By multiplying both parts of Equation (23) by $2{\left(v_i\right)}_j^{\kappa +1}$, we have:

$$\begin{array}{c}\left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(w_i\right)}_{jl}^{\kappa }\right]2{\left(v_i\right)}_{jl}^{\kappa +1}+{}_i\rho _{aj}\left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]2{\left(v_i\right)}_{jl}^{\kappa +1}\\ + \Delta t{}_{i}q_j{\left(v_i\right)}_{jl}^{\kappa +1}2{\left(v_i\right)}_{jl}^{\kappa +1}=0, i=\overline{1,m}.\end{array}$$

(24)

We separately transform each of the difference Equation (24) terms as follows:

$$\begin{array}{l}\left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(w_i\right)}_{jl}^{\kappa }\right]2{\left(v_i\right)}_{jl}^{\kappa +1}={\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2-{\left[{\left(w_i\right)}_{jl}^{\kappa }\right]}^2+{\left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(w_i\right)}_{jl}^{\kappa }\right]}^2;\\ \left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]2{\left(v_i\right)}_{jl}^{\kappa +1}={\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2+{\left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2.\end{array}$$

Note that these equalities are proved using direct verification. Taking into account these transformations, (24) can be represented as follows:

$$\begin{array}{c}{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2-{\left[{\left(w_i\right)}_{jl}^{\kappa }\right]}^2+{}_i\rho _{aj}\left\{{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2\right\}\\ + {\left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(w_i\right)}_{jl}^{\kappa }\right]}^2+{}_i\rho _{aj}\left\{{\left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2\right\}+2\Delta t{}_{i}q_j{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2=0.\end{array}$$

The last identity is represented in the following form:

$${}_i\varsigma _{jl}^k=-{}_i\rho _{aj}\left\{{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2\right\}-{}_i\omega _{jl}^{k+1}$$

(25)

where

$${}_i\varsigma _{jl}^k\triangleq {\left[{\left(v_i\right)}_j^{\kappa +1}\right]}^2-{\left[{\left(w_i\right)}_j^{\kappa }\right]}^2,$$

$${}_i\omega _{jl}^{k+1}={\left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(w_i\right)}_{jl}^{\kappa }\right]}^2+{}_i\rho _{aj}{\left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2+2\Delta t{}_{i}q_j{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2.$$

Consider the following possible cases:

(1)

Case I: ${}_{i}q_j\ge 0, \forall i,j.$

(2)

Case II: ${}_{i}q_j<0, \forall i,j.$

Note, that in Case I, ${}_i\omega _{jl}^{k+1}\ge 0$. Taking into account in the first case from (25) for ${}_i\varsigma _{jl}^k$, we have

$${}_i\varsigma _{jl}^k\le -{}_i\rho _{aj}\left\{{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2\right\}.$$

By applying this inequality for ${}_i\varsigma _{jl}^k$ to the expression $\frac{L_i^{\kappa +1}-W_i^{\kappa }}{\Delta t}$, we obtain:

$$\begin{array}{l}\frac{L_i^{\kappa +1}-W_i^{\kappa }}{\Delta t}=\\ \Delta y\frac{\Delta x}{\Delta t}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}\frac{{\mu }_{yi}}{{\left(c_i\right)}_l}{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2-{\left[{\left(w_i\right)}_{jl}^{\kappa }\right]}^2\right\}}}\\ \le \Delta y{\mu }_{xi}\frac{{\mu }_{yi}}{{\left(c_i\right)}_l}{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2\right\}, i=\overline{1,m}. }}\end{array}$$

(26)

In Case II,

$${}_i\omega _{jl}^{k+1}={}_i\overline{\omega }_{jl}^{k+1}+2\Delta t{}_{i}q_j{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2,$$

where

$${}_i\overline{\omega }_{jl}^{k+1}\triangleq {\left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(v_i\right)}_{jl}^{\kappa }\right]}^2+{}_i\rho _j{\left[{\left(v_i\right)}_{jl}^{\kappa +1}-{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2\ge 0.$$

Therefore, the following inequality is valid:

$$-{}_i\omega _{jl}^{k+1}\le 2\Delta t\left|{}_{i}q_j\right|{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2.$$

Considering this fact for ${}_i\varsigma _{jl}^k$ instead of equality, we have the following inequality:

$${}_i\varsigma _{jl}^k\le -{}_i\rho _{aj}\left\{{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2\right\}+2\Delta t\left|{}_{i}q_j\right|{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2.$$

Applying this inequality for ${}_i\varsigma _{jl}^k$ to the expression $\frac{L_i^{\kappa +1}-W_i^{\kappa }}{\Delta t}$, we obtain

$$\begin{array}{l}\frac{L_i^{\kappa +1}-W_i^{\kappa }}{\Delta t}=\Delta y\frac{\Delta x}{\Delta t}{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}\frac{{\mu }_{yi}}{{\left(c_i\right)}_l}{}_{i}e_{l-1}{}_{i}e_{j-1}{}_i\varsigma _{jl}^k}}\\ \le \Delta y{\mu }_{xi}{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_l}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2\right\}}}\\ +2\Delta y\Delta x{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{xi}}{{\left(a_i\right)}_j}\frac{{\mu }_{yi}}{{\left(c_i\right)}_l}\left|{}_{i}q_j\right|{}_{i}e_{l-1}{}_{i}e_{j-1}{}_i\left[{\left(v_i\right)}_j^{\kappa +1}\right]^2}}\\ =\Delta y{\mu }_{xi}{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_l}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2\right\}}}+2\overline{q}{L}_i^{\kappa +1}, i=\overline{1,m}.\end{array}$$

(27)

Inequality (22) follows inequalities (26) and (27), and hence, it is the proof of Lemma 3. □

According to Lemma 3 for each $L_i^{\kappa +1}, \forall i=\overline{1,m}$, we have inequality (22).

