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].
3. Results
Using
, we denote the set of real matrices in dimension
. In order to formulate the stability theorem, we first introduce function
, which is similar to that in [
2] (see page 86, Formula (3.6)); it is defined by
where
denotes the set of diagonal
real matrices with strictly positive diagonal elements,
is strictly positive number and
for
for .
Also the following composite matrix consisting of boundary matrices
is denoted by
:
And now, we formulate the following basic stability theorem.
Theorem 1. Let and the discrete Lyapunov function is defined using Formula (12). If matrices and of the parameters of the boundary conditions (10) obey the inequality , (dissipative conditions of boundary conditions), then numerical solution for difference boundary value problems (7)–(11) is exponentially stable in the 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:
Lemma 1. Let conditions (13) be fulfilled. Then, for the solution of difference scheme (7), the following inequality is valid: Proof. Denote the Courant number by
Taking into account notation (16), difference scheme (7) can be written as:
Taking into account the form of record (17) of difference scheme (7), we obtain the following expression for
:
In the right part of the expression for
, by substituting values
according to Formula (17), we have
From the last part, taking into account (16), we have
. Hence,
. Using an obvious inequality
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.
Proof. Using the difference differentiation formula, we obtain
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:
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
:
Lemma 2 is proved. □
From Lemma 2, taking into account equality (19), we obtain
or
By doing the same again, we can obtain the following inequalities in the cases
:
Summing up the left and right sides of inequalities (20) and (21), respectively, we obtain
Lemma 3. For any solution of difference Equation (8), the following inequality holdsHere, Proof. Denote the Courant number
by
for the first
m difference equations of scheme (7). Then, the first
m difference equations of scheme (7) will take the following form:
By multiplying both parts of Equation (23) by
, we have:
We separately transform each of the difference Equation (24) terms as follows:
Note that these equalities are proved using direct verification. Taking into account these transformations, (24) can be represented as follows:
The last identity is represented in the following form:
where
Consider the following possible cases:
- (1)
Case I:
- (2)
Case II:
Note, that in Case I,
. Taking into account in the first case from (25) for
, we have
By applying this inequality for
to the expression
, we obtain:
In Case II,
where
Therefore, the following inequality is valid:
Considering this fact for
instead of equality, we have the following inequality:
Applying this inequality for
to the expression
, we obtain
Inequality (22) follows inequalities (26) and (27), and hence, it is the proof of Lemma 3. □
According to Lemma 3 for each , we have inequality (22).
By summing
from
to
of the corresponding left and right parts of inequality (22), we obtain:
For convenience, we use the matrix form of inequality (28):
where
Similarly, it is easy to obtain the following inequality for
:
Here,
let us present inequality (30) in matrix form:
where
Lemma 4. The following equalities are valid.
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.andHere, Proof. Let us prove inequality (32). Let be
. Then, taking into account the first equality of Lemma 4 from inequality (28), we have:
Similarly, by taking into account the first equality of Lemma 4 from the second part of inequality (28), for
, we have (32).
We move on to the proof of inequality (33). Let
. Then, taking into account the second equality of Lemma 4 from inequality (30), we have:
Now, let
. Similarly, taking into account the second equality of Lemma 4 from the second part of inequality (30), we have:
Lemma 5 is proved. □
We introduce the following notation:
We select parameter in such a way that . Then, the following lemma is valid:
Lemma 6. Let the conditions of the theorem be satisfied. Then, the following inequality holds
where
is the maximum eigenvalue of the diagonal matrix
Here, and are defined in (29) and (31), correspondingly.
Proof. By applying inequalities for quadratic forms
, we have
Here,
Obviously, after substituting the values of the parameters
, the expression
takes the form
The right part of the expression
consists of two quadratic forms. By applying boundary conditions (10), we calculate the first quadratic form of the right part
:
Taking into account the last transformation, the expression
reduces to the form:
Next, we use the inequality
, which is true for any
and
.
Then, for the expression
, we obtain the following estimate from above
For the third quadratic form of the right part (35), the following estimate is valid:
Here, as noted above, we denote the maximum eigenvalue of the matrix
by
(see formulation of Lemma 6).
According to the assumption of the theorem, the dissipative condition of the boundary conditions
is fulfilled. Therefore (see the definition of dissipativity [
2]), there are strictly positive defined diagonal matrices
with dimensions
and
, respectively, such that
In (35), by choosing parameters
such that
,
and taking into account (36) for the vectors
, satisfying boundary conditions (10), we have
Here,
Consider the expression of the quadratic form
for
:
Since
, then
is a strictly negatively defined quadratic form with respect to
and
. Then, continuity
remains a strictly negatively defined quadratic form with a sufficiently small
. Therefore, for any
in the solutions for the system (7)–(11), we have the following chains of inequalities:
Hence, by virtue of (37), the validity of estimate (34) is easily determined. Lemma 6 is proved. □
We present inequality (34) in the form
where
Note that
. Let a sequence of real numbers satisfy inequality (38) for
and
. Then, according to Lemma 5 of [
16] (a discrete version of Gronwall’s lemma), the following inequality is true:
It should be noted that with a sufficiently small
, the following inequality is obtained
. Then, according to the formula of the partial sum of terms of infinite geometric progression with denominator
, the following equality is fair
Taking into account this equality from (39), we obtain the following inequality
Here,
Consider matrix
and positive numbers
and
, which are determined using the formula:
where
is an identity matrix. Then, due to the imposed conditions on the data of the problem, we have:
Hence, it follows from (40) that
Therefore,
where
.
Thus, the numerical solution of this mixed problem is stable according to Lyapunov in the norm.
Therefore, if , then the solution of difference problems (7)–(11) is Lyapunov-stable. The theorem is proved.