Numerical solution of nonlinear parabolic systems by block monotone iterations

TL;DR

This paper develops and analyzes block monotone iterative methods, based on Jacobi and Gauss-Seidel schemes, for efficiently solving coupled nonlinear parabolic systems with monotone reaction functions.

Contribution

It introduces a block monotone iterative framework for nonlinear parabolic systems, proving convergence and solution existence using upper and lower solution sequences.

Findings

Block Gauss-Seidel converges at least as fast as block Jacobi.

Monotone sequences guarantee existence of solutions for quasi-monotone systems.

The methods effectively handle coupled nonlinear parabolic problems.

Abstract

This paper deals with investigating numerical methods for solving coupled system of nonlinear parabolic problems. We utilize block monotone iterative methods based on Jacobi and Gauss--Seidel methods to solve difference schemes which approximate the coupled system of nonlinear parabolic problems, where reaction functions are quasi-monotone nondecreasing or nonincreasing. In the view of upper and lower solutions method, two monotone upper and lower sequences of solutions are constructed, where the monotone property ensures the theorem on existence of solutions to problems with quasi-monotone nondecreasing and nonincreasing reaction functions. Construction of initial upper and lower solutions is presented. The sequences of solutions generated by the block Gauss--Seidel method converge not slower than by the block Jacobi method.

Equations

$$u_{\alpha ,t}-L_{\alpha }{u}_{\alpha }(x,y,t)+f_{\alpha }(x,y,t,u)=0,(x,y,t)\in Q_T=\omega \times (0,T],$$

$$\alpha =1,2,\omega =\{(x,y):0<x<l_1,0<y<l_2\},$$

$$u(x,y,t)=g(x,y,t),(x,y,t)\in \partial Q_T,$$

$$u(x,y,0)=\psi (x,y),(x,y)\in \overline{\omega },$$

$$L_{\alpha }{u}_{\alpha }(x,y,t)={\epsilon }_{\alpha }(u_{\alpha ,xx}+u_{\alpha ,yy})-v_{\alpha ,1}(x,y,t)u_{\alpha ,x}-v_{\alpha ,2}(x,y,t)u_{\alpha ,y},$$

$$\alpha =1,2,$$

$$-\frac{\partial f_{\alpha }}{\partial u_{{\alpha }^{\prime }}}\ge 0,(x,y,t)\in {\overline{Q}}_T,{\alpha }^{\prime }\ne \alpha ,\alpha ,{\alpha }^{\prime }=1,2,$$

$$-\frac{\partial f_{\alpha }}{\partial u_{{\alpha }^{\prime }}}\le 0,(x,y,t)\in {\overline{Q}}_T,{\alpha }^{\prime }\ne \alpha ,\alpha ,{\alpha }^{\prime }=1,2,$$

$$f_{\alpha}(x,y,t,u_{\alpha},u_{\alpha^{\prime}})=\left\{\begin{array}[]{ll}f_{1}(x,y,t,u_{1},u_{2}),\quad\alpha=1,\\ f_{2}(x,y,t,u_{1},u_{2}),\quad\alpha=2,\end{array}\right.\alpha\neq\alpha^{\prime}.$$

$$\hat{u}(x,y,t)\le \tilde{u}(x,y,t),(x,y,t)\in {\overline{Q}}_T,$$

$${\hat{u}}_{\alpha ,t}-\text{mbox}{L}_{\alpha }{\hat{u}}_{\alpha }+f_{\alpha }(x,y,t,\hat{u})\le 0\le {\tilde{u}}_{\alpha ,t}-\text{mbox}{L}_{\alpha }{\tilde{u}}_{\alpha }+f_{\alpha }(x,y,t,\tilde{u}),$$

$$(x,y,t)\in Q_T,\hat{u}(x,y,t)\le g(x,y,t)\le \tilde{u}(x,y,t),(x,y,t)\in \partial Q_T,$$

$$\hat{u}(x,y,0)\le \psi (x,y)\le \tilde{u}(x,y,0),(x,y)\in \overline{\omega },\alpha =1,2,$$

$$\hat{u}(x,y,t)\le \tilde{u}(x,y,t),(x,y,t)\in {\overline{Q}}_T,$$

$${\hat{u}}_{\alpha ,t}-\text{mbox}{L}_{\alpha }{\hat{u}}_{\alpha }+f_{\alpha }(x,y,t,{\hat{u}}_{\alpha },{\tilde{u}}_{{\alpha }^{\prime }})\le 0\le {\tilde{u}}_{\alpha ,t}-\text{mbox}{L}_{\alpha }{\tilde{u}}_{\alpha }+f_{\alpha }(x,y,t,{\tilde{u}}_{\alpha },{\hat{u}}_{{\alpha }^{\prime }}),$$

$$(x,y,t)\in Q_T,\hat{u}(x,y,t)\le g(x,y,t)\le \tilde{u}(x,y,t),(x,y,t)\in \partial Q_T,$$

$$\hat{u}(x,y,0)\le \psi (x,y)\le \tilde{u}(x,y,0),(x,y)\in \overline{\omega },\alpha =1,2,$$

$$⟨\hat{u},\tilde{u}⟩=\left\{u(x,y,t):\hat{u}(x,y,t)\le u(x,y,t)\le \tilde{u}(x,y,t),(x,y,t)\in {\overline{Q}}_T\right\}.$$

$$0\le \frac{\partial f_{\alpha }(x,y,t,u)}{\partial u_{\alpha }}\le c_{\alpha }(x,y,t),u\in ⟨\hat{u},\tilde{u}⟩,(x,y,t)\in {\overline{Q}}_T,\alpha =1,2,$$

$$-\frac{\partial f_{\alpha }(x,y,t,u)}{\partial u_{{\alpha }^{\prime }}}\ge 0,u\in ⟨\hat{u},\tilde{u}⟩,(x,y,t)\in {\overline{Q}}_T,{\alpha }^{\prime }\ne \alpha ,\alpha ,{\alpha }^{\prime }=1,2,$$

$$-\frac{\partial f_{\alpha }(x,y,t,u)}{\partial u_{{\alpha }^{\prime }}}\le 0,u\in ⟨\hat{u},\tilde{u}⟩,(x,y,t)\in {\overline{Q}}_T,{\alpha }^{\prime }\ne \alpha ,\alpha ,{\alpha }^{\prime }=1,2.$$

$${\overline{\Omega }}^{hx}=\{x_i,i=0,1,\dots ,N_x;x_0=0,x_{N_x}=l_1;hx=x_{i+1}-x_i\},$$

$${\overline{\Omega }}^{hy}=\{y_j,j=0,1,\dots ,N_y;y_0=0,y_{N_y}=l_2;hy=y_{j+1}-y_j\},$$

$${\overline{\Omega }}^{\tau }=\{t_m,m=0,1,\dots ,N_{\tau };t_0=0,t_{N_{\tau }}=T;\tau =t_m-t_{m-1}\}.$$

$$\left({\mathcal{L}}_{\alpha }^h(p,t_m)+{\tau }^{-1}\right)U_{\alpha }(p,t_m)+f_{\alpha }(p,t_m,U)-{\tau }^{-1}{U}_{\alpha }(p,t_{m-1})=0,$$

$$(p,t_m)\in {\Omega }^{h\tau }={\Omega }^h\times {\Omega }^{\tau },U(p,t_m)=g(p,t_m),(p,t_m)\in \partial {\Omega }^{h\tau },$$

$$U(p,0)=\psi (p),p\in {\overline{\Omega }}^h,$$

$${\mathcal{L}}_{\alpha }^h(p,t_m)U_{\alpha }(p,t_m)=$$

$$+v_{\alpha ,1}(p,t_m)D_x^1{U}_{\alpha }(p,t_m)$$

$$+v_{\alpha ,2}(p,t_m)D_y^1{U}_{\alpha }(p,t_m),\alpha =1,2,$$

$$D_x^2{U}_{\alpha }(x_i,y_j,t_m)=\frac{U_{\alpha ,i-1,j,m}-2U_{\alpha ,ij,m}+U_{\alpha ,i+1,j,m}}{h_x^2},$$

$$D_y^2{U}_{\alpha }(x_i,y_j,t_m)=\frac{U_{\alpha ,i,j-1,m}-2U_{\alpha ,ij,m}+U_{\alpha ,i,j+1,m}}{h_y^2},$$

$$D_x^1{U}_{\alpha }(x_i,y_j,t_m)=\frac{U_{\alpha ,ij,m}-U_{\alpha ,i-1,j,m}}{h_x},$$

$$D_y^1{U}_{\alpha }(x_i,y_j,t_m)=\frac{U_{\alpha ,ij,m}-U_{\alpha ,i,j-1,m}}{h_y},\alpha =1,2,$$

$$\displaystyle\Big{(}\mathcal{L}_{\alpha}^{h}(p,t_{m})+\left(\tau^{-1}+k_{\alpha}(p,t_{m})\right)I\Big{)}W_{\alpha}(p,t_{m})=\varphi_{\alpha}(p,t_{m}),\quad p\in\Omega^{h},$$

$$\alpha =1,2,U(p,t_m)=g(p,t_m),p\in \partial {\Omega }^h,$$

$$\left({\mathcal{L}}_{\alpha }^h(p,t_m)+({\tau }^{-1}+k_{\alpha }(p,t_m))I\right)W_{\alpha }(p,t_m)\ge 0(\le 0),p\in {\Omega }^h,$$

$$W_{\alpha }(p,t_m))\ge 0(\le 0),p\in \partial {\Omega }^h,$$

$$\Vert W_{\alpha }(\cdot ,t_m){\Vert }_{{\overline{\Omega }}^h}\le \max \left\{\Vert g_{\alpha }(\cdot ,t_m){\Vert }_{\partial {\Omega }^h},{\Vert \frac{{\phi }_{\alpha }(\cdot ,t_m)}{k_{\alpha }(\cdot ,t_m)+{\tau }^{-1}}\Vert }_{{\Omega }^h}\right\},\alpha =1,2,$$

$$\Vert g_{\alpha }(\cdot ,t_m){\Vert }_{\partial {\Omega }^h}=\underset{p\in \partial {\Omega }^h}{\max }|g_{\alpha }(p,t_m)|,$$

$${\Vert \frac{{\phi }_{\alpha }(\cdot ,t_m)}{k_{\alpha }(\cdot ,t_m)+{\tau }^{-1}}\Vert }_{{\Omega }^h}=\underset{p\in {\Omega }^h}{\max }\left|\frac{{\phi }_{\alpha }(p,t_m)}{k_{\alpha }(p,t_m)+{\tau }^{-1}}\right|.$$

Full text

1 Introduction

Several problems in the chemical, physical and engineering sciences are characterized by coupled systems of nonlinear parabolic equations [4]. In this paper, we construct block monotone iterative methods for solving the coupled system of nonlinear parabolic equations

[TABLE]

where $u=(u_{1},u_{2})$, $g=(g_{1},g_{2})$, $f=(f_{1},f_{2})$, $\psi=(\psi_{1},\psi_{2})$, $\partial Q_{T}=\partial\omega\times(0,T]$, and $\partial\omega$ is the boundary of $\omega$. The differential operators $L_{\alpha}$, $\alpha=1,2$, are defined by

[TABLE]

where $\varepsilon_{\alpha}$, $\alpha=1,2$, are positive constants diffusion coefficients. It is assumed that the functions $f_{\alpha}$, $g_{\alpha}$, $v_{\alpha}$, $\alpha=1,2$, are smooth in their respective domains.

The aim of this paper is to construct and investigate block monotone iterative methods based on Jacobi and Gauss–Seidel methods for solving coupled systems of nonlinear parabolic equations with quasi-monotone nondecreasing or quasi-monotone nonincreasing reaction functions $f_{\alpha}$, $\alpha=1,2$, which satisfy the inequalities

[TABLE]

when $f_{\alpha}$, $\alpha=1,2$, are quasi-monotone nondecreasing, and

[TABLE]

when $f_{\alpha}$, $\alpha=1,2$, are quasi-monotone nonincreasing.