By summing $i$ from $1$ to $m$ of the corresponding left and right parts of inequality (22), we obtain:

$$\begin{array}{l}\frac{{}_{1}L^{k+1}-{}_{1}W^k}{\Delta t}\le \\ \{\begin{array}{c}{\displaystyle \sum _{i=1}^m{\mu }_{xi}\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2\right\}, q_{\min }\ge 0; }}\\ {\displaystyle \sum _{i=1}^m{\mu }_{xi}\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2\right\}}+2\overline{q}{\displaystyle \sum _{i=1}^m{L}_i^{\kappa +1}}}, q_{\max }<0;\end{array}\\ i=\overline{1,m}. \end{array}$$

(28)

For convenience, we use the matrix form of inequality (28):

$$\begin{array}{l}\frac{{}_{1}L^{k+1}-{}_{1}W^k}{\Delta t}\le \\ \{\begin{array}{c}\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\left[\left({\mu }_j^{+}{\left(v^I\right)}_{j-1l}^{\kappa +1},{\left(v^I\right)}_{j-1l}^{\kappa +1}\right)-\left({\mu }_j^{+}{\left(v^I\right)}_{jl}^{\kappa +1},{\left(v^I\right)}_{jl}^{\kappa +1}\right)\right]}} , if q_{\min }\ge 0; \\ \Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\left[\left({\mu }_j^{+}{\left(v^I\right)}_{j-1l}^{\kappa +1},{\left(v^I\right)}_{j-1l}^{\kappa +1}\right)-\left({\mu }_j^{+}{\left(v^I\right)}_{jl}^{\kappa +1},{\left(v^I\right)}_{jl}^{\kappa +1}\right)\right]}}2\overline{q}{}_{1}L^{k+1}, if q_{\max }<0;\end{array}\end{array}$$

where

$${\mu }_j^{+}=diag\left({}_{1}e_{j-1}{\mu }_{x1}\frac{{\mu }_{y1}}{{\left(c_1\right)}_j},\dots ,{}_{m}e_{j-1}{\mu }_m\frac{{\mu }_{ym}}{{\left(c_m\right)}_j}\right), j=\overline{1,J-1} .$$

(29)

Similarly, it is easy to obtain the following inequality for ${}_{2}L^k$:

$$\begin{array}{l}\frac{{}_{2}L^{k+1}-{}_{2}L^k}{\Delta t}\le \\ \{\begin{array}{c}{\displaystyle \sum _{i=m+1}^n{\mu }_{xi}\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_j}{}_{i}e_{l-1}{}_{i}e_{j+1}\left\{{\left[{\left(v_i\right)}_{j+1l}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2\right\}, }}} if q_{\min }\ge 0; \\ {\displaystyle \sum _{i=m+1}^n{\mu }_{xi}\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_j}{}_{i}e_{l-1}{}_{i}e_{j+1}\left\{{\left[{\left(v_i\right)}_{j+1l}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2\right\}+2\overline{\overline{q}}{}_{2}L^{k+1}, }}}if q_{\max }<0;\end{array}\end{array}$$

(30)

Here, $\overline{\overline{q}}=\underset{\begin{array}{l}1\le j\le J-1\\ m+1\le i\le n\end{array}}{\max }\left|{}_{i}q_j\right|, q_{\max }=\underset{\begin{array}{l}1\le j\le J-1\\ m+1\le i\le n\end{array}}{\max }\left({}_{i}q_j\right), q_{\min }=\underset{\begin{array}{l}1\le j\le J-1\\ m+1\le i\le n\end{array}}{\min }\left({}_{i}q_j\right).$ let us present inequality (30) in matrix form:

$$\begin{array}{l}\frac{{}_{2}L^{k+1}-{}_{2}L^k}{\Delta t}\le \\ \{\begin{array}{c}\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\left[\left({\mu }_j^{-}{\left(v^{II}\right)}_{j+1l}^{\kappa +1},{\left(v^{II}\right)}_{j+1l}^{\kappa +1}\right)-\left({\mu }_j^{-}{\left(v^{II}\right)}_{jl}^{\kappa +1},{\left(v^{II}\right)}_{jl}^{\kappa +1}\right)\right]}}, if q_{\min }\ge 0; \\ \Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\left[\left({\mu }_j^{-}{\left(v^{II}\right)}_{j+1l}^{\kappa +1},{\left(v^{II}\right)}_{j+1l}^{\kappa +1}\right)-\left({\mu }_j^{-}{\left(v^{II}\right)}_{jl}^{\kappa +1},{\left(v^{II}\right)}_{jl}^{\kappa +1}\right)\right]}}+2\overline{\overline{q}}{}_{2}L^{k+1}, if q_{\max }<0;\end{array}\end{array}$$

where

$${\mu }_j^{-}=diag\left({}_{m+1}e_{j+1}{\mu }_{x m+1}\frac{{\mu }_{y m+1}}{{\left(c_{m+1}\right)}_j},\dots ,{}_{n}e_{j+1}{\mu }_n\frac{{\mu }_{y n}}{{\left(c_n\right)}_j}\right), j=\overline{1,J-1} .$$

(31)

Lemma 4. 

