Global existence and stability of temporal periodic solution to non-isentropic compressible Euler equations with a source term

1.   Introduction

In mathematics and various sciences, the changes of output and input of nonlinear systems are out of proportion. Most of the systems involved in life are essentially nonlinear, so solving nonlinear problems has attracted various scientists. Scholars have proposed more efficient iterative methods for solving nonlinear systems. One of the most famous iterative methods for solving nonlinear systems is Newton's method ^[1],

for $k = 0, 1, 2, ...,$ $x_{0}$ is the starting point. The Newton's method is second order convergent and effective in solving some nonlinear systems.

With the advancement of computers and numerical algebra, scholars have developed many iterative methods based on Newton's method that are more efficient than second-order Newton's method for solving nonlinear problems ^{[2,3,4,5,6,7]}. In addition, when the Jacobian matrix cannot be calculated for nonlinear systems, some effective derivative free methods can also solve nonlinear systems well (see ^{[8,9,10,11,12]}). We propose an eighth order iterative method with high computational efficiency, which is suitable for solving large systems of equations ^[13]. The specific iteration format is as follows

where $M^{(k)} = \Gamma_{k}(F^{'}(x^{(k)})-F^{'}(y^{(k)}))$, and $\Gamma_{k} = [F^{'}(x^{(k)})]^{-1}$.

The theoretical results of local convergence and semilocal convergence of the iterative method are also important in the study. Local convergence requires the existence of the assumed solution and the initial value is close enough to the solution. Semilocal convergence does not require the existence of an assumed solution, but the selection of initial values also needs to meet certain conditions (see ^{[14,15,16,17,18]}). Therefore, for some systems that cannot be analyzed and solved, the results of semilocal convergence cannot only prove the convergence of iterative sequences, but also prove the existence of solutions of these systems, so as to obtain the existence domain and uniqueness domain of system solutions; for further study (see ^{[19,20,21,22]}). Based on this, we perform a semilocal convergence analysis on the method (1.2) .

This paper consists of five sections. In Section $2$ of the paper, the recurrence relation is explained. The semilocal convergence of the iterative method (1.2) is proved in Section $3$. In Section $4$, the numerical experiments of two nonlinear systems are completed. Finally, the conclusion of this paper is made.

2.   Recurrence relations

In this section, let $X$ and $Y$ be Banach spaces and let $F : \Omega \subseteq X \rightarrow Y$ be a twice differetiable nonlinear Fréchet operator in an open $\Omega$ ^[23]. Let us assume that the inverse of the Jacobian matrix of the system in the iteration (1.2) is $\Gamma_{0} \in \mathcal{L}(Y, X)$, which is the set of linear operation from $Y$ to $X$.

Moreover, in order to obtain the semilocal convergence result for this iterative method (1.2) , Kantorovich conditions are assumed:

$(M_{1}) \quad \Vert \Gamma_{0} \Vert \leq \beta$,

$(M_{2}) \quad \Vert \Gamma_{0}F(x_{0}) \Vert \leq \eta$,

$(M_{3}) \quad \Vert F^{'}(x)-F^{'}(y) \Vert \leq K \Vert x-y \Vert$,

where $K$, $\beta$, $\eta$ are non-negative real numbers. For the sake of simplicity, we denote $a_{0} = K \beta \eta$ and define the sequence

where we use the following auxiliary functions

and

These functions play a key role in the analysis that will be performed next.

Preliminary results. In order to get the difference of the first two elements in the iterative method (1.2) , we have

The Taylor series expansion of $F$ around $x_{0}$ evaluated in $y_{0}$ is

where the term $F(x_{0})+F^{'}(x_{0})(y_{0}-x_{0})$ is equal to zero, since it comes from a Newton's step. With the change $x = x_{0}+t(y_{0}-x_{0})$, we get

Then,

Taking norms and applying Lipschitz condition, we get

so that,

Using a method similar to (2.5), we get $w_{0}-y_{0}$

So,

Next, the next step analysis

Using Taylor's expansion of $F(w_{0})$ around $x_{0}$ and applying Lipschitz condition, we obtain

By applying Banach's lemma, one has

where

Then, as far as $a_{0}(h(a_{0})) < 1$ (by taking $a_{0} < 0.45807$), Banach's lemma guarantees that

exists and

so

Based on the above analysis, we can obtain the following theorem.

Theorem 1. For $k \ge 1$, the following conditions are valid:

$(O1_{k}) \quad \Vert \Gamma_{k} \Vert \leq f(a_{k-1}) \Vert \Gamma_{k-1} \Vert$,

$(O2_{k}) \quad \Vert y_{k}-x_{k} \Vert = \Vert \Gamma_{k} F(x_{k}) \Vert \leq f(a_{k-1})g(a_{k-1}) \Vert y_{k-1}-x_{k-1} \Vert$,

$(O3_{k}) \quad K \, \Vert \Gamma_{k} \Vert \, \Vert y_{k}-x_{k} \Vert \leq a_{k}$,

$(O4_{k}) \quad \Vert x_{k}-x_{k-1} \Vert \leq h(a_{k-1}) \Vert y_{k-1}-x_{k-1} \Vert$.