2 Properties of solutions to system (1)

We introduce the following notation:

[TABLE]

Two vector functions $\widetilde{u}(x,y,t)=(\widetilde{u}_{1},\widetilde{u}_{2})$ and $\widehat{u}(x,y,t)=(\widehat{u}_{1},\widehat{u}_{2})$, are called ordered upper and lower solutions to (1), if they satisfy the inequalities

[TABLE]

when the reaction functions $f_{\alpha}$, $\alpha=1,2$, are quasi-monotone nondecreasing, and if they satisfy the inequalities

[TABLE]

when the reaction functions $f_{\alpha}$, $\alpha=1,2$, are quasi-monotone nonincreasing.

For a given ordered upper $\widetilde{u}$ and lower $\widehat{u}$ solutions, a sector $\langle\widehat{u},\widetilde{u}\rangle$ is defined as follows:

[TABLE]

In the sector $\langle\widehat{u},\widetilde{u}\rangle$, the vector function $f(x,y,t,u)$ is assumed to satisfy the constraints

[TABLE]

where $c_{\alpha}(x,y,t)$, $\alpha=1,2$, are nonnegative bounded functions.

The vector function $f(x,y,t,u)$ is called quasi-monotone nondecreasing in the sector $\langle\widehat{u},\widetilde{u}\rangle$ if it satisfies the conditions

[TABLE]

and $f(x,y,t,u)$ is called quasi-monotone nonincreasing if it satisfies the conditions

[TABLE]

Theorem 1**.**

Let $\widetilde{u}$ and $\widehat{u}$ be ordered upper and lower solutions of problem (1), $f$ in (1) be quasi-monotone nondecreasing (6) or quasi-monotone nonincreasing (7) in the sector $\langle\widehat{u},\widetilde{u}\rangle$ and satisfy (5). Then problem (1) has a unique solution in the sector $\langle\widehat{u},\widetilde{u}\rangle$.

The proof of the theorem can be found in Theorems 8.3.1 and 8.3.2, [4].

3 The nonlinear difference scheme

3.1 The statement of the nonlinear difference

scheme

On $\overline{\omega}$ and [0,T], we introduce a rectangular mesh $\overline{\Omega}^{h}=\overline{\Omega}^{hx}\times\overline{\Omega}^{hy}$ and $\overline{\Omega}^{\tau}$, such that

[TABLE]

For a mesh function $U(p,t_{m})=(U_{1}(p,t_{m}),U_{2}(p,t_{m}))$, $(p,t_{m})\in\overline{\Omega}^{h}\times\overline{\Omega}^{\tau}$, $p=(x_{i},y_{j})$, we use the implicit difference scheme

[TABLE]

[TABLE]

where $\partial\Omega^{h}$ is the boundary of $\Omega^{h}$. It is assumed that the functions $v_{\alpha,1}(p,t_{m})$ and $v_{\alpha,2}(p,t_{m})$ , $(p,t_{m})\in\overline{\Omega}^{h\tau}$, $\alpha=1,2$, are nonnegative, $D^{2}_{x}U_{\alpha}(p,t_{m})$, $D^{2}_{y}U_{\alpha}(p,t_{m})$ and $D^{1}_{x}U_{\alpha}(p,t_{m})$, $D^{1}_{y}U_{\alpha}(p,t_{m})$, $\alpha=1,2$, are, respectively, the central difference and backward difference approximations to the second and first derivatives:

[TABLE]

where $U_{\alpha,ij,m}\equiv U_{\alpha}(x_{i},y_{j},t_{m})$.

Remark 1*.*

An approximation of the first derivatives $u_{x}$ and $u_{y}$ depends on the signs of $v_{\alpha,1}(x,y,t)$ and $v_{\alpha,2}(x,y,t)$ , $\alpha=1,2$. When $v_{\alpha,1}(x,y,t)$ and $v_{\alpha,2}(x,y,t)$, $\alpha=1,2$, are nonpositive, then $u_{x}$ and $u_{y}$ are approximated by the forward difference formula. The first derivatives $u_{x}$ and $u_{y}$ are approximated by using both forward or backward difference formulae when $v_{\alpha,1}(x,y,t)$ and $v_{\alpha,2}(x,y,t)$, $\alpha=1,2$, have variable signs.

On each time level $t_{m}$, $m\geq 1$, we introduce the linear problems

[TABLE]

where $I$ is the identity operator and $k_{\alpha}(p,t_{m})$, $\alpha=1,2$, are nonnegative bounded mesh functions. We now formulate the maximum principle for the difference operators $\mathcal{L}_{\alpha}^{h}(p,t_{m})+(\tau^{-1}+k_{\alpha}(p,t_{m}))I$, $\alpha=1,2$, and give an estimate of the solution to (3.1).

Lemma 1**.**

• (i)

If $W_{\alpha}(p,t_{m})$, $\alpha=1,2$, satisfy the conditions

[TABLE]

then $W_{\alpha}(p,t_{m})\geq 0\ (\leq 0),\quad p\in\overline{\Omega}^{h}$.

• (ii)

The following estimates of the solution to (3.1) hold

[TABLE]

where

[TABLE]

The proof of the lemma can be found in [1], [5].

Remark 2*.*

In this remark we discuss the mean-value theorem for vector-valued functions. Introduce the following notation:

[TABLE]

Assume that $\mathcal{F}_{\alpha}(x,y,t,u_{\alpha},u_{\alpha^{\prime}})$, $\alpha=1,2$, are smooth functions, then we have

[TABLE]

[TABLE]

where $h_{\alpha}(x,y,t)$ lies between $u_{\alpha}(x,y,t)$ and $w_{\alpha}(x,y,t)$, and $h_{\alpha^{\prime}}(x,y,t)$ lies between $u_{\alpha^{\prime}}(x,y,t)$ and $w_{\alpha^{\prime}}(x,y,t)$, $\alpha=1,2$.

3.2 Quasi-monotone nondecreasing reaction functions

On each time level $t_{m}\in\Omega^{\tau}$, $m\geq 1$, the vector mesh functions

[TABLE]

[TABLE]

are called ordered upper and lower solutions of (9), if they satisfy the inequalities

[TABLE]

For a given pair of ordered upper and lower solutions $\widetilde{U}(p,t_{m})$ and $\widehat{U}(p,t_{m})$, we define the sector

[TABLE]

In the sector $\langle\widehat{U}(t_{m}),\widetilde{U}(t_{m})\rangle$, the vector function $f(p,t_{m},U)$ is assumed to satisfy the constraints

[TABLE]

[TABLE]

where $c_{\alpha}(p,t_{m})$, $\alpha=1,2$, are nonnegative bounded functions. Reaction functions, which satisfy (17), are called quasi-monotone nondecreasing.

We introduce the notation

[TABLE]

where $c_{\alpha}(p,t_{m})$, $\alpha=1,2$, are defined in (16), and give a monotone property of $\Gamma_{\alpha}$, $\alpha=1,2$.

Lemma 2**.**

Suppose that $U=(U_{1},U_{2})$ and $V=(V_{1},V_{2})$, are any functions in $\langle\widehat{U}(t_{m}),\widetilde{U}(t_{m})\rangle$, where $U\geq V$, and assume that (16), (17) are satisfied. Then

[TABLE]

where $(p,t_{m})$ is suppressed in (19).

Proof.

From (18), we have

[TABLE]

For $\alpha=1$ in (3.2), using the mean-value theorem (13), we obtain

[TABLE]

where

[TABLE]

From here, (16), (17) and taking into account that $U_{\alpha}\geq V_{\alpha}$, $\alpha=1,2$, we conclude (19) for $\alpha=1$. Similarly, we can prove (19) for $\alpha=2$. ∎

3.3 Quasi-monotone nonincreasing reaction functions

On each time level $t_{m}\in\Omega^{\tau}$, $m\geq 1$, the vector mesh functions

[TABLE]

[TABLE]

are called ordered upper and lower solutions of (9), if they satisfy the inequalities

[TABLE]

The upper $\widetilde{U}(p,t_{m})$ and lower $\widehat{U}(p,t_{m})$ solutions are dependent of each other and calculated simultaneously.

We assume that in the sector $\langle\widehat{U},\widetilde{U}\rangle$ defined in (15), the vector function $f(p,t_{m},U)$ in (9), satisfies the constraints (16) and

[TABLE]

Reaction functions, which satisfy (22), are called quasi-monotone nonincreasing. We give a monotone property of $\Gamma_{\alpha}$, $\alpha=1,2$, in the case of quasi-monotone nonincreasing reaction functions, where $\Gamma_{\alpha}$, $\alpha=1,2$, are defined in (18).

Lemma 3**.**

Suppose that $U=(U_{1},U_{2})$ and $V=(V_{1},V_{2})$, are any functions in $\langle\widehat{U}(t_{m},\widetilde{U}(t_{m})\rangle$, where $U\geq V$, and assume that (16) and (22) are satisfied. Then

[TABLE]

where $(p,t_{m})$ is suppressed in (23).

Proof.

From (18), we have

[TABLE]

For $\alpha=1$ in (3.3), using the mean-value theorem (13), we obtain

[TABLE]

where

[TABLE]

From here, (16), (22) and taking into account that $U_{\alpha}\geq V_{\alpha}$, $\alpha=1,2$, we conclude that

[TABLE]

Similarly, we can prove that

[TABLE]

∎

4 The case of quasi-monotone nondecreasing reaction functions

4.1 The statement of the block nonlinear difference scheme

Write down the difference scheme (9) at an interior mesh point $(x_{i},y_{j})\in\Omega^{h}$ in the form

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $G_{\alpha,ij,m}^{*}$ is associated with the boundary function $g_{\alpha}(x_{i},y_{j},t_{m})$. On each time level $m$, $m\geq 1$, we define column vectors and diagonal matrices by

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $L_{\alpha,1,m}U_{\alpha,0,m}$ is included in $G_{\alpha,1,m}^{*}$, and $R_{\alpha,N_{x}-1,m}U_{\alpha,N_{x},m}$ is included in $G_{\alpha,N_{x},m}^{*}$. Then the difference scheme (9) may be written in the form

[TABLE]

[TABLE]

with the tridiagonal matrix $A_{\alpha,i,m}$ in the form

[TABLE]

Matrices $L_{\alpha,i,m}$ and $R_{\alpha,i,m}$ contain the coupling coefficients of a mesh point, respectively, to the mesh point of the left line and the mesh point of the right line.

We introduce the notation for the residuals of the nonlinear difference scheme (26) in the form

[TABLE]

where

[TABLE]

4.2 Block monotone Jacobi and Gauss-Seidel methods

We now present the block monotone Jacobi and block monotone Gauss–Seidel methods for the nonlinear difference scheme (9) when the reaction functions are quasi-monotone nondecreasing based on the method of upper and lower solutions. We define functions $c_{\alpha,m}$, $\alpha=1,2$, $m\geq 1$, in the following form

[TABLE]

where $c_{\alpha,ij,m}$, $ij\in\overline{\Omega}^{h}$, $\alpha=1,2$, are defined in (16). On each time level $t_{m}$, $m\geq 1$, the upper $\{\overline{U}^{(n)}_{\alpha,i,m}\}$ and lower $\{\underline{U}^{(n)}_{\alpha,i,m}\}$, $\alpha=1,2$, sequences of solutions are calculated by the following block Jacobi and block Gauss-Seidel methods

[TABLE]

[TABLE]

[TABLE]

where $U_{i,m}^{(n-1)}=(U_{1,i,m}^{(n-1)},U_{2,i,m}^{(n-1)})$, $\mathcal{G}_{\alpha,i,m}\left(U_{\alpha,i,m}^{(n-1)},U_{\alpha,i,m-1},U_{\alpha^{\prime},i,m}^{(n-1)}\right)$, $\alpha^{\prime}\neq\alpha$, $\alpha,\alpha^{\prime}=1,2$, are defined in (27), $\mathbf{0}$ is zero column vector with the $N_{x}-1$ components, and $U_{\alpha,i,m}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, are the approximate solutions on time level $m\geq 1$, where $n_{m}$ is a number of iterations on time level $m\geq 1$. For $\eta=0$ and $\eta=1$, we have, respectively, the block Jacobi and block Gauss–Seidel methods.