The following equalities are valid.

$$\begin{array}{l} {\displaystyle \sum _{j=1}^{J-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2\right\}=}{}_{i}e_0{\left[{\left(v_i\right)}_{0l}^{\kappa +1}\right]}^2-{}_{i}e_{J-1}{\left[{\left(v_i\right)}_{J-1l}^{\kappa +1}\right]}^2\\ = -\Delta x\frac{\nu }{{\left(a_i\right)}_j}{\displaystyle \sum _{j=1}^{J-1}{}_{i}e_{j-1}{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2}, i=\overline{1,m}.\end{array}$$

$$\begin{array}{l}{\displaystyle \sum _{j=1}^{J-1}{}_{i}e_{j+1}\left\{{\left[{\left(v_i\right)}_{j+1l}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2\right\}= }{}_{i}e_J{\left[{\left(v_i\right)}_{Jl}^{\kappa +1}\right]}^2-{}_{i}e_1{\left[{\left(v_i\right)}_{1l}^{\kappa +1}\right]}^2\\ =-\Delta x\frac{\nu }{{\left(a_i\right)}_{j+1}}{\displaystyle \sum _{j=1}^{J-1}{}_{i}e_{j+1}{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2}, i=\overline{m+1,n}.\end{array}$$

Proof. 

The proof of Lemma 4 is carried out similarly to the proof of Lemma 2. □

Lemma 5. 

Let the conditions of the theorem be fulfilled. Then, the following inequalities are valid.

$$\frac{{}_{1}L^{k+1}-{}_{1}L^k}{\Delta t}\le \{\begin{array}{cc}{B}^I-\nu {}_{1}L^k ,\hfill & if q_{\min }\ge 0;\hfill \\ B^I-\nu {}_{1}L^k+2\overline{q}{}_{1}L^{k+1},\hfill & if q_{\max }<0\hfill \end{array}$$

(32)

and

$$\frac{{}_{2}L^{k+1}-{}_{2}L^k}{\Delta t}\le \{\begin{array}{cc}{B}^{II}-\nu {}_{2}L^{k+1},\hfill & if q_{\min }\ge 0;\hfill \\ B^{II}-\nu {}_{2}L^{k+1}+2\overline{\overline{q}}{}_{2}L^{k+1},& if q_{\max }<0.\hfill \end{array}$$

(33)

Here,

$$B^I=\left({\mu }_1^{+}{\left(v^I\right)}_{0l}^{\kappa +1},{\left(v^I\right)}_{0l}^{\kappa +1}\right)-\left({\mu }_{J-1}^{+}{\left(v^I\right)}_{J-1l}^{\kappa +1},{\left(v^I\right)}_{J-1l}^{\kappa +1}\right),$$

$$B^{II}=\left({\mu }_J^{-}{\left(v^{II}\right)}_{Jl}^{\kappa +1},{\left(v^{II}\right)}_{Jl}^{\kappa +1}\right) -\left({\mu }_1^{-}{\left(v^{II}\right)}_{1l}^{\kappa +1},{\left(v^{II}\right)}_{1l}^{\kappa +1}\right).$$

Proof. 

Let us prove inequality (32). Let be $q_{\min }\ge 0$. Then, taking into account the first equality of Lemma 4 from inequality (28), we have:

$$\begin{array}{l}\frac{{}_{1}L^{k+1}-{}_{1}W^k}{\Delta t}\le {\displaystyle \sum _{i=1}^m{\mu }_{xi}\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_j}}{}_{i}e_{l-1}{}_{i}e_{j-1}\left\{{\left[{\left(v_i\right)}_{j-1l}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2\right\}}}\\ ={\displaystyle \sum _{i=1}^m{\mu }_{xi}\Delta y{\displaystyle \sum _{l=1}^{L-1}{}_{i}e_{l-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_j}\left({}_{i}e_0{\left[{\left(v_i\right)}_{0l}^{\kappa +1}\right]}^2-{}_{i}e_{J-1}{\left[{\left(v_i\right)}_{J-1l}^{\kappa +1}\right]}^2-\Delta x\frac{\nu }{{\left(a_i\right)}_j}{\displaystyle \sum _{j=1}^{J-1}{}_{i}e_{j-1}{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2}\right)}}\\ =B^I-\nu {}_{1}L^{k+1}.\end{array}$$

Similarly, by taking into account the first equality of Lemma 4 from the second part of inequality (28), for $q_{\max }<0$, we have (32).

We move on to the proof of inequality (33). Let $q_{\min }\ge 0$. Then, taking into account the second equality of Lemma 4 from inequality (30), we have:

$$\begin{array}{l}\frac{{}_{2}L^{k+1}-{}_{2}L^k}{\Delta t}\le {\displaystyle \sum _{i=m+1}^n{\mu }_{xi}\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_j}{}_{i}e_{l-1}{}_{i}e_{j+1}\left\{{\left[{\left(v_i\right)}_{j+1l}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2\right\} }}}\\ ={\displaystyle \sum _{i=m+1}^n{\mu }_{xi}\Delta y{\displaystyle \sum _{l=1}^{L-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_j}{}_{i}e_{l-1}\left({}_{i}e_J{\left[{\left(v_i\right)}_{Jl}^{\kappa +1}\right]}^2-{}_{i}e_1{\left[{\left(v_i\right)}_{1l}^{\kappa +1}\right]}^2-\Delta x\frac{\nu }{{\left(a_i\right)}_{j+1}}{\displaystyle \sum _{j=1}^{J-1}{}_{i}e_{j+1}{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2}\right)}}\\ =B^{II}-\nu {}_{2}L^{k+1}. \end{array}$$