Proof. The above theorem is proven through induction. Starting with $k = 1$, (2.16) proved the $(O1_{1})$.

$(O2_{1})$: The Taylor's expansion of $F(x_{1})$ around $y_{0}$, we can get

So, we must to have $x_{1}-y_{0}$

And bounding its norm, the following inequality is obtained

Then, using (2.17)–(2.19), the $\Vert F(x_{1}) \Vert$ is bounded

Therefore, by applying $(O1_{1})$, we get

That is,

where,

$(O3_{1})$: Using $(O1_{1})$ and $(O2_{1})$,

$(O4_{1})$: For $k = 1$ it has been proven in (2.16).

The proof of $(O1_{k+1})$, $(O2_{k+1})$, $(O3_{k+1})$ and $(O4_{k+1})$ is based on the same method of proving that the inductive assumption with $(O1_{k})$, $(O2_{k})$, $(O3_{k})$ and $(O4_{k})$ as $k \ge 1$ holds true.

3.   Semilocal convergence analysis

According to the convergence property of ${x_{k}}$ sequence in Banach space, we need to prove that this sequence is a Cauchy sequence. Based on the auxiliary function, we can obtain the following results.

Lemma 1. According $h(x), f(x)$ and $g(x)$, we have:

$i$. $f(x)$ is inceasing and $f(x) > 1$ for $x\in (0, 0.45807)$,

$ii$. $h(x)$ and $g(x)$ are increasing for $x\in (0, 0.45807)$.

The above lemma can be calculated from the Section 2, and the process is omitted.

Lemma 2. The $f(x)$ and $g(x)$ defined by (2.3) and (2.4). Then

$i$. $f(a_{0})g(a_{0}) < 1$ for $a_{0} < 0.252232$,

$ii$. $f(a_{0})^{2}g(a_{0}) < 1$ for $a_{0} < 0.21715$,

$iii$. the sequence ${a_{k}}$ is decreasing and $a_{k} < 0.21715$ for $k > 0$.

Proof. It is straightforword that $i$, $ii$ are satisfied. As $f(a_{0})^{2}g(a_{0}) < 1$, then by construction of $a_{k}$, it is a dereasing sequence. So $a_{k} < a_{0} \leq 0.21715$, for all $k \ge 1$.

Theorem 2. Let X, Y be Banach spaces and $F:\Omega \subseteq X \rightarrow Y$ be a nonlinear twice differentiable Fréchet operator in an open set domain $\Omega$. Assume that $\Gamma_{0} = [F^{'}(x_{0})]^{-1}$ exists in $x_{0} \in \Omega$ and meet the conditions of $(M_{1})-(M_{3})$. Let be $a_{0} = K\beta\eta$, and assume that $a_{0} < 0.21$. The sequence $\{x_{k}\}$ defined in $(2.1)$ and starting in $x_{0}$ converges to the solution $x^{*}$ of $F(x) = 0$, if $B_{e}(x_{0}, R\eta) = {x\in X: \Vert x-x_{0} \Vert < R \eta} \subset \Omega$ where $R = \frac{h(a_{0})}{1-f(a_{0})g(a_{0})}$. In the case, the iterates $\{x_{k}\}$ and $\{y_{k}\}$ are contained in $B_{e}(x_{0}, R\eta)$ and $x^{*} \in B_{e}(x_{0}, R\eta)$. In addition, the $x^{*}$ is the only solution of equation $F(x) = 0$ in $B_{n}(x_{0}, \frac{2}{K\beta}-R\eta) \cap \Omega$.

Proof. By recursively applying $(O4_{k})$, we can write

Then,

As $h(x)$ is increasing and $a_{k}$ dreasing, it can be stated that

Moreover, according Lemmas 1 and 2, by using the expression for the partial sum of a geometrical series,

So, the Cauchy sequence if and only if $f(a_{0})g(a_{0}) < 1$ (Lemma $2$).

For $k = 0$,

when $m\rightarrow \infty$, we get the radius od convergence $R\eta = \frac{h(a_{0})}{1-f(a_{0})g(a_{0})}\eta$.

Let's prove that $x^{*}$ is the solution of $F(x) = 0$ starting from the boundary of $\Vert F^{'} (x_{n}) \Vert$,

Then, acorrding $M_{2}$ and (3.1)

as $h(x)$, $f(x)$ and $g(x)$ are increasing and $a_{k}$ is the decreasing sequence,

Taking into account that $\Vert F^{'}(x_{k}) \Vert$ is bounded and $(f(a_{0})g(a_{0}))^{k}$ tends to zero when $k \rightarrow \infty$, we conclude that $\Vert F(x_{k}) \Vert \rightarrow 0$. As $F$ is continuous in $\Omega$, then $F(x^{*}) = 0$.