Remark 3*.*

Similar to Remark 2, we discuss the mean-value theorem for mesh vector-functions. Assume that $F_{\alpha}(x,y,t,u_{\alpha},u_{\alpha^{\prime}})$, $i=0,1,\ldots,N_{x}$, $\alpha\neq\alpha^{\prime}$, $\alpha,\alpha^{\prime}=1,2$, are smooth functions. In the notation of $F_{\alpha,i,m}(U_{\alpha,i,m},U_{\alpha^{\prime},i,m})$ in (27), we have

[TABLE]

[TABLE]

where $Y_{\alpha,i,m}$ lie between $U_{\alpha,i,m}$ and $V_{\alpha,i,m}$, and $Y_{\alpha^{\prime},i,m}$ lie between $U_{\alpha^{\prime},i,m}$ and $V_{\alpha^{\prime},i,m}$, $i=0,1,\ldots,N_{x}$, $\alpha^{\prime}\neq\alpha$, $\alpha,\alpha^{\prime}=1,2$, $m\geq 1$. The partial derivatives $\frac{\partial F_{\alpha,i,m}}{\partial u_{\alpha}}$ and $\frac{\partial F_{\alpha,i,m}}{\partial u_{\alpha^{\prime}}}$, are the diagonal matrices

[TABLE]

[TABLE]

where $\frac{\partial f_{\alpha,ij,m}}{\partial u_{\alpha}}$ and $\frac{\partial f_{\alpha,ij,m}}{\partial u_{\alpha^{\prime}}}$, $j=1,\ldots,N_{y}-1$, are calculated, respectively, at $Y_{\alpha,i,m}$ and $Y_{\alpha^{\prime},i,m}$, $i=1,2,\ldots,N_{x}-1$.

Theorem 2**.**

Let $f(p,t_{m},U)$ in (9) satisfy (16) and (17), where $\widetilde{U}(p,t_{m})=(\widetilde{U}_{1}(p,t_{m}),\widetilde{U}_{2}(p,t_{m}))$ and $\widehat{U}(p,t_{m})=(\widehat{U}_{1}(p,t_{m}),\widehat{U}_{2}(p,t_{m}))$ are ordered upper and lower solutions (14) of (9) . Then the upper $\{\overline{U}_{\alpha,i,m}^{(n)}\}$ and lower $\{\underline{U}_{\alpha,i,m}^{(n)}\}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, sequences generated by (29), with $\overline{U}^{(0)}(p,t_{m})=\widetilde{U}(p,t_{m})$ and $\underline{U}^{(0)}(p,t_{m})=\widehat{U}(p,t_{m})$, converge monotonically, such that,

[TABLE]

Proof.

We consider the case of Gauss-Seidel method $\eta=1$, and the case of the Jacobi method can be proved by a similar manner. On first time level $m=1$, since $\overline{U}^{(0)}$ is an upper solution (14) with respect to $U_{\alpha}(p,0)=\psi_{\alpha}(p)$, from (29), we have

[TABLE]

where $I$ is the identity matrix. For $i=1$ in (32), taking into account that $L_{\alpha,i,1}\geq\mathbf{0}$, $i=1,2,\ldots,N_{x}-1$, and $\overline{Z}^{(1)}_{\alpha,0,1}\leq\mathbf{0}$, it follows that $\left(A_{\alpha,1,1}+c_{\alpha,1}I\right)\overline{Z}_{\alpha,1,1}^{(1)}\leq\mathbf{0}$. Taking into account that $d_{\alpha,ij}>0$, $b_{\alpha,ij}$, $t_{\alpha,ij}\geq 0$, $\alpha=1,2$, in (4.1) and $A_{\alpha,i,1}$ are strictly diagonal dominant matrix, we conclude that $A_{\alpha,i,1}$, $i=1,2,\ldots,N_{x}-1$, $\alpha=1,2$, are $M$-matrices and $A_{\alpha,i,1}^{-1}\geq\emph{O}$ (Corollary 3.20, [6]), which leads to $(A_{\alpha,i,1}+c_{\alpha,1}I)^{-1}\geq\emph{O}$, where O is the ($N_{y}-1)\times(N_{y}-1$) null matrix. From here, we obtain

[TABLE]

Taking into account that $\overline{Z}_{\alpha,1,1}^{(1)}\leq\mathbf{0}$, for $i=2$ in (32), in a similar manner, we conclude that

[TABLE]

By induction on $i$, we can prove that

[TABLE]

Similarly, for the lower solution $\underline{U}^{(0)}=\widehat{U}$, we have

[TABLE]

We now prove that $\overline{U}^{(1)}_{\alpha,i,1}$ and $\underline{U}^{(1)}_{\alpha,i,1}$, are ordered upper and lower solutions (14) with respect to the column vector $U_{\alpha,i,0}=\psi_{\alpha,i}$, where the column vector $\psi_{\alpha,i}$ is associated with the initial function $\psi(x,y)$ from (1). Let $W^{(1)}_{\alpha,i,1}=\overline{U}^{(1)}_{\alpha,i,1}-\underline{U}^{(1)}_{\alpha,i,1}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, from (29) for $\alpha=1$, we have

[TABLE]

By the mean-value theorem (30), we have

[TABLE]

[TABLE]

where $\underline{U}_{\alpha,i,1}^{(0)}\leq Q_{\alpha,i,1}^{(0)}\leq\overline{U}_{\alpha,i,1}^{(0)}$, $i=1,2,\ldots,N_{x}-1$, $\alpha=1,2$, and

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From here, we conclude that $\frac{\partial F_{1,i,1}}{\partial u_{1}}$, $\frac{\partial F_{1,i,1}}{\partial u_{2}}$ satisfy (16) and (17). From here and (4.2), we have

[TABLE]

[TABLE]

From here, (16), (17), taking into account that $W_{\alpha,i,1}^{(0)}\geq\mathbf{0}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, and $R_{1,i,1}\geq\emph{O}$, we obtain

[TABLE]

[TABLE]

Taking into account that $(A_{1,i,1}+c_{1,1}I)^{-1}\geq\emph{O}$ (Corollary 3.20, [6]), $i=1,2,\ldots,N_{x}-1$, for $i=1$ in (37) and $W_{1,0,1}^{(1)}=\mathbf{0}$, we conclude that $W_{1,1,1}^{(1)}\geq\mathbf{0}$. For $i=2$ in (37), using $L_{1,2,1}\geq\emph{O}$ and $W_{1,1,1}^{(1)}\geq\mathbf{0}$, we obtain $W_{1,2,1}^{(1)}\geq\mathbf{0}$. Thus, by induction on $i$, we prove that

[TABLE]

By a similar argument, we can prove that

[TABLE]

Thus, we prove (14a). We now prove (14b). From (29) for $\alpha=1$ and using the mean-value theorem (30), we conclude that

[TABLE]

where

[TABLE]

From (33), (34), taking into account that $\underline{U}^{(1)}_{\alpha,i,1}\leq\overline{U}^{(1)}_{\alpha,i,1}$, $\alpha=1,2$, we conclude that $\frac{\partial F_{1,i,1}}{\partial u_{1}}$ and $\frac{\partial F_{1,i,1}}{\partial u_{2}}$ satisfy (16) and (17). From (16), (17), (33) and taking into account that $R_{1,i,1}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, we conclude that

[TABLE]

Similarly, we obtain

[TABLE]

which means that $\overline{U}_{\alpha,i,1}^{(1)}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, are upper solution (14b) on $m=1$. By a similar manner, we can prove that

[TABLE]

which means that $\underline{U}_{\alpha,i,1}^{(1)}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, are lower solutions (14b) on $m=1$. By induction on $n$, we prove (31) on the first time level $m=1$.

On the second time level $m=2$, taking into account that $\overline{U}_{\alpha,i,2}^{(0)}=\widetilde{U}_{\alpha,i,2}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, from (9), we obtain

[TABLE]

where $\overline{U}_{\alpha,i,1}$, $i=1,2,\ldots,N_{x}-1$, $\alpha=1,2$, are the approximate solutions on the first time level $m=1$, which defined in (29). From here, taking into account that from (31), we have $\overline{U}_{\alpha,i,1}\leq\widetilde{U}_{\alpha,i,1}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, it follows that

[TABLE]

which means that $\overline{U}_{\alpha,i,2}^{(1)}=\widetilde{U}_{\alpha,i,2}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, are upper solutions with respect to $\overline{U}_{\alpha,i,1}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$. Similarly, we can obtain that

[TABLE]

which means that $\underline{U}_{\alpha,i,2}^{(1)}=\widehat{U}_{\alpha,i,2}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, are lower solutions with respect to $\underline{U}_{\alpha,i,1}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$.

From here and (29), on the second time level $m=2$, we obtain

[TABLE]

Taking into account that $d_{\alpha,ij}>0$, $b_{\alpha,ij}$, $t_{\alpha,ij}\geq 0$, $(x_{i},y_{j})\in\Omega^{h}$, $\alpha=1,2$, in (4.1) and $A_{\alpha,i,2}$, $i=1,2,\ldots,N_{x}-1$, $\alpha=1,2$, are strictly diagonal dominant matrix, we conclude that $A_{\alpha,i,2}+c_{\alpha,2}I$, $i=1,2,\ldots,N_{x}-1$, $\alpha=1,2$, are $M$-matrices and $(A_{\alpha,i,2}+c_{\alpha,2}I)^{-1}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, $\alpha=1,2$, (Corollary 3.20, [6]). From here, for $i=1$ in (40), taking into account that $L_{\alpha,i,2}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, and $\overline{Z}^{(1)}_{\alpha,0,2}\leq\mathbf{0}$ from (29), we obtain that

[TABLE]

From here, for $i=2$ in (40), we conclude that

[TABLE]

By induction on $i$, we can prove that

[TABLE]

Similarly, for the lower case, we can prove that

[TABLE]

The proof that $\overline{U}^{(1)}_{\alpha,i,2}$ and $\underline{U}^{(1)}_{\alpha,i,2}$, $\alpha=1,2$, are ordered upper and lower solutions (14) repeats the proof on the first time level $m=1$. By induction on $n$, we can prove $(\ref{bp31})$ for $m=1$. By induction on $m$, we can prove (31) for $m\geq 1$.

∎

4.3 Existence and uniqueness of a solution to the nonlinear difference scheme (26)

In the following theorem, we prove the existence of a solution to (26) based on Theorem 2.

Theorem 3**.**

Let $f(p,t_{m},U)$ satisfy (16), where $\widetilde{U}_{\alpha,i,m}$ and $\widehat{U}_{\alpha,i,m}$, $i=0,1\ldots,N_{x}$, $\alpha=1,2$, $m\geq 1$, be ordered upper and lower solutions (14) to (26). Then a solution of the nonlinear difference scheme (26) exists in $\langle\widehat{U}(t_{m}),\widetilde{U}(t_{m})\rangle$, $m\geq 1$.

Proof.

We consider the upper case of the Gauss–Seidel method $(\eta=1)$ in (29). On the first time level $t_{1}$, from (31), we conclude that $\lim\overline{U}^{(n)}_{\alpha,i,1}=\overline{V}_{\alpha,i,1}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, as $n\rightarrow\infty$ exists, and

[TABLE]

where $\overline{U}^{(0)}_{\alpha,i,1}=\widetilde{U}_{\alpha,i,1}$. Similar to (38), we have

[TABLE]

where

[TABLE]

From here and (4.3), we conclude that $\overline{V}_{\alpha,i,1}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, solve (26) on the first time level $m=1$.

By the assumption of the theorem that $\widetilde{U}_{\alpha,i,2}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, are upper solutions and from (4.3), it follows that $\widetilde{U}_{\alpha,i,2}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, are upper solutions with respect to $\overline{V}_{\alpha,i,1}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$. Indeed, from (4.3), we have

[TABLE]

Using a similar argument as in (4.3), we can prove that the limits

[TABLE]

exist and solve (26) on the second time level $m=2$.

By induction on $m$, $m\geq 1$, we can prove that

[TABLE]

are solutions of the nonlinear difference scheme (26). Similarly, we can prove that $\underline{V}_{\alpha,i,m}$ defined by

[TABLE]

are solutions to the nonlinear difference scheme (26). ∎

We now assume that the reaction functions $f_{\alpha}$, $\alpha=1,2$, satisfy the two-sided constrains

[TABLE]

[TABLE]

[TABLE]

where $c_{\alpha}(p,t_{m})$ is defined in (16), $q_{\alpha}(p,t_{m})$ and $\underline{c}_{\alpha}(p,t_{m})$, $\alpha=1,2$, are, respectively, nonnegative bounded and bounded functions. It is assumed that the time step $\tau$ satisfies the assumptions

[TABLE]

the notation of the discrete norm from (11) is in use. When $\beta_{m}=0$, $m\geq 1$, then there is no restriction on $\tau$.