Now, let $q_{\max }<0$. Similarly, taking into account the second equality of Lemma 4 from the second part of inequality (30), we have:

$$\begin{array}{l}\frac{{}_{2}L^{k+1}-{}_{2}L^k}{\Delta t}\le {\displaystyle \sum _{i=m+1}^n{\mu }_{xi}\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_j}{}_{i}e_{l-1}{}_{i}e_{j+1}\left\{{\left[{\left(v_i\right)}_{j+1l}^{\kappa +1}\right]}^2-{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2\right\}+2\overline{\overline{q}}{}_{2}L^{k+1} }}}\\ ={\displaystyle \sum _{i=m+1}^n{\mu }_{xi}\Delta y{\displaystyle \sum _{l=1}^{L-1}\frac{{\mu }_{yi}}{{\left(c_i\right)}_j}{}_{i}e_{l-1}\left({}_{i}e_J{\left[{\left(v_i\right)}_{Jl}^{\kappa +1}\right]}^2-{}_{i}e_1{\left[{\left(v_i\right)}_{1l}^{\kappa +1}\right]}^2-\Delta x\frac{\nu }{{\left(a_i\right)}_{j+1}}{\displaystyle \sum _{j=1}^{J-1}{}_{i}e_{j+1}{\left[{\left(v_i\right)}_{jl}^{\kappa +1}\right]}^2}\right)}}\\ +2\overline{\overline{q}}{}_{2}L^{k+1}=B^{II}-\nu {}_{2}L^{k+1}+2\overline{\overline{q}}{}_{2}L^{k+1}.\end{array}$$

Lemma 5 is proved. □

We introduce the following notation:

$$\begin{array}{l}{\nu }^I=\{\begin{array}{c}\nu , if q_{\min }\ge 0,\\ \nu -2\overline{q}, if q_{\max }<0,\end{array} {\nu }^{II}=\{\begin{array}{c}\nu , if q_{\min }\ge 0,\\ \nu -2\overline{\overline{q}}, if q_{\max }<0.\end{array}\\ {\nu }_{\min }=\min \left({\nu }^I,{\nu }^{II}\right).\end{array}$$

We select parameter $\nu$ in such a way that ${\nu }_{\min }>0$. Then, the following lemma is valid:

Lemma 6. 

Let the conditions of the theorem be satisfied. Then, the following inequality holds

$$\frac{L^{\kappa +1}-L^{\kappa }}{\Delta t}<-{\nu }_{\min }{L}^{\kappa +1}+\left(1+\frac{1}{\gamma }\right){\mu }_{\max }\left(\nu \right)\underset{0\le s<k}{\sup }{\displaystyle \sum _{i=1}^n\left({\left|{\left({\psi }_i\right)}^s\right|}^2\right)},\kappa =\overline{0,N-1}$$

(34)

where ${\mu }_{\max }\left(\nu \right)$ is the maximum eigenvalue of the diagonal matrix

$$\mu =\left(\begin{array}{cc}{\mu }_{J-1}^{-}& 0\\ 0& {\mu }_1^{+}\end{array}\right),\mu >0.$$

Here, ${\mu }_1^{+}$ and ${\mu }_{J-1}^{-}$ are defined in (29) and (31), correspondingly.

Proof. 

By applying inequalities for quadratic forms ${}_{1}L^{\kappa }, {}_{2}L^{\kappa }$, we have

$$\begin{array}{c}\frac{L^{\kappa +1}-L^{\kappa }}{\Delta t}=\frac{{}_{1}L^{k+1}+{}_{2}L^{k+1}-{}_{1}L^k-{}_{2}L^k}{\Delta t}=\frac{{}_{1}L^{k+1}-{}_{1}L^k}{\Delta t}+\frac{{}_{2}L^{k+1}-{}_{2}L^k}{\Delta t}\\ \le B^I-{\nu }_{\min }{}_{1}L^{k+1}+B^{II}-{\nu }_{\min }{}_{2}L^{k+1}=-{\nu }_{\min }{L}^{\kappa +1}+W^{\kappa }\left(\nu \right).\end{array}$$

Here,

$$W^{\kappa }\left(\nu \right)=B^I+B^{II}.$$

Obviously, after substituting the values of the parameters $B^I,B^{II}$, the expression $W^{\kappa }\left(\nu \right)$ takes the form

$$W^{\kappa }\left(\nu \right)= \left(\left[\begin{array}{c}{\mu }_1^{+}{\left(v^I\right)}_0^{\kappa +1}\\ {\mu }_{J-1}^{-}{\left(v^{II}\right)}_J^{\kappa +1}\end{array}\right],\left[\begin{array}{c}{\left(v^I\right)}_0^{\kappa +1}\\ {\left(v^{II}\right)}_J^{\kappa +1}\end{array}\right]\right)-\left(\left[\begin{array}{c}{\mu }_{J-1}^{+}{\left(v^I\right)}_{J-1}^{\kappa +1}\\ {\mu }_1^{-}{\left(v^{II}\right)}_1^{\kappa +1}\end{array}\right],\left[\begin{array}{c}{\left(v^I\right)}_{J-1}^{\kappa +1}\\ {\left(v^{II}\right)}_1^{\kappa +1}\end{array}\right]\right)$$