Finally, the uniqueness of $x^{*}$ in $B(x_{0}, \frac{2}{K\beta}-R\eta) \cap \Omega$.

In order to guarantee that $y^{*}-x^{*} = 0$ it is necesssary to prove that operator $\int_{0}^{1}F^{'}(x^{*}+t(y^{*}-x^{*}))dt$ is invertible. Applying hypothesis $(M_{3})$,

By the Banach lemma, the intergal operator is invertible and hence $y^{*} = x^{*}$.

4.   Numerical results

In this section, we provide some numerical examples to illustrate the theoretical results introduced earlier.

Example 1. Hammerstein equation is a kind of important nonlinear integral equation ^[24], which is given as follows:

where $x\in \mathbb{C}[0, 1], s, t\in [0, 1]$, with the kernel $N$ is

To solve (4.1) we transform it into a syste of nonlinear equations through a discretization process. We approximate the integral appearing in Eq (4.1) by using Gauss-Legendre quadrature,

being $t_{j}$ and $w_{j}$ the nodes and the weights of the Gauss-Legendre polynomial. Denoting the approximation of $x(t_{j})$ as $x_{i}, i = 1, ..., 7$, then we estimate (4.1) with the nonlinear system of equations

where

So, the system can be rewritten as

where $F$ if a nonlinear operator in the Banach space $\mathbb{R}^{L}$, and $F^{'}$ is its Fréchet derivative in $\mathcal{L}(\mathbb{R}^{L}, \mathbb{R}^{L})$.

According the method (1.2), we will use it to solve the nonlinear systems.

Taking $x_{0} = (1.8, 1.8, ..., 1.8)^{T}, L = 7$ and the infinity norm, we get

The above results satisfy the semilocal convergence condition, so this method can be applied to the system. Thus, we guarantee the existence of the solution in $B_{e}(x_{0}, 0.5646)$, and the uniqueness in $B_{n}(x_{0}, 22.4874)$. Table 1 shows the the radius of the existence domain and the radius of the unique domain under different initial values. For $x_{0i} > 1.87, i = 1, 2, ..., 7$, convergence conditions are not satisfied and, therefore, the convergence is not guaranteed.

Using the iterative method (1.2) to solve (4.2), the exact solution is

Example 2. Let $\mathcal{X} = \mathcal{Y} = \mathbb{R}^{2}$ be equipped with the max-norm. Choose: $x_{0} = (0.9, 0.9)^{T}$, $s \in \left[\right.0, \frac{1}{2} \left.\right)$. Let $s = 0.49$, define function $F$ by

The fréchet-derivative of operator $F$ is given by

Taking $x_{0} = (0.9, 0.9)^{T}$ and the infinity norm, we get

The convergence conditions are met and consequently the method can be applied to the system. The existence domain of the solution is $B_{e}(x_{0}, 0.9101)$, and the uniqueness domain is $B_{n}(x_{0}, 1.2192)$.

Taking $x_{0} = (0.73, 0.73)^{T}$ and the infinity norm, then

The existence domain of the solution is $B_{e}(x_{0}, 0.5095)$, and the uniqueness domain is $B_{n}(x_{0}, 0.943534)$.

When the initial value satisfies the Kantorovich condition and the range of $a_{0}$ obtained, the initial value within that range is taken to solve the system. Iterative method (1.2) for solving nonlinear $(4.4)$ with roots of $x^{*} = (0.7884, 0.7884)^{T}$.

Similar results can be obtained in Tables 2 and 3, that is, under the Kantorovich condition, by selecting different initial values, we can converge to a unique solution. When the initial value is closer to the root, the error estimate is lower. This semilocal convergence that can prove the existence and uniqueness of solutions under certain assumptions is very valuable.

5.   Conclusions

In this paper, the semilocal convergence of the eighth order iterative method (1.2) is studied. By analyzing the behavior of the iterative method under the Kantorovich condition, the Lipschitz condition is applied to the first derivative, and the theory of semilocal convergence of the iterative method is obtained by using the recurrence relation. The existence and uniqueness domain of the solution of the nonlinear system is obtained. In the experimental part, a classical Hammerstein nonlinear integral equation and a matrix function are solved. The experimental results are consistent with expectations, and the high-precision approximation of the system solution also proves the effectiveness of the method numerically.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 61976027), the Natural Science Foundation of Liaoning Province (Nos. 2022-MS-371, 2023-MS-296), Educational Commission Foundation of Liaoning Province of China (Nos. LJKMZ20221492, LJKMZ20221498) and the Key Project of Bohai University (No. 0522xn078).

Conflict of interest

The authors declare no conflicts of interest.