Theorem 4**.**

Suppose that functions $f_{\alpha}(p,t_{m},U)$, $\alpha=1,2$, satisfy (45) and (46), where $\widetilde{U}(p,t_{m})$ and $\widehat{U}(p,t_{m})$ are ordered upper and lower solutions (14) of (9). Let assumptions in (47) on time step $\tau$ be satisfied. Then the nonlinear difference scheme (9) has a unique solution.

Proof.

To prove the uniqueness of a solution to the nonlinear difference scheme (9), it suffices to prove that

[TABLE]

where $\overline{V}_{\alpha}(p,t_{m})$ and $\underline{V}_{\alpha}(p,t_{m})$ are the solutions to the nonlinear difference scheme (9), which are defined in Theorem 3. From (31) and Theorem 3, we obtain

[TABLE]

Letting $W_{\alpha}(p,t_{m})=\overline{V}_{\alpha}(p,t_{m})-\underline{V}_{\alpha}(p,t_{m})$, from (9), we have

[TABLE]

Using the mean-value theorem (13), we obtain

[TABLE]

From here and (4.3), it follows that the partial derivatives satisfy (45) and (46). If $\underline{c}_{m}\geq 0$, from (52) for $m=1$, using (11), (45), (46) and taking into account that $W_{\alpha}(p,0)=0$, we conclude that

[TABLE]

where

[TABLE]

From here, by the assumption on $\tau$ in (47) and taking into account that $W(t_{m})\geq 0$, we conclude that $W(t_{1})=0$.

On the second time level $m=2$, from (52) and taking into account that $W(t_{1})=0$, by a similar manner, we obtain that $W(t_{2})=0$. By induction on $m$, we prove that $W(t_{m})=0$, $m\geq 1$. Thus, we prove the theorem when $\underline{c}_{m}\geq 0$.

If $\underline{c}_{m}<0$, from (52) for $m=1$, using (11), (45) and (46), we conclude that

[TABLE]

From here, by the assumption on $\tau$ in (47) and taking into account that $W(t_{m})\geq 0$, we conclude that $W(t_{1})=0$.

On the second time level $m=2$, from (52) and taking into account that $W(t_{1})=0$, by a similar manner, we obtain that $W(t_{2})=0$. By induction on $m$, we prove that $W(t_{m})=0$, $m\geq 1$. Thus, we prove the theorem. ∎

4.4 Convergence analysis

A stopping test for the block monotone iterative methods (29) is chosen in the following form

[TABLE]

[TABLE]

where $\mathcal{G}_{\alpha}\left(U_{\alpha}^{(n)}(p,t_{m}),U_{\alpha}(p,t_{m-1}),U_{\alpha^{\prime}}^{(n)}(p,t_{m})\right)$, $\alpha^{\prime}\neq\alpha$, $\alpha,\alpha^{\prime}=1,2$, are defined in (27), $U^{(n)}_{\alpha}(p,t_{m})$, $p\in\Omega^{h}$, $\alpha=1,2$, are generated by (29), and $\delta$ is a prescribed accuracy. On each time level $t_{m}$, $m\geq 1$, we set up $U_{\alpha}(p,t_{m})=U^{(n_{m})}_{\alpha}(p,t_{m})$, $p\in\Omega^{h}$, $\alpha=1,2$, $\alpha=1,2$, such that $m_{n}$ is the minimal subject to (53).

Instead of (45), we now assume that

[TABLE]

where $q_{m}$ is defined in (47).

Remark 4*.*

The assumption $\frac{\partial f_{\alpha}(p,t_{m},U)}{\partial u_{\alpha}}\geq q_{m}>0$, in (4.4) can always be obtain by a change of variables. Indeed, we introduce the following functions $z_{\alpha}(x,y,t)=\exp^{-\lambda t}u_{\alpha}(x,y,t)$, $\alpha=1,2$, where $\lambda$ is a constant. Now, $z=(z_{1},z_{2})$ satisfy (1) with

[TABLE]

instead of $f_{\alpha}$, $\alpha=1,2$, and we have

[TABLE]

Thus, if $\lambda\geq\max_{m\geq 1}\left(q_{m},|\underline{c}_{m}|\right)$, where $q_{m}$ and $\underline{c}_{m}$ are defined in (47), then from this, (45) and (46), we conclude that $\frac{\partial f^{*}_{\alpha}}{\partial z_{\alpha}}$ satisfies (4.4).

Theorem 5**.**

Let $\widetilde{U}(p,t_{m})$ and $\widehat{U}(p,t_{m})$ be ordered upper and lower solutions (14) of (9). Suppose that functions $f_{\alpha}(p,t_{m},U)$, $\alpha=1,2$, satisfy (46) and (4.4). Then for the sequence of solutions $\{U^{(n)}\}$ generated by (29), (53), we have the following estimate

[TABLE]

where $U_{\alpha}(p,t_{m})=U_{\alpha}^{(n_{m})}(p,t_{m})$, $n_{m}$ is a minimal number of iterations subject to (53), and $U_{\alpha}^{*}(p,t_{m})$, $\alpha=1,2$, $m\geq 1$, are the unique solutions to the nonlinear difference scheme (9).

Proof.

We consider the case of an upper sequence. On a time level $t_{m}$, $m\geq 1$, from (9) for $\overline{U}_{\alpha}(p,t_{m})$ and $U^{*}_{\alpha}(p,t_{m})$ , we have

[TABLE]

[TABLE]

Letting $W_{\alpha}(p,t_{m})=U_{\alpha}(p,t_{m})-U_{\alpha}^{*}(p,t_{m})$, $p\in\overline{\Omega}^{h}$, $\alpha=1,2$, $m\geq 1$, from here and using the mean-value theorem, we obtain

[TABLE]

where

[TABLE]

The partial derivatives satisfy (46) and (4.4). From here, (46) and (4.4), by using (11), we obtain that

[TABLE]

where the notation of the norm from (11) is in use. From here, in the notation $W_{m}=\max_{\alpha=1,2}W_{\alpha}(t_{m})$, we have

[TABLE]

Taking into account that

[TABLE]

it follows that

[TABLE]

From here, taking into account that $W_{0}=0$, by induction on $m$, we obtain that

[TABLE]

Since $\sum_{\rho=1}^{m}\tau\leq T$, we prove the theorem. ∎

Theorem 6**.**

Let the assumptions in Theorem 5 be satisfied. Then for the sequence of solutions $\{U^{(n)}\}$ generated by (29), (53), the following estimate holds

[TABLE]

where the notation of the norm from (11) is in use, $U_{\alpha}(p,t_{m})=U_{\alpha}^{(n_{m})}(p,t_{m})$, $\alpha=1,2$, $m\geq 1$, $n_{m}$ is the minimal number of iterations subject to the stopping test (53), $u_{\alpha}^{*}(x,y,t)$, $\alpha=1,2$, are the exact solutions to (1), and $E_{\alpha}(p,t_{m})$, $\alpha=1,2$, $m\geq 1$, are the truncation errors of the exact solutions $u_{\alpha}^{*}(x,y,t)$, $\alpha=1,2$, on the nonlinear difference scheme (9).

Proof.

We denote $V(p,t_{m})=u^{*}(p,t_{m})-U^{*}(p,t_{m})$, where the mesh vector function $U^{*}(p,t_{m})$ is the unique solution of the nonlinear difference scheme (9). From (9), by using the mean-value theorem, we obtain that

[TABLE]

where $Y_{\alpha}(p,t_{m})$, $\alpha=1,2$ lie between $u^{*}_{\alpha}(p,t_{m})$ and $U^{*}_{\alpha}(p,t_{m})$, $\alpha=1,2$. From here, (46), (4.4), by using (11), it follows that

[TABLE]

Letting $V_{m}=\max_{\alpha=1,2}\|V_{\alpha}(\cdot,t_{m})\|_{\overline{\Omega}^{h}}$, $m\geq 1$, we have

[TABLE]

Thus, taking into account that

[TABLE]

we conclude

[TABLE]

Since $V_{0}=0$, for $m=1$ in (57), we have

[TABLE]

For $m=2$ in (57), we obtain

[TABLE]

By induction on $m$, we can prove that

[TABLE]

Since $\sum_{\rho=1}^{m}\tau\leq T$, where $T$ is the final time, we have

[TABLE]

We estimate the left hand side in (56) as follows

[TABLE]

where $U_{\alpha}^{*}(p,t_{m})$, $\alpha=1,2$, are the exact solutions of (9). From here, (55) and (58), we prove (56). ∎

Remark 5*.*

The truncation errors $E_{\alpha}(p,t_{m})$, $\alpha=1,2$, $m\geq 1$, for the nonlinear difference scheme (9) are given in the form

[TABLE]

where $E_{m}$ is defined in (56), $\tau$ and $h$ are, respectively, the time and space steps, $\kappa=1$ in the case of one-sided difference approximations of $u_{\alpha,x}$, $u_{\alpha,y}$, $\alpha=1,2$, and $\kappa=2$ in the case of central difference approximations of these derivatives.

4.5 Construction of upper and lower solutions

To start the monotone iterative methods (29), on each time level $t_{m}$, $m\geq 1$, initial iterations are needed. In this section, we discuss the construction of initial iterations $\widetilde{U}_{\alpha}(p,t_{m})$ and $\widehat{U}_{\alpha}(p,t_{m})$, $\alpha=1,2$.

4.5.1 Bounded $f_{u}$

Assume that the functions $f_{\alpha}$, $g_{\alpha}$ and $\psi_{\alpha}$, $\alpha=1,2$, in (1) satisfy the conditions

[TABLE]

where $M_{\alpha}$, $\alpha=1,2$, are positive constants. From (14b) and (59), we obtain that the functions

[TABLE]

are lower solutions of (9).

We introduce the linear problems

[TABLE]

Theorem 7**.**

Let assumptions in (59) be satisfied. Then $\widehat{U}$ and $\widetilde{U}$ from, respectively, (60) and (4.5.1) , are ordered lower and upper solutions to (9), such that

[TABLE]

Proof.

From (59) and (4.5.1), by the maximum principle (11), we conclude (62) for $m=1$. By induction on $m$, we can prove (62) for $m\geq 1$.

We now show that $\widetilde{U}_{\alpha}(p,t_{m})$, $\alpha=1,2$, are upper solutions (14) to (9). We present the left hand side of (9) in the form

[TABLE]

Using (4.5.1), for $m\geq 1$, we obtain that

[TABLE]

From here and (59), we conclude that

[TABLE]

Since $\widetilde{U}_{\alpha}(p,t_{m})$, $\alpha=1,2$, satisfy the boundary and initial conditions, we prove that $\widetilde{U}_{\alpha}(p,t_{m})$, $\alpha=1,2$, are upper solutions to (9). From here and (62), we conclude that $\widehat{U}$ and $\widetilde{U}$ from, respectively, (60) and (4.5.1), are ordered lower and upper solutions to (9). ∎

4.5.2 Constant upper and lower solutions

Let the functions $f_{\alpha}$, $g_{\alpha}$ and $\psi_{\alpha}$, $\alpha=1,2$, in (1) satisfy the conditions

[TABLE]

where $K_{1}$, $K_{2}$ are positive constants, and $K=(K_{1},K_{2})$. The mesh functions $\widehat{U}_{\alpha}(p,t_{m})$, $\alpha=1,2$, from (60) are lower solutions to (9).

In the following lemma, we prove that the mesh functions

[TABLE]

are upper solutions to (9).

Theorem 8**.**

Suppose that the assumptions in (4.5.2) are satisfied. Then the mesh functions $\widehat{U}_{\alpha}(p,t_{m})$ and $\widetilde{U}_{\alpha}(p,t_{m})$ from, respectively, (60) and (65), are ordered lower and upper solutions to (9) and satisfy (62).

Proof.