The right part of the expression $W^{\kappa }\left(\nu \right)$ consists of two quadratic forms. By applying boundary conditions (10), we calculate the first quadratic form of the right part $W^{\kappa }\left(\nu \right)$:

$$\begin{array}{l}\left(\left[\begin{array}{c}{\mu }_1^{+}{\left(v^I\right)}_0^{\kappa +1}\\ {\mu }_{J-1}^{-}{\left(v^{II}\right)}_J^{\kappa +1}\end{array}\right],\left[\begin{array}{c}{\left(v^I\right)}_0^{\kappa +1}\\ {\left(v^{II}\right)}_J^{\kappa +1}\end{array}\right]\right)=\left(\left[\begin{array}{c}{\mu }_1^{+}s{\left(v^{\mathrm{II}}\right)}_1^{\kappa +1}\\ {\mu }_{J-1}^{-}r{\left(v^{\mathrm{I}}\right)}_{J-1}^{\kappa +1}\end{array}\right],\left[\begin{array}{c}s{\left(v^{\mathrm{II}}\right)}_1^{\kappa +1}\\ r{\left(v^{\mathrm{I}}\right)}_{J-1}^{\kappa +1}\end{array}\right]\right)\\ +2 \left(\left[\begin{array}{c}{\mu }_1^{+}s{\left(v^{\mathrm{II}}\right)}_1^{\kappa +1}\\ {\mu }_{J-1}^{-}r{\left(v^{\mathrm{I}}\right)}_{J-1}^{\kappa +1}\end{array}\right],\left[\begin{array}{c}{\left({\psi }^{\mathrm{I}}\right)}^{\kappa +1}\\ {\left({\psi }^{\mathrm{II}}\right)}^{\kappa +1}\end{array}\right]\right)+ \left(\left[\begin{array}{c}{\mu }_1^{+}{\left({\psi }^{\mathrm{I}}\right)}^{\kappa +1}\\ {\mu }_{J-1}^{-}{\left({\psi }^{\mathrm{II}}\right)}^{\kappa +1}\end{array}\right],\left[\begin{array}{c}{\left({\psi }^{\mathrm{I}}\right)}^{\kappa +1}\\ {\left({\psi }^{\mathrm{II}}\right)}^{\kappa +1}\end{array}\right]\right)\end{array}$$

Taking into account the last transformation, the expression $W^{\kappa }\left(\nu \right)$ reduces to the form:

$$\begin{array}{l} W^{\kappa }\left(\nu \right)= \left(\left[\begin{array}{c}{\mu }_1^{+}s{\left(v^{\mathrm{II}}\right)}_1^{\kappa +1}\\ {\mu }_{J-1}^{-}r{\left(v^{\mathrm{I}}\right)}_{J-1}^{\kappa +1}\end{array}\right],\left[\begin{array}{c}s{\left(v^{\mathrm{II}}\right)}_1^{\kappa +1}\\ r{\left(v^{\mathrm{I}}\right)}_{J-1}^{\kappa +1}\end{array}\right]\right)-\left(\left[\begin{array}{c}{\mu }_{J-1}^{+}{\left(v^I\right)}_{J-1}^{\kappa +1}\\ {\mu }_1^{-}{\left(v^{II}\right)}_1^{\kappa +1}\end{array}\right],\left[\begin{array}{c}{\left(v^I\right)}_{J-1}^{\kappa +1}\\ {\left(v^{II}\right)}_1^{\kappa +1}\end{array}\right]\right)\\ +2\left(\left[\begin{array}{c}{\mu }_1^{+}s{\left(v^{\mathrm{II}}\right)}_1^{\kappa +1}\\ {\mu }_{J-1}^{-}r{\left(v^{\mathrm{I}}\right)}_{J-1}^{\kappa +1}\end{array}\right],\left[\begin{array}{c}{\left({\psi }^{\mathrm{I}}\right)}^{\kappa +1}\\ {\left({\psi }^{\mathrm{II}}\right)}^{\kappa +1}\end{array}\right]\right)+ \left(\left[\begin{array}{c}{\mu }_1^{+}{\left({\psi }^{\mathrm{I}}\right)}^{\kappa +1}\\ {\mu }_{J-1}^{-}{\left({\psi }^{\mathrm{II}}\right)}^{\kappa +1}\end{array}\right],\left[\begin{array}{c}{\left({\psi }^{\mathrm{I}}\right)}^{\kappa +1}\\ {\left({\psi }^{\mathrm{II}}\right)}^{\kappa +1}\end{array}\right]\right).\end{array}$$

Next, we use the inequality $2pq\le \gamma p^2+\frac{1}{\gamma }{q}^2$, which is true for any $p,q$ and $\gamma >0$.