It is clear from (60) and (65), that $0\leq\widehat{U}_{\alpha}(p,t_{m})\leq\widetilde{U}_{\alpha}(p,t_{m})$, $p\in\overline{\Omega}^{h}$, $\alpha=1,2$, $m\geq 1$. We now show that $\widetilde{U}_{\alpha}(p,t_{m})$, $\alpha=1,2$, are upper solutions (14) to (9).

Using (65), we write the left hand side of (9) for $m=1$ in the form

[TABLE]

From here and (4.5.2) , we conclude that

[TABLE]

For $m\geq 2$, from (4.5.2) and (65), we have

[TABLE]

Since $\widetilde{U}_{\alpha}(p,t_{0})$, $\alpha=1,2$, satisfy the initial conditions and $\widetilde{U}_{\alpha}(p,t_{m})\geq g_{\alpha}(p,t_{m})$, $p\in\partial\Omega^{h}$, $\alpha=1,2$, at $m\geq 1$, we prove that $\widetilde{U}_{\alpha}(p,t_{m})$, $\alpha=1,2$, are upper solutions to (9). From here and (62), we conclude that $\widehat{U}$ and $\widetilde{U}$ from, respectively, (60) and (65), are ordered lower and upper solutions to (9). ∎

4.6 Applications

4.6.1 Gas-liquid interaction model

Consider the gas-liquid interaction model [4], where a dissolved gas A and a dissolved reactant B interact in a bounded diffusion medium $\omega$. The chemical reaction scheme is given by $A+k_{1}B\rightarrow k_{2}P$ and is called the second order reaction, where $k_{1}$ and $k_{2}$ are the reaction rates and P is the product. Denote by $z_{1}(x,y,t)$ and $z_{2}(x,y,t)$ the concentrations of the dissolved gas $A$ and the reactant $B$, respectively. Then the above reactant scheme is governed by (1) with $L_{\alpha}z_{\alpha}=\varepsilon_{\alpha}\triangle z_{\alpha}$, $f_{\alpha}=\sigma_{\alpha}z_{1}z_{2}$, $\alpha=1,2$, where $\sigma_{1}$ is the reaction rate and $\sigma_{2}=k_{1}\sigma_{1}$. By choosing a suitable positive constant $\varrho_{1}>0$ and letting $u_{1}=\varrho_{1}-z_{1}\geq 0$, $u_{2}=z_{2}$, we have

[TABLE]

and system (1) is reduced to

[TABLE]

where $g^{*}_{1}=\varrho_{1}-g_{1}\geq 0$, $g_{2}\geq 0$ on $\partial\omega$ and $\psi_{\alpha}\geq 0$, $\alpha=1,2$, in $\overline{\omega}^{h}$. It is clear from (66) that $f_{\alpha}$, $\alpha=1,2$, are quasi-monotone nondecreasing in the rectangle

[TABLE]

for any positive constant $\varrho_{2}$.

The nonlinear difference scheme (9) is reduced to

[TABLE]

where $f_{\alpha}$, $\alpha=1,2$, are defined in (66). Since the reaction functions $f_{\alpha}$, $\alpha=1,2$, satisfy the assumptions in (4.5.2), with $K_{\alpha}$, $\alpha=1,2$ are given by

[TABLE]

it follows that the mesh functions $\widehat{U}_{\alpha}(p,t_{m})$ and $\widetilde{U}_{\alpha}(p,t_{m})$ from, respectively, (60) and (65) are ordered lower and upper solutions to (67).

From (66), in the sector $\langle\widehat{U}(t_{m}),\widetilde{U}(t_{m})\rangle=\langle 0,K_{\alpha}\rangle$, we have

[TABLE]

and the assumptions in (16) and (17) are satisfied with

[TABLE]

From here and (68), we conclude that Theorem 2 holds for the discrete gas-liquid interaction model (67).

4.6.2 The Volterra-Lotka competition model

In the Volterra-Lotka competition model [4] with the effect of dispersion between two competing species in an ecological systems, the model is governed by (1) with reaction functions are given by

[TABLE]

where $u_{1}$ and $u_{2}$ are the populations of two competing species, the parameters $a_{\alpha}$, $\alpha=1,2$, are positive constants which describe the interaction of the two species. We assume that $a_{\alpha}$, $\alpha=1,2$, satisfy the inequality

[TABLE]

System (1) is reduced to

[TABLE]

The nonlinear difference scheme (9) is reduced to

[TABLE]

where $f_{\alpha}$, $\alpha=1,2$, are defined in (69). We take $(M_{1},M_{2})$ and $(0,0)$ as ordered upper and lower solutions (14) to (4.6.2), where $M_{\alpha}$, $\alpha=1,2$, are positive constants and chosen in the following forms

[TABLE]

It is clear that $(M_{1},M_{2})$ and $(0,0)$ satisfy (14a) and (14c). Now we prove (14b). From (63), it follows that $M_{\alpha}$, $\alpha=1,2$, must satisfy the inequalities

[TABLE]

From here, we conclude that $M_{\alpha}$, $\alpha=1,2$, must satisfy the inequalities

[TABLE]

By using (72), it is clear that the inequalities in (73) are satisfied. Thus, we prove (14).

In the sector $\langle\widehat{U}(t_{m}),\widetilde{U}(t_{m})\rangle=\langle\mathbf{0},M\rangle$, $M=(M_{1},M_{2})$, we have

[TABLE]

From here, the assumptions in (16) and (17) are satisfied with

[TABLE]

and we conclude that Theorem 2 holds for the Volterra-Lotka competition model (4.6.2) with $(\widetilde{U}_{1},\widetilde{U}_{2})=(M_{1},M_{2})$ and $(\widehat{U}_{1},\widehat{U}_{2})=(0,0)$.

5 Comparison of the block monotone Jacobi and block monotone Gauss–Seidel methods

The following theorem shows that the block monotone Gauss–Seidel method (29), $(\eta=1)$, converge not slower than the block monotone Jacobi method (29), $(\eta=0)$.

Theorem 9**.**

Let $f(p,t_{m},U)$ in (9) satisfy (16) and (17), where $\widetilde{U}(p,t_{m})=(\widetilde{U}_{1}(p,t_{m}),\widetilde{U}_{2}(p,t_{m}))$ and $\widehat{U}(p,t_{m})=(\widehat{U}_{1}(p,t_{m}),\widehat{U}_{2}(p,t_{m}))$ are ordered upper and lower solutions (14) of (9). Suppose that $\{(\overline{U}^{(n)}_{\alpha,i,m})_{J},(\underline{U}^{(n)}_{\alpha,i,m})_{J}\}$ and $\{(\overline{U}^{(n)}_{\alpha,i,m})_{GS},(\underline{U}^{(n)}_{\alpha,i,m})_{GS}\}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, $m\geq 1$, are, respectively, the sequences generated by the block monotone Jacobi method (29), $(\eta=0)$ and the block monotone Gauss–Seidel method (29), $(\eta=1)$, where $(\overline{U}^{(0)})_{J}=(\overline{U}^{(0)})_{GS}=\widetilde{U}$ and $(\underline{U}^{(0)})_{J}=(\underline{U}^{(0)})_{GS}=\widehat{U}$, then

[TABLE]

Proof.

From (29), we have

[TABLE]

[TABLE]

From here, letting $\underline{W}^{(n)}_{\alpha,i,m}=\left(\underline{U}^{(n)}_{\alpha,i,m}\right)_{GS}-\left(\underline{U}^{(n)}_{\alpha,i,m}\right)_{J}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, $m\geq 1$, we have

[TABLE]

By using Theorem 2, we have $\left(\underline{U}^{(n)}_{\alpha,i,m}\right)_{GS}\geq\left(\underline{U}^{(n-1)}_{\alpha,i,m}\right)_{GS}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, $m\geq 1$. From here and (5), we conclude that

[TABLE]

Taking into account that $(A_{\alpha,i,m}+c_{\alpha,m}I)^{-1}\geq\emph{O}$, $L_{\alpha,i,m}\geq\emph{O}$, $R_{\alpha,i,m}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, $\alpha=1,2$, $m\geq 1$, for $n=1$ in (5), on the first time level $m=1$, in view of $(\underline{U}^{(0)}_{\alpha,i,m})_{GS}=(\underline{U}^{(0)}_{\alpha,i,m})_{J}$ and $\underline{W}^{(0)}_{\alpha,i,m}=\mathbf{0}$, we conclude that

[TABLE]

For $n=2$ in (5) and using notation (18), we obtain

[TABLE]

Taking into account that $(A_{\alpha,i,1}+c_{\alpha,1}I)^{-1}\geq\emph{O}$, $L_{\alpha,i,1}\geq\emph{O}$, $R_{\alpha,i,1}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, $\alpha=1,2$, and $\underline{W}^{(1)}_{\alpha,i,1}\geq\mathbf{0}$, by using (19), we have

[TABLE]

By induction on $n$, we prove that

[TABLE]

On the second time level $m=2$, taking into account that $(A_{\alpha,i,2}+c_{\alpha,2}I)^{-1}\geq\emph{O}$, $L_{\alpha,i,2}\geq\emph{O}$, $R_{\alpha,i,2}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, $\alpha=1,2$, $\underline{W}^{(0)}_{\alpha,i,2}=\mathbf{0}$ and $\underline{W}_{\alpha,i,1}\geq\mathbf{0}$, from (5), we have

[TABLE]

For $n=2$ in (5) and using notation (18), we obtain

[TABLE]

Taking into account that $(A_{\alpha,i,2}+c_{\alpha,2}I)^{-1}\geq\emph{O}$, $L_{\alpha,i,2}\geq\emph{O}$, $R_{\alpha,i,2}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, $\alpha=1,2$, and $\underline{W}^{(1)}_{\alpha,i,2}\geq\mathbf{0}$, by using (19), we have

[TABLE]

By induction on $n$, we prove that

[TABLE]

By induction on $m$, we prove that

[TABLE]

Thus, we prove (9) for lower solutions. By following the same manner, we can prove (9) for upper solutions. ∎

6 The case of quasi-monotone nonincreasing reaction functions

6.1 The statement of the block nonlinear difference scheme

We consider the same block nonlinear difference scheme discussed in section 4.1 which is given by (26).

6.2 Block monotone Jacobi and Gauss-Seidel methods

We now present the block monotone Jacobi and block monotone Gauss–Seidel methods for the nonlinear difference scheme (26) in the case of quasi-monotone nonincreasing reaction functions (22).

For solving the nonlinear difference scheme (26), on each time level $t_{m}$, $m\geq 1$, we calculate either the sequence $\{\overline{U}_{1,i,m}^{(n)},\underline{U}_{2,i,m}^{(n)}\}$, or the sequence $\{\underline{U}_{1,i,m}^{(n)},\overline{U}_{2,i,m}^{(n)}\}$, $i=0,1,\ldots,N_{x}$, $m\geq 1$, by the block Jacobi and block Gauss-Seidel methods. In the case of $\{\overline{U}_{1,i,m}^{(n)},\underline{U}_{2,i,m}^{(n)}\}$, we have

[TABLE]

[TABLE]

where $c_{\alpha,m}$, $\alpha=1,2$, $m\geq 1$, are defined in (28), the residuals \mathcal{G}_{\alpha,i,m}\Big{(}\overline{U}_{\alpha,i,m}^{(n-1)},\\ \overline{U}_{\alpha,i,m-1},\underline{U}_{\alpha^{\prime},i,m}^{(n-1)}\Big{)}, $\mathcal{G}_{\alpha,i,m}\left(\underline{U}_{\alpha,i,m}^{(n-1)},\underline{U}_{\alpha,i,m-1},\overline{U}_{\alpha^{\prime},i,m}^{(n-1)}\right)$ are defined in (27), $\mathbf{0}$ is zero column vector with $N_{x}-1$ components. The column vectors $U_{\alpha,i,m}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, are the approximate solutions on time level $m\geq 1$, where $n_{m}$ is a number of iterations on time level $m\geq 1$. For $\eta=0$ and $\eta=1$, we have, respectively, the block Jacobi and block Gauss–Seidel methods.

Theorem 10**.**

Let $f(p,t_{m},U)$ in (9) satisfy (16) and (22), where $\widetilde{U}(p,t_{m})=(\widetilde{U}_{1}(p,t_{m}),\widetilde{U}_{2}(p,t_{m}))$ and $\widehat{U}(p,t_{m})=(\widehat{U}_{1}(p,t_{m}),\widehat{U}_{2}(p,t_{m}))$ are ordered upper and lower solutions (21) of (9). Then the sequences $\{\overline{U}_{1,i,m}^{(n)},\underline{U}_{2,i,m}^{(n)}\}$ and $\{\underline{U}_{1,i,m}^{(n)},\overline{U}_{2,i,m}^{(n-1)}\}$ generated by (77), with $\overline{U}^{(0)}(p,t_{m})=\widetilde{U}(p,t_{m})$ and $\underline{U}^{(0)}(p,t_{m})=\widehat{U}(p,t_{m})$ are ordered upper and lower solutions and converge monotonically, such that,

[TABLE]

Proof.

We consider the case of Gauss-Seidel method $\eta=1$, and the case of the Jacobi method can be proved by a similar manner. On first time level $m=1$, since $\overline{U}^{(0)}$ and $\underline{U}^{(0)}$ are ordered upper and lower solution (21) with respect to $U_{\alpha}(p,0)=\psi_{\alpha}(p)$, from (77a) and (77c), we have

[TABLE]

where $I$ is the identity matrix. For $i=1$ in (6.2), taking into account that $L_{\alpha,i,1}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, and $\overline{Z}^{(1)}_{1,0,1}\leq\mathbf{0}$, $\underline{Z}^{(1)}_{2,0,1}\geq\mathbf{0}$, we have $\left(A_{1,1,1}+c_{1,1}I\right)\overline{Z}_{1,1,1}^{(1)}\leq\mathbf{0}$, $\left(A_{2,1,1}+c_{2,1}I\right)\underline{Z}_{2,1,1}^{(1)}\geq\mathbf{0}$. Taking into account that $d_{\alpha,ij}>0$, $b_{\alpha,ij}$, $t_{\alpha,ij}\geq 0$, $\alpha=1,2$, in (4.1) and $A_{\alpha,i,1}$ are strictly diagonal dominant matrix, we conclude that $A_{\alpha,i,1}$, $i=1,2,\ldots,N_{x}-1$, $\alpha=1,2$, are $M$-matrices and $A_{\alpha,i,1}^{-1}\geq\emph{O}$ (Corollary 3.20, [6]), which leads to $(A_{\alpha,i,1}+c_{\alpha,1}I)^{-1}\geq\emph{O}$, where O is the ($N_{y}-1)\times(N_{y}-1$) null matrix. From here, we obtain that

[TABLE]

From here, for $i=2$ in (6.2), in a similar manner, we conclude that

[TABLE]

By induction on $i$, we can prove that

[TABLE]

From (77b) and (77c), by a similar manner, we prove that

[TABLE]

We now prove that $\overline{U}^{(1)}_{\alpha,i,1}$ and $\underline{U}^{(1)}_{\alpha,i,1}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, satisfy (21a) with respect to the column vector $U_{\alpha,i,0}=\psi_{\alpha,i}$, $i=0,1,\ldots,N_{x}$. Let $W^{(1)}_{\alpha,i,1}=\overline{U}^{(1)}_{\alpha,i,1}-\underline{U}^{(1)}_{\alpha,i,1}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, from (77), we have

[TABLE]

Using notation (18) with $(U_{1},U_{2})=(\overline{U}_{1,i,1}^{(0)},\underline{U}_{2,i,1}^{(0)})$ and $(V_{1},V_{2})=(\underline{U}_{1,i,1}^{(0)},\\ \overline{U}_{2,i,1}^{(0)})$, we present the above problem in the form

[TABLE]

From (23), taking into account that $R_{\alpha,i,1}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, and $W_{\alpha,i,1}^{(0)}\geq\mathbf{0}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, we conclude that

[TABLE]

For $i=1$ in (82), taking into account that $L_{\alpha,i,1}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, $W_{\alpha,0,1}^{(1)}=\mathbf{0}$, and $(A_{\alpha,i,1}+c_{\alpha,1}I)^{-1}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, $\alpha=1,2$, (Corollary 3.20, [6]), we have

[TABLE]

For $i=2$ in (82), taking into account that $W_{\alpha,1,1}^{(1)}\geq\mathbf{0}$, by a similar manner, we obtain

[TABLE]

By induction on $i$, we can prove that

[TABLE]

Now, by induction on $n$, we can prove that

[TABLE]

Thus, we prove (21a) on the first time level $m=1$. We now prove (21b). From (77a) and using (30), we obtain

[TABLE]

where

[TABLE]

From (80), (81) and taking into account that $W^{(1)}_{\alpha,i,1}\geq 0$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, it follows that the partial derivatives in (6.2) satisfy (16) and (22). From (16), (22), (80), (81), (6.2) and taking into account that $R_{1,i,1}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, we conclude that

[TABLE]

Similarly, we conclude that

[TABLE]

By a similar argument, from (77b), we prove that

[TABLE]

Thus, from (84)–(86), it follows (21b) on the first time level $m=1$. By induction on $n$, we can prove (78) on the first time level $m=1$.

On the second time level $m=2$, from (77a) and (78), we have $\overline{U}_{1,i,1}\leq\widetilde{U}_{1,i,1}$, $i=0,1,\ldots,N_{x}$. Thus, it follows that

[TABLE]

which means that $\widetilde{U}_{1,i,2}$ and $\widehat{U}_{2,i,2}$, $i=0,1,\ldots,N_{x}$, are, respectively, upper and lower solutions with respect to $\overline{U}_{1,i,1}$ and $\underline{U}_{1,i,1}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$.

Similarly, we can obtain that

[TABLE]

which means that $\widehat{U}_{1,i,2}$ and $\widetilde{U}_{2,i,2}$, $i=0,1,\ldots,N_{x}$, are, respectively, lower and upper solutions with respect to $\underline{U}_{1,i,1}$ and $\overline{U}_{2,i,1}$, $i=0,1,\ldots,N_{x}$.

From (77a) and (77c), we have

[TABLE]

where $I$ is the identity matrix. For $i=1$ in (6.2), taking into account that $L_{\alpha,i,2}\geq\emph{O}$, $i=1,\ldots,N_{x}-1$, and $\overline{Z}^{(1)}_{1,0,2}\leq\mathbf{0}$, $\underline{Z}^{(1)}_{2,0,2}\geq\mathbf{0}$, we have $\left(A_{1,1,2}+c_{1,2}I\right)\overline{Z}_{1,1,2}^{(1)}\leq\mathbf{0}$, $\left(A_{2,1,2}+c_{2,2}I\right)\underline{Z}_{2,1,2}^{(1)}\geq\mathbf{0}$. Taking into account that $d_{\alpha,ij}>0$, $b_{\alpha,ij}$, $t_{\alpha,ij}\geq 0$, $\alpha=1,2$, in (4.1) and $A_{\alpha,i,2}$ are strictly diagonal dominant matrix, we conclude that $A_{\alpha,i,2}$, $i=1,2,\ldots,N_{x}-1$, $\alpha=1,2$, are $M$-matrices and $A_{\alpha,i,2}^{-1}\geq\emph{O}$ (Corollary 3.20, [6]), which leads to $(A_{\alpha,i,2}+c_{\alpha,2}I)^{-1}\geq\emph{O}$, where O is the ($N_{y}-1)\times(N_{y}-1$) null matrix. From here, we obtain that

[TABLE]

From here, for $i=2$ in (6.2), in a similar manner, we conclude that

[TABLE]

By induction on $i$, we can prove that

[TABLE]

By a similar argument, for $\{\underline{U}^{(n)}_{1,i,2},\overline{U}^{(n)}_{2,i,2}\}$, from (77b) and (77c), we can prove that

[TABLE]

The proof that $\overline{U}^{(1)}_{\alpha,i,2}$ and $\underline{U}^{(1)}_{\alpha,i,2}$, $\alpha=1,2$, are ordered upper and lower solutions (21) repeats the proof on the first time level $m=1$. By induction on $n$, we can prove (78) on the second time level $m=2$. By induction on $m$, we can prove (78) for $m\geq 1$. ∎

6.3 Existence and uniqueness of a solution to the nonlinear difference scheme (26)

In the following theorem, we prove the existence of a solution to (26) based on Theorem 10.

Theorem 11**.**

Let $f(p,t_{m},U)$ satisfy (16), where $\widetilde{U}_{\alpha,i,m}$ and $\widehat{U}_{\alpha,i,m}$, $i=0,1\ldots,N_{x}$, $\alpha=1,2$, $m\geq 1$, be ordered upper and lower solutions (21) to (26). Then a solution of the nonlinear implicit difference scheme (26) exists in $\langle\widehat{U}(t_{m}),\widetilde{U}(t_{m})\rangle$, $m\geq 1$.

Proof.

We consider the Gauss–Seidel method $(\eta=1)$ in (77). On the first time level $t_{1}$, from (78), we conclude that $\lim\overline{U}^{(n)}_{\alpha,i,1}=\overline{V}_{\alpha,i,1}$, $\lim\underline{U}^{(n)}_{\alpha,i,1}=\underline{V}_{\alpha,i,1}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$ as $n\rightarrow\infty$ exist, and

[TABLE]

where $\overline{U}^{(0)}_{\alpha,i,1}=\widetilde{U}_{\alpha,i,1}$, $\underline{U}^{(0)}_{\alpha,i,1}=\widehat{U}_{\alpha,i,1}$. Similar to (6.2), we have

[TABLE]

By taking the limit of both side of (6.3) and using (6.3), we conclude that

[TABLE]

Similarly, we have

[TABLE]

In a similar manner, we can prove that

[TABLE]

From (90)–(6.3), we conclude that $\overline{V}_{1,i,1}$, $\underline{V}_{2,i,1}$ and $\underline{V}_{1,i,1}$, $\overline{V}_{2,i,1}$, $i=0,1,\ldots,\\ N_{x}$, solve (26).

By the assumption of the theorem that $\widetilde{U}_{\alpha,i,2}$, $\widehat{U}_{\alpha,i,2}$ $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, are ordered upper and lower solutions and from (6.3), it follows that $\widetilde{U}_{\alpha,i,2}$ and $\widehat{U}_{\alpha,i,2}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, are upper and lower solutions with respect to, respectively, $\overline{V}_{\alpha,i,1}$ and $\underline{V}_{\alpha,i,1}$ $i=0,1,\ldots,N_{x}$, $\alpha=1,2$. Indeed from (77a) and (6.3), we have

[TABLE]

[TABLE]

By a similar manner, from (77b) and (6.3), we can prove that

[TABLE]

Using a similar argument as in (6.3), we can prove that the limits

[TABLE]

exist and solve (26) on the second time level $m=2$.

By induction on $m$, $m\geq 1$, we can prove that

[TABLE]

Thus, $(\overline{V}_{1,i,m},\underline{V}_{2,i,m})$ and $(\underline{V}_{1,i,m},\overline{V}_{2,i,m})$, $i=0,1,\ldots,N_{x}$, $m\geq 1$, are solutions of the nonlinear difference scheme (26). ∎

We now assume that the reaction functions $f_{\alpha}$, $\alpha=1,2$, satisfy (45) and the two-sided constrains

[TABLE]

[TABLE]

where $q_{\alpha}(p,t_{m})$, $\alpha=1,2$, are nonnegative bounded functions. It is assumed that the time step $\tau$ satisfies the assumptions in (47).

Theorem 12**.**

Suppose that functions $f_{\alpha}(p,t_{m},U)$, $\alpha=1,2$, satisfy (45) and (93), where $\widetilde{U}(p,t_{m})$ and $\widehat{U}(p,t_{m})$ are ordered upper and lower solutions (21) of (9). Let assumption (47) on time step $\tau$ be satisfied. Then the nonlinear difference scheme (9) has a unique solution.

Proof.