$$\begin{array}{l}2\left(\left[\begin{array}{c}{\mu }_1^{+}s{\left(v^{\mathrm{II}}\right)}_1^{\kappa +1}\\ {\mu }_{J-1}^{-}r{\left(v^{\mathrm{I}}\right)}_{J-1}^{\kappa +1}\end{array}\right],\left[\begin{array}{c}{\left({\psi }^{\mathrm{I}}\right)}^{\kappa +1}\\ {\left({\psi }^{\mathrm{II}}\right)}^{\kappa +1}\end{array}\right]\right)\le \gamma \left(\left[\begin{array}{c}{\mu }_1^{+}s{\left(v^{\mathrm{II}}\right)}_1^{\kappa +1}\\ {\mu }_{J-1}^{-}r{\left(v^{\mathrm{I}}\right)}_{J-1}^{\kappa +1}\end{array}\right],\left[\begin{array}{c}s{\left(v^{\mathrm{II}}\right)}_1^{\kappa +1}\\ r{\left(v^{\mathrm{I}}\right)}_{J-1}^{\kappa +1}\end{array}\right]\right)\\ +\frac{1}{\gamma }\left(\left[\begin{array}{c}{\mu }_1^{+}{\left({\psi }^{\mathrm{I}}\right)}^{\kappa +1}\\ {\mu }_{J-1}^{-}{\left({\psi }^{\mathrm{II}}\right)}^{\kappa +1}\end{array}\right],\left[\begin{array}{c}{\left({\psi }^{\mathrm{I}}\right)}^{\kappa +1}\\ {\left({\psi }^{\mathrm{II}}\right)}^{\kappa +1}\end{array}\right]\right).\end{array}$$

Then, for the expression $W^{\kappa }\left(\nu \right)$, we obtain the following estimate from above

$$\begin{array}{l} W^{\kappa }\left(\nu \right)= \left(1+\gamma \right) \left(\left[\begin{array}{c}{\mu }_1^{+}s{\left(v^{\mathrm{II}}\right)}_1^{\kappa +1}\\ {\mu }_{J-1}^{-}r{\left(v^{\mathrm{I}}\right)}_{J-1}^{\kappa +1}\end{array}\right],\left[\begin{array}{c}s{\left(v^{\mathrm{II}}\right)}_1^{\kappa +1}\\ r{\left(v^{\mathrm{I}}\right)}_{J-1}^{\kappa +1}\end{array}\right]\right)-\left(\left[\begin{array}{c}{\mu }_{J-1}^{+}{\left(v^I\right)}_{J-1}^{\kappa +1}\\ {\mu }_1^{-}{\left(v^{II}\right)}_1^{\kappa +1}\end{array}\right],\left[\begin{array}{c}{\left(v^I\right)}_{J-1}^{\kappa +1}\\ {\left(v^{II}\right)}_1^{\kappa +1}\end{array}\right]\right)\\ +\left(1+\frac{1}{\gamma }\right)\left(\left[\begin{array}{c}{\mu }_1^{+}{\left({\psi }^{\mathrm{I}}\right)}^{\kappa +1}\\ {\mu }_{J-1}^{-}{\left({\psi }^{\mathrm{II}}\right)}^{\kappa +1}\end{array}\right],\left[\begin{array}{c}{\left({\psi }^{\mathrm{I}}\right)}^{\kappa +1}\\ {\left({\psi }^{\mathrm{II}}\right)}^{\kappa +1}\end{array}\right]\right).\end{array}$$

(35)

For the third quadratic form of the right part (35), the following estimate is valid:

$$\left(\left[\begin{array}{c}{\mu }_1^{+}{\left({\psi }^{\mathrm{I}}\right)}^{\kappa +1}\\ {\mu }_{J-1}^{-}{\left({\psi }^{\mathrm{II}}\right)}^{\kappa +1}\end{array}\right],\left[\begin{array}{c}{\left({\psi }^{\mathrm{I}}\right)}^{\kappa +1}\\ {\left({\psi }^{\mathrm{II}}\right)}^{\kappa +1}\end{array}\right]\right)\le {\mu }_{\max }\left(\nu \right)\underset{0\le s<k}{\sup }{\displaystyle \sum _{i=1}^n\left({\left|{\left({\psi }_i\right)}^s\right|}^2\right)}.$$

(36)

Here, as noted above, we denote the maximum eigenvalue of the matrix $\mu$ by ${\mu }_{\max }\left(\nu \right)$ (see formulation of Lemma 6).

According to the assumption of the theorem, the dissipative condition of the boundary conditions ${\rho }_2(R)<1$ is fulfilled. Therefore (see the definition of dissipativity [2]), there are strictly positive defined diagonal matrices $D_0, D_1$ with dimensions $m$ and $n-m$, respectively, such that

$$\Vert \left(1+\gamma \right)\Delta R{\Delta }^{-1}\Vert <1, гд\mathrm{e} \Delta \triangleq \mathrm{diag}\text{{}{D}_0, D_1\}$$

In (35), by choosing parameters ${\mu }_1^{+}, {\mu }_1^{-}$ such that ${\mu }_1^{+}=D_0^2$, ${\mu }_1^{-}=D_1^2$ and taking into account (36) for the vectors $v_j^{\kappa }$, satisfying boundary conditions (10), we have