To prove the uniqueness of a solution to the nonlinear difference scheme (9), it suffices to prove that

[TABLE]

where $(\overline{V}_{1}(p,t_{m}),\underline{V}_{2}(p,t_{m}))$ and $(\underline{V}_{1}(p,t_{m}),\overline{V}_{2}(p,t_{m}))$, $p\in\overline{\Omega}^{h}$, $m\geq 1$, are the solutions to the nonlinear difference scheme (9), which are defined in the proof of Theorem 11. From (78) and Theorem 11, we obtain

[TABLE]

Letting $W_{\alpha}(p,t_{m})=\overline{V}_{\alpha}(p,t_{m})-\underline{V}_{\alpha}(p,t_{m})$, from (9), we have

[TABLE]

Using the mean-value theorem (13), we obtain

[TABLE]

From here and (6.3), it follows that the partial derivatives satisfy (45) and (93). If $\underline{c}_{1}\geq 0$ in (47), from (95) for $m=1$, using (11), (45), (93) and taking into account that $W_{\alpha}(p,0)=0$, we conclude that

[TABLE]

where

[TABLE]

From here, by the assumption on $\tau$ in (47) and taking into account that $W(t_{m})\geq 0$, we conclude that $W(t_{1})=0$.

If $\underline{c}_{1}<0$ in (47), from (95) for $m=1$, using (11), (45) and (93), we conclude that

[TABLE]

From here, by the assumption on $\tau$ in (47) and taking into account that $W(t_{m})\geq 0$, we conclude that $W(t_{1})=0$.

By induction on $m$, we can prove that $W(t_{m})=0$, $m\geq 1$. Thus, we prove the theorem. ∎

6.4 Convergence analysis

For the sequences $\{\overline{U}^{(n)}_{1,i,m},\underline{U}^{(n)}_{2,i,m}\}$ and $\{\underline{U}^{(n)}_{1,i,m},\overline{U}^{(n)}_{2,i,m}\}$ generated by (77), we introduce the notation

[TABLE]

[TABLE]

where the residuals $\mathcal{G}_{\alpha}\left(U_{\alpha}^{(n)}(p,t_{m}),U_{\alpha}(p,t_{m-1}),U_{\alpha^{\prime}}^{(n)}(p,t_{m})\right)$, $\alpha^{\prime}\neq\alpha$, $\alpha,\alpha^{\prime}\\ =1,2$, are defined in (27), the notation of the norm from (11) is in use.

A stopping test for the block monotone iterative methods (77) is chosen in the following form

[TABLE]

where $\mathcal{G}_{\alpha}(t_{m})$, $\alpha=1,2$, are defined in (96), $\delta$ is a prescribed accuracy. On each time level $t_{m}$, $m\geq 1$, we set up $U_{\alpha}(p,t_{m})=U^{(n_{m})}_{\alpha}(p,t_{m})$, $p\in\Omega^{h}$, $\alpha=1,2$, such that $m_{n}$ is the minimal number of iterations subject to (97).

Theorem 13**.**

Let $\widetilde{U}(p,t_{m})$ and $\widehat{U}(p,t_{m})$ be ordered upper and lower solutions (21) of (9). Suppose that functions $f_{\alpha}(p,t_{m},U)$, $\alpha=1,2$, satisfy (4.4) and (93). Then for the sequences of solutions $\{\overline{U}_{1}^{(n)}(p,t_{m}),\underline{U}^{(n)}_{2}(p,t_{m})\}$ and $\{\underline{U}_{1}^{(n)}(p,t_{m}),\overline{U}^{(n)}_{2}(p,t_{m})\}$ generated by (77), (97), the following estimate holds

[TABLE]

where $U_{\alpha}(p,t_{m})=U_{\alpha}^{(n_{m})}(p,t_{m})$, $n_{m}$ is a minimal number of iterations subject to (97), and $U_{\alpha}^{*}(p,t_{m})$, $\alpha=1,2$, $m\geq 1$, are the unique solutions to the nonlinear difference scheme (9).

Proof.

We consider the case of the sequence $\{\overline{U}_{1}^{(n)}(p,t_{m}),\underline{U}^{(n)}_{2}(p,t_{m})\}$. On a time level $t_{m}$, $m\geq 1$, from (9) for $\overline{U}_{1}(p,t_{m})$, $\underline{U}_{2}(p,t_{m})$ and $U^{*}_{\alpha}(p,t_{m})$, $\alpha=1,2$ , we have

[TABLE]

[TABLE]

[TABLE]

Letting $\overline{W}_{1}(p,t_{m})=\overline{U}_{1}(p,t_{m})-U_{1}^{*}(p,t_{m})$ and $\underline{W}_{2}(p,t_{m})=U_{2}^{*}(p,t_{m})-\underline{U}_{2}(p,t_{m})$, $p\in\overline{\Omega}^{h}$, $m\geq 1$, from here and using the mean-value theorem, we obtain

[TABLE]

[TABLE]

where

[TABLE]

From here, (4.4) and (93), by using (11), we obtain that

[TABLE]

where the notation of the norm from (11) is in use. Letting $W(t_{m})=\max\left\{\|\overline{W}_{1}(\cdot,t_{m})\|_{\overline{\Omega}^{h}},\|\underline{W}_{2}(\cdot,t_{m})\|_{\overline{\Omega}^{h}}\right\}$, from (6.4), we have

[TABLE]

Taking into account that

[TABLE]

it follows that

[TABLE]

From here, taking into account that $W(t_{0})=0$, by induction on $m$, we obtain that

[TABLE]

Thus, we conclude that

[TABLE]

By a similar argument, for the sequence $\{\underline{U}_{1}^{(n)}(p,t_{m}),\overline{U}^{(n)}_{2}(p,t_{m})\}$, we can prove that

[TABLE]

Thus, we prove the theorem. ∎

Theorem 14**.**

Let the assumptions in Theorem 13 be satisfied. Then for the sequences $\{\overline{U}_{1}^{(n)}(p,t_{m}),\underline{U}^{(n)}_{2}(p,t_{m})\}$ and $\{\underline{U}_{1}^{(n)}(p,t_{m}),\overline{U}^{(n)}_{2}(p,t_{m})\}$ generated by (77), (97), the following estimate holds

[TABLE]

where $U_{\alpha}(p,t_{m})=U_{\alpha}^{(n_{m})}(p,t_{m})$, $\alpha=1,2$, $m\geq 1$, $n_{m}$ is the minimal number of iterations subject to the stopping test (97), $u_{\alpha}^{*}(x,y,t)$, $\alpha=1,2$, are the exact solutions to (1), and $E_{\alpha}(p,t_{m})$, $\alpha=1,2$, $m\geq 1$, are the truncation errors of the exact solutions $u_{\alpha}^{*}(x,y,t)$, $\alpha=1,2$, on the nonlinear difference scheme (9).

Proof.

We denote $V(p,t_{m})=u^{*}(p,t_{m})-U^{*}(p,t_{m})$, where the mesh vector function $U^{*}(p,t_{m})$ is the unique solution of the nonlinear difference scheme (9). From (9), by using the mean-value theorem, we obtain that

[TABLE]

where $Y_{\alpha}(p,t_{m})$, $\alpha=1,2$ lie between $u^{*}_{\alpha}(p,t_{m})$ and $U^{*}_{\alpha}(p,t_{m})$, $\alpha=1,2$. From here, (4.4) and (93), by using notation (11), it follows that

[TABLE]

Letting $V_{m}=\max_{\alpha=1,2}\|V_{\alpha}(\cdot,t_{m})\|_{\overline{\Omega}^{h}}$, $m\geq 1$, we have

[TABLE]

From here and taking into account that

[TABLE]

we conclude

[TABLE]

Since $V_{0}=0$, for $m=1$ in (101), we have

[TABLE]

For $m=2$ in (101), we obtain

[TABLE]

and by induction on $m$, we can prove that

[TABLE]

Since $\sum_{\rho=1}^{m}\tau\leq T$, where $T$ is the final time, we have

[TABLE]

We estimate the left hand side in (100) as follows

[TABLE]

where $U_{\alpha}^{*}(p,t_{m})$, $\alpha=1,2$, are the exact solutions of (9). From here and (102), we prove (100). ∎

6.5 Construction of upper and lower solutions

To start the monotone iterative methods (77), on each time level $t_{m}$, $m\geq 1$, initial iterations are needed. In this section, we discuss the construction of initial iterations $\widetilde{U}_{\alpha}(p,t_{m})$ and $\widehat{U}_{\alpha}(p,t_{m})$, $\alpha=1,2$.

6.5.1 Bounded $f_{u}$

Assume that the functions $f_{\alpha}$, $g_{\alpha}$ and $\psi_{\alpha}$, $\alpha=1,2$, in (1) satisfy the conditions

[TABLE]

where $M_{\alpha}$, $\alpha=1,2$, are positive constants and $0_{\alpha}$ means $u_{\alpha}(x,y,t)=0$. We introduce the functions

[TABLE]

and the linear problems

[TABLE]

Theorem 15**.**

*Let the assumptions in (6.5.1) be satisfied. Then $\widehat{U}_{\alpha}$, $\alpha=1,2$, from (104) and solutions $\widetilde{U}_{\alpha}$, $\alpha=1,2$, of the linear problems (105) are ordered lower and upper solutions (21) to (9). *

Proof.

From (6.5.1) and (105) with $m=1$, by using the maximum principle in Lemma 1, we obtain that

[TABLE]

From here and (105) with $m=2$, by using the maximum principle in Lemma 1, we have

[TABLE]

By induction on $m$, we can prove that

[TABLE]

From here and (104), we prove (21a).

We now prove (21b) for $(\widetilde{U}_{1}(p,t_{m}),\widehat{U}_{2}(p,t_{m}))$. We present the left hand side of (21b) in the form

[TABLE]

Using (105) for $m\geq 1$, we obtain that

[TABLE]

From here and using (6.5.1), it follows that

[TABLE]

Similarly, we can prove that

[TABLE]

Thus, we prove (21b) for $(\widetilde{U}_{1}(p,t_{m}),\widehat{U}_{2}(p,t_{m}))$. By following a similar argument, we can prove (21b) for $(\widehat{U}_{1}(p,t_{m}),\widetilde{U}_{2}(p,t_{m}))$, that is,

[TABLE]

Since $\widetilde{U}_{\alpha}(p,t_{m})$, $\alpha=1,2$, satisfy the boundary and initial conditions (21c), $\widehat{U}_{\alpha}(p,0)$, $\alpha=1,2$, satisfy the initial condition and $\widehat{U}_{\alpha}(p,t_{m})=0\leq g_{\alpha}(p,t_{m})$, $p\in\partial\Omega^{h}$, $\alpha=1,2$, we conclude that $\widehat{U}_{\alpha}(p,t_{m})$ and $\widetilde{U}_{\alpha}(p,t_{m})$, $\alpha=1,2$, from, respectively, (104) and (105), are ordered lower and upper solutions (21) to (9). ∎

6.5.2 Constant upper and lower solutions

Let the functions $f_{\alpha}$, $g_{\alpha}$ and $\psi_{\alpha}$, $\alpha=1,2$, in (1) satisfy the conditions

[TABLE]

where $K_{1}$, $K_{2}$ are positive constants. Introduce the mesh functions

[TABLE]

Theorem 16**.**

Suppose that the assumptions in (6.5.2) are satisfied. Then the mesh functions $\widehat{U}_{\alpha}(p,t_{m})$ and $\widetilde{U}_{\alpha}(p,t_{m})$, $\alpha=1,2$, from, respectively, (104) and (108), are ordered lower and upper solutions (21) to (9).

Proof.

From (104) and (108), it is clear that the inequalities in (21a) are satisfied. We now prove (21b) for $(\widetilde{U}_{1}(p,t_{m}),\widehat{U}_{2}(p,t_{m}))$. Using (108), we write the left hand side of (21b) for $m=1$ in the form

[TABLE]

From here and (6.5.2), we conclude that

[TABLE]

From (6.5.2) and (108), using (21b) for $m\geq 2$, we have

[TABLE]

Similarly, we can prove

[TABLE]

Thus, we prove (21b) for $(\widetilde{U}_{1}(p,t_{m}),\widehat{U}_{2}(p,t_{m}))$.