$$\begin{array}{cc}\hfill W^{\kappa }\left(\nu \right)& \le -\left(\left[\begin{array}{c}{\mu }_{J-1}^{+}{D}_0^{-2}{D}_0{\left(v^I\right)}_{J-1}^{\kappa +1}\\ D_1{\left(v^{II}\right)}_1^{\kappa +1}\end{array}\right],\left[\begin{array}{c}{D}_0{\left(v^I\right)}_{J-1}^{\kappa +1}\\ D_1{\left(v^{II}\right)}_1^{\kappa +1}\end{array}\right]\right)\hfill \\ & +\left(1+\gamma \right) \left(\left[\begin{array}{c}{D}_0^{2}s{D}_1^{-1}{D}_1{\left(v^{II}\right)}_1^{\kappa +1}\\ {\mu }_{J-1}^{-}r{D}_0^{-1}{D}_0{\left(v^I\right)}_{J-1}^{\kappa +1}\end{array}\right],\left[\begin{array}{c}s{D}_1^{-1}{D}_1{\left(v^{II}\right)}_1^{\kappa +1}\\ r{D}_0^{-1}{D}_0{\left(v^I\right)}_{J-1}^{\kappa +1}\end{array}\right]\right)\hfill \\ & +\left(1+\frac{1}{\gamma }\right){\mu }_{\max }\left(\nu \right)\underset{0\le s<k}{\sup }{\displaystyle \sum _{i=1}^n\left({\left|{\left({\psi }_i\right)}^s\right|}^2\right)}=W_1{}^k\left(\nu \right)+W_2{}^k\left(\nu \right).\hfill \end{array}$$

(37)

Here,

$$W_1{}^k\left(\nu \right)=-\left(\Omega (\nu )\left(\begin{array}{c}{D}_0{\left(v^I\right)}_{J-1}^{\kappa +1}\\ D_1{\left(v^{II}\right)}_1^{\kappa +1}\end{array}\right),\left(\begin{array}{c}{D}_0{\left(v^I\right)}_{J-1}^{\kappa +1}\\ D_1{\left(v^{II}\right)}_1^{\kappa +1}\end{array}\right)\right),$$

$$W_2{}^k\left(\nu \right)=\left(1+\frac{1}{\gamma }\right){\mu }_{\max }\left(\nu \right)\underset{0\le s<k}{\sup }{\displaystyle \sum _{i=1}^n\left({\left|{\left({\psi }_i\right)}^s\right|}^2\right)},$$

$$\begin{array}{cc}\hfill \Omega (\nu )& =\left(\begin{array}{cc}{\mu }_{J-1}^{+}{D}_0^{-2}& 0\\ 0& E\end{array}\right)\hfill \\ & -\left(1+\gamma \right){\left(\begin{array}{cc}0& D_{0}s{D}_1^{-1}\\ D_{1}r{D}_0^{-1}& 0\end{array}\right)}^T \left(\begin{array}{cc}0& D_{0}s{D}_1^{-1}\\ {\mu }_{J-1}^{-}{D}_1^{-1}r{D}_0^{-1}& 0\end{array}\right)\end{array}$$

Consider the expression of the quadratic form $W_1^k\left(\nu \right)$ for $v=0$:

$$\begin{array}{ll}\hfill W_1{}^k\left(0\right)& =-\left(\Omega (0)\left(\begin{array}{c}{D}_0{\left(v^I\right)}_{J-1}^{\kappa +1}\\ D_1{\left(v^{II}\right)}_1^{\kappa +1}\end{array}\right),\left(\begin{array}{c}{D}_0{\left(v^I\right)}_{J-1}^{\kappa +1}\\ D_1{\left(v^{II}\right)}_1^{\kappa +1}\end{array}\right)\right)\hfill \\ & =-\left(\left(I-\left(1+\gamma \right){(\Delta R{\Delta }^{-1})}^T(\Delta R{\Delta }^{-1})\right)\left(\begin{array}{c}{D}_0{\left(v^I\right)}_{J-1}^{\kappa +1}\\ D_1{\left(v^{II}\right)}_1^{\kappa +1}\end{array}\right),\left(\begin{array}{c}{D}_0{\left(v^I\right)}_{J-1}^{\kappa +1}\\ D_1{\left(v^{II}\right)}_1^{\kappa +1}\end{array}\right)\right) .\hfill \end{array}$$

Since $\Vert \left(1+\gamma \right)\Delta R{\Delta }^{-1}\Vert <1$, then $W_1{}^k\left(0\right)$ is a strictly negatively defined quadratic form with respect to ${\left(v^I\right)}_{J-1}^{\kappa +1}$ and ${\left(v^{II}\right)}_1^{\kappa +1}$. Then, continuity $W_1{}^k\left(v\right)$ remains a strictly negatively defined quadratic form with a sufficiently small $v>0$. Therefore, for any $k$ in the solutions for the system (7)–(11), we have the following chains of inequalities:

$$\frac{L^{\kappa +1}-L^{\kappa }}{\Delta t}\le -{\nu }_{\min }{L}^{\kappa +1}+W^{\kappa }\left(\nu \right)\le -{\nu }_{\min }{L}^{\kappa +1}+W_1{}^k\left(\nu \right)+W_2{}^k\left(\nu \right)\le -{\nu }_{\min }{L}^{\kappa +1}+W_2{}^k\left(\nu \right) ,$$

Hence, by virtue of (37), the validity of estimate (34) is easily determined. Lemma 6 is proved. □

We present inequality (34) in the form

$$L^{\kappa +1}\le \alpha L^{\kappa }+\beta ,$$

(38)

where

$$\alpha =1+\overline{\alpha }, \overline{\alpha }=\frac{-\Delta t\cdot {\nu }_{\min }}{1+\Delta t\cdot {\nu }_{\min }}, \beta =\frac{\Delta t}{1+\Delta t\cdot {\nu }_{\min }}\left(1+\frac{1}{\gamma }\right){\mu }_{\max }\left(\nu \right)\underset{0\le s<k}{\sup }{\displaystyle \sum _{i=1}^n\left({\left|{\left({\psi }_i\right)}^s\right|}^2\right)}.$$