By a similar argument, we can prove (21b) for $(\widehat{U}_{1}(p,t_{m}),\widetilde{U}_{2}(p,t_{m}))$, that is,

[TABLE]

Since $\widetilde{U}_{\alpha}(p,t_{0})$, $\widehat{U}_{\alpha}(p,t_{0})$ , $\alpha=1,2$, satisfy the initial condition and $\widetilde{U}_{\alpha}(p,t_{m})\\ \geq g_{\alpha}(p,t_{m})$, $\widehat{U}_{\alpha}(p,t_{m})\leq g_{\alpha}(p,t_{m})$, $p\in\partial\Omega^{h}$, $\alpha=1,2$, at $m\geq 1$, we conclude that $\widehat{U}$ and $\widetilde{U}$ from, respectively, (104) and (108), are ordered lower and upper solutions (21) to (9). ∎

6.6 Applications

6.7 The Belousov-Zhabotinskii reaction diffusion system

The Belousov-Zhabotinskii reaction diffusion model [4] includes the metal-ion-catalyzed oxidation by bromate ion of brominated organ materials. the chemical reaction scheme is given by

[TABLE]

where $A_{1}$ and $A_{2}$ are constants which represent reactants, $P_{1}$ and $P_{2}$ are products, $\lambda$ is the stoichiometric factor, and $X$, $Y$ and $Z$ are, respectively, the concentrations of the intermediates HBrO2 (bromous acid), B$\mbox{r}^{-}$ (bromide ion) and Ce(IV)(cerium). A simplified system of two equations [2] of the above reactant scheme is governed by (1) with $L_{\alpha}u_{\alpha}=\varepsilon_{\alpha}\triangle u_{\alpha}$, $\alpha=1,2$, where $u_{1}$ and $u_{2}$ represent, respectively, the concentrations $X$ and $Y$. The reaction functions are given by

[TABLE]

where $a$, $b$, $\sigma_{\alpha}$, $\alpha=1,2$, are positive constants. It is clear from (109) that $f_{\alpha}$, $\alpha=1,2$, are quasi-monotone nonincreasing functions (22). The nonlinear difference scheme (9) is reduced to

[TABLE]

where $f_{\alpha}$, $\alpha=1,2$, are defined in (109). To satisfy the assumptions in (6.5.2), we choose constants $K_{\alpha}$, $\alpha=1,2$, in the following form

[TABLE]

it follows that the mesh functions $\widehat{U}_{\alpha}(p,t_{m})$ and $\widetilde{U}_{\alpha}(p,t_{m})$ from, respectively, (104) and (108) are ordered lower and upper solutions to (6.7).

From (109), in the sector $\langle\widehat{U}(t_{m}),\widetilde{U}(t_{m})\rangle=\langle 0,K_{\alpha}\rangle$, we have

[TABLE]

and the assumptions in (16) and (22) are satisfied with

[TABLE]

From here, (104) and (108), we conclude that Theorem 10 holds for the Belousov-Zhabotinskii reaction diffusion model (6.7).

6.8 Enzyme-substrate reaction diffusion model

In the enzyme-substrate model [4], the chemical reaction scheme is given by $E+S\rightleftharpoons ES\rightarrow E+P$, where $E$, $S$ and $P$ are, respectively, enzyme, substrate and reaction product. Denote by $u_{1}(x,y,t)$ and $u_{2}(x,y,t)$ the concentrations of $S$ and $E$, respectively. Then the above reactant scheme is governed by (1) with $L_{\alpha}u_{\alpha}=\varepsilon_{\alpha}\triangle u_{\alpha}$, $\alpha=1,2$. The reaction functions are given by

[TABLE]

where a positive constant $E_{0}$ is the total enzyme, $a_{\alpha}>0$, $b_{\alpha}>0$, $\alpha=1,2$, are reaction constants. It is clear from (111) that $f_{\alpha}$, $\alpha=1,2$, are quasi-monotone nonincreasing functions (22). The nonlinear difference scheme (9) is reduced to

[TABLE]

where $f_{\alpha}$, $\alpha=1,2$, are defined in (111).

Introduce the linear problem

[TABLE]

We now prove that $(V(p,t_{m}),E_{0})$ and $(0,0)$ are ordered upper and lower solutions (21) to (6.8). Firstly, we prove that $V(p,t_{m})\geq 0$. From (6.8), for $m=1$, we obtain that

[TABLE]

From here and taking into account that $\psi_{1}(p)\geq 0$, we have

[TABLE]

Using the maximum principle in Lemma 1, we conclude that

[TABLE]

From here and (6.8), for $m=2$, by a similar manner, we conclude that

[TABLE]

By induction on $m$, we can prove that $V(p,t_{m})\geq 0$, $p\in\overline{\Omega}^{h}$, $m\geq 1$. From here, taking into account that the total enzyme $E_{0}>0$ and (104), it follows that $(V(p,t_{m}),E_{0})$ and $(0,0)$ satisfy (21a). We now prove (21b) for $(\widetilde{U}_{1}(p,t_{m}),\widehat{U}_{2}(p,t_{m}))=(V(p,t_{m}),0)$. From (21b), by using (6.8), we obtain that

[TABLE]

From here, (111) and (6.8), we conclude that

[TABLE]

Similarly, we prove that

[TABLE]

Thus, we prove (21b) for $(\widetilde{U}_{1}(p,t_{m}),\widehat{U}_{2}(p,t_{m}))=(V(p,t_{m}),0)$.

Now, from (21b) for $(\widehat{U}_{1}(p,t_{m}),\widetilde{U}_{2}(p,t_{m}))=(0,E_{0})$, we have

[TABLE]

From here and (111), we conclude that

[TABLE]

Similarly, we obtain that

[TABLE]

Thus, we prove (21b) for $(\widehat{U}_{1}(p,t_{m}),\widetilde{U}_{2}(p,t_{m}))=(0,E_{0})$.

Taking into account that the total enzyme $E_{0}$ satisfies $E_{0}\geq u_{2}$, we conclude that $(\widetilde{U}_{1},\widetilde{U}_{2})=(V,E_{0})$ and $(\widehat{U}_{1},\widehat{U}_{2})=(0,0)$ satisfy (21c). Thus, we prove that $(\widetilde{U}_{1},\widetilde{U}_{2})=(V,E_{0})$ and $(\widehat{U}_{1},\widehat{U}_{2})=(0,0)$ are ordered upper and lower solutions (21) to (6.8). From (111), in the sector $\langle\widehat{U}(t_{m}),\widetilde{U}(t_{m})\rangle$, $\widehat{U}=(0,0)$, $\widetilde{U}=(V,E_{0})$, we have

[TABLE]

Thus, the assumptions in (16) and (22) are satisfied with

[TABLE]

From here, (104) and (6.8), we conclude that Theorem 10 holds for the enzyme-substrate reaction diffusion model (6.8).

7 Comparison of the block monotone Jacobi and block monotone Gauss–Seidel methods

The following theorem shows that the block monotone Gauss–Seidel method (77), $(\eta=1)$, converges not slower than the block monotone Jacobi method (29), $(\eta=0)$.

Theorem 17**.**

Let $f(p,t_{m},U)$ in (9) satisfy (16) and (22), where $\widetilde{U}(p,t_{m})=(\widetilde{U}_{1}(p,t_{m}),\widetilde{U}_{2}(p,t_{m}))$ and $\widehat{U}(p,t_{m})=(\widehat{U}_{1}(p,t_{m}),\widehat{U}_{2}(p,t_{m}))$ are ordered upper and lower solutions (21) of the nonlinear difference scheme (9). Suppose that $\{(\overline{U}^{(n)}_{1,i,m})_{J},(\underline{U}^{(n)}_{2,i,m})_{J}\}$, $\{(\underline{U}^{(n)}_{1,i,m})_{J},(\overline{U}^{(n)}_{2,i,m})_{J}\}$ and $\{(\overline{U}^{(n)}_{1,i,m})_{GS},(\underline{U}^{(n)}_{2,i,m})_{GS}\}$, $\{(\underline{U}^{(n)}_{1,i,m})_{GS},(\overline{U}^{(n)}_{2,i,m})_{GS}\}$ , $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, $m\geq 1$, are, respectively, the sequences generated by the block monotone Jacobi method (77), $(\eta=0)$ and the block monotone Gauss–Seidel method (77), $(\eta=1)$, where $(\overline{U}^{(0)})_{J}=(\overline{U}^{(0)})_{GS}=\widetilde{U}$ and $(\underline{U}^{(0)})_{J}=(\underline{U}^{(0)})_{GS}=\widehat{U}$, then

[TABLE]

Proof.

We consider the case of the sequences $\{(\overline{U}^{(n)}_{1,i,m})_{J},(\underline{U}^{(n)}_{2,i,m})_{J}\}$ and $\{(\overline{U}^{(n)}_{1,i,m})_{GS},(\underline{U}^{(n)}_{2,i,m})_{GS}\}$. From (29), we have

[TABLE]

[TABLE]

From here, letting $W^{(n)}_{\alpha,i,m}=\left(U^{(n)}_{\alpha,i,m}\right)_{GS}-\left(U^{(n)}_{\alpha,i,m}\right)_{J}$, $i=0,1,\ldots,N_{x}$, $\alpha=1,2$, $m\geq 1$, we have

[TABLE]

By using Theorem 10, we have $\left(\overline{U}^{(n)}_{1,i,m}\right)_{GS}\leq\left(\overline{U}^{(n-1)}_{1,i,m}\right)_{GS}$, $i=0,1,\ldots,N_{x}$, $m\geq 1$. From here and (7), we obtain

[TABLE]

Taking into account that $(A_{1,i,m}+c_{1,m}I)^{-1}\geq\emph{O}$, $L_{1,i,m}\geq\emph{O}$, $R_{1,i,m}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, $m\geq 1$, for $n=1$ in (7), on the first time level $m=1$, in view of $(\overline{U}^{(0)}_{1,i,m})_{GS}=(\overline{U}^{(0)}_{1,i,m})_{J}$ and $\overline{W}^{(0)}_{1,i,m}=\mathbf{0}$, we conclude that

[TABLE]

For $n=2$ in (7) and using notation (18), we obtain

[TABLE]

Taking into account that $(A_{1,i,1}+c_{1,1}I)^{-1}\geq\emph{O}$, $L_{1,i,1}\geq\emph{O}$, $R_{1,i,1}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, and $\overline{W}^{(1)}_{1,i,1}\leq\mathbf{0}$, by using (23), we have

[TABLE]

where $U$ and $V$ in (23) are taken in the form

[TABLE]

By induction on $n$, we can prove that

[TABLE]

Similarly, by using the property $\left(\underline{U}^{(n-1)}_{2,i,m}\right)_{GS}\leq\left(\underline{U}^{(n)}_{2,i,m}\right)_{GS}$ in Theorem 10, we prove that

[TABLE]

On the second time level $m=2$, taking into account that $(A_{1,i,2}+c_{1,2}I)^{-1}\geq\emph{O}$, $L_{1,i,2}\geq\emph{O}$, $R_{1,i,2}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, $\overline{W}^{(0)}_{1,i,2}=\mathbf{0}$ and $\overline{W}_{1,i,1}\leq\mathbf{0}$, from (7), we have

[TABLE]

For $n=2$ in (7) and using notation (18), we obtain

[TABLE]

Taking into account that $(A_{1,i,2}+c_{1,2}I)^{-1}\geq\emph{O}$, $L_{1,i,2}\geq\emph{O}$, $R_{1,i,2}\geq\emph{O}$, $i=1,2,\ldots,N_{x}-1$, $\overline{W}^{(1)}_{1,i,2}\leq\mathbf{0}$ and $\overline{W}_{1,i,1}\leq\mathbf{0}$, by using (23), we have

[TABLE]

where $U$ and $V$ in (23) are taken similar to (117) with $m=2$.

By induction on $n$, we can prove that

[TABLE]

By induction on $m$, we can prove that

[TABLE]

By a similar argument, we can prove that

[TABLE]

Thus, we prove (17) for $\{(\overline{U}^{(n)}_{1,i,m})_{J},(\underline{U}^{(n)}_{2,i,m})_{J}\}$ and $\{(\overline{U}^{(n)}_{1,i,m})_{GS},(\underline{U}^{(n)}_{2,i,m})_{GS}\}$. By a similar manner, we can prove (17) for $\{(\underline{U}^{(n)}_{1,i,m})_{J},(\overline{U}^{(n)}_{2,i,m})_{J}\}$ and

$\{(\underline{U}^{(n)}_{1,i,m})_{GS},(\overline{U}^{(n)}_{2,i,m})_{GS}\}$. ∎