Note that $\overline{\alpha }<0$. Let a sequence of real numbers satisfy inequality (38) for $\alpha >0$ and $\beta >0$. Then, according to Lemma 5 of [16] (a discrete version of Gronwall’s lemma), the following inequality is true:

$$L^{\kappa }\le {\alpha }^k{L}^0+\beta {\displaystyle \sum _{i=1}^k{\alpha }^{k-i}}.$$

(39)

It should be noted that with a sufficiently small $\Delta t$, the following inequality is obtained $0<\alpha <1$. Then, according to the formula of the partial sum of terms of infinite geometric progression with denominator $\left|\alpha \right|<1$, the following equality is fair

$$\beta {\displaystyle \sum _{i=1}^k{\alpha }^{k-i}}=\beta +\beta {\alpha }^1+\beta {\alpha }^2+\cdots +\beta {\alpha }^{k-1}=\beta \frac{1-{\alpha }^k}{1-\alpha }.$$

Taking into account this equality from (39), we obtain the following inequality

$$L^{\kappa }\le {\alpha }^k\left(L^0-\overline{\beta }\right)+\overline{\beta }.$$

Here,

$$\overline{\beta }=\frac{\beta }{\overline{\alpha }}=\frac{\left(1+\frac{1}{\gamma }\right){\mu }_{\max }\left(\nu \right)\underset{0\le s<k}{\sup }{\displaystyle \sum _{i=1}^n\left({\left|{\left({\psi }_i\right)}^s\right|}^2\right)}}{{\nu }_{\min }}.$$

Consider matrix

$$\begin{array}{l} {\mu }_j=\left(\begin{array}{cc}{\mu }_j^{+}& 0\\ 0& {\mu }_j^{-}\end{array}\right), K_j=K\left(x_j\right), {\left|K\right|}_j= \left(\begin{array}{cc}{K}_j^{+}& 0\\ 0& K_j^{-}\end{array}\right), \\ K_j^{\pm }=K^{\pm }\left(x_j\right), {\left|K\right|}_j^{-1}=diag\left(\frac{1}{a_1\left(x_j\right)},\cdots ,\frac{1}{a_n\left(x_j\right)}\right), j=\overline{1,J-1} .\end{array}$$

and positive numbers $C_1$ and $C_2$, which are determined using the formula:

$$C_1=\underset{\begin{array}{l}1\le i\le n\\ 1\le j\le J-1\end{array}}{\min }\left\{{\lambda }_{ij}:\left|{\left|K\right|}_j^{-1}{\mu }_j-{\lambda }_{ij}E\right|=0\right\} ,C_2=\underset{\begin{array}{l}1\le i\le n\\ 1\le j\le J-1\end{array}}{\max }\left\{{\lambda }_{ij}:\left|{\left|K\right|}_j^{-1}{\mu }_j-{\lambda }_{ij}E\right|=0\right\},$$

where $E$ is an identity matrix. Then, due to the imposed conditions on the data of the problem, we have:

$$C_{1}E\le {\left|K\right|}_j^{-1}{\mu }_j\le C_{2}E, j=\overline{1,J-1}.$$

Hence, it follows from (40) that

$$C_1\Delta x\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\left(v_{jl}^{\kappa },v_{jl}^{\kappa }\right)}}\le L^{\kappa }; L^0\le C_2\Delta x\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\left({\left(v_0\right)}_{jl},{\left(v_0\right)}_{jl}\right)}}.$$

Therefore,

$${\Vert v^k\Vert }_{l^2}^2\le {\alpha }^k\left(C{\Vert v^0\Vert }^2-\frac{\overline{\beta }}{C_1}\right)+\frac{\overline{\beta }}{C_1},$$

where $C=\frac{C_2}{C_1}, {\Vert v^k\Vert }_{l^2}^2\triangleq \Delta x\Delta y{\displaystyle \sum _{l=1}^{L-1}{\displaystyle \sum _{j=1}^{J-1}\left(v_{jl}^{\kappa },v_{jl}^{\kappa }\right)}}$.

Thus, the numerical solution $v_{jl}^{\kappa }$ of this mixed problem is stable according to Lyapunov in the $l^2$ norm.

Therefore, if ${\rho }_2(R)<1$, then the solution of difference problems (7)–(11) is Lyapunov-stable. The theorem is proved.

4. Discussion

The primary focus of this work is to reduce the complex initial-boundary difference problem to the sequential solution of simpler structured initial-boundary difference problems. This approach facilitates the development of simple, flexible and cost-efficient difference schemes:

• To solve the mixed problem for a linear two-dimensional symmetric hyperbolic system with dissipative boundary conditions incorporating control functions, a new explicit–implicit direction splitting method has been introduced.
• Leveraging the Lyapunov stability theory, we have proven a theorem on the exponential stability of the numerical solution of the upwind difference scheme split across spatial variables. Additionally, a discrete Lyapunov function for the numerical solution of the initial-boundary difference problem has been constructed. We derived sufficient algebraic conditions for the exponential stability of the numerical solution.
• We obtained an a priori estimate for the numerical solution of the initial-boundary difference problem, enabling us to gauge the numerical solution using initial data functions and control functions in boundary conditions. This estimation confirms the continuous dependence of the numerical solution on control functions in boundary conditions, thus allowing control over the hyperbolic system.

Given the primarily theoretical nature of this research, our main goal was to pinpoint sufficient stability conditions for the proposed upwind explicit–implicit scheme. In upcoming studies, we aim to apply the devised numerical scheme to more practical scenarios.