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1 Introduction

In this paper we give a numerical treatment for the following semi-linear singularly perturbed parabolic system:

$$\displaystyle{ \frac{\partial u_{i}} {\partial t} -\varepsilon _{i}\frac{\partial ^{2}u_{i}} {\partial x^{2}} + f_{i}(x,t,u) = 0,\quad (x,t) \in \omega \times (0,T], }$$
(1)
$$\displaystyle{u_{i}(0,t) = 0,\quad u_{i}(1,t) = 0,\quad t \in [0,T],}$$
$$\displaystyle{u_{i}(x,0) =\psi _{i}(x),\quad x \in \overline{\omega },\quad \omega = (0,1),\quad i = 1,2,}$$

where 0 < ɛ 1 ≤ ɛ 2 ≤ 1, u ≡ (u 1, u 2), the functions f i and ψ i , i = 1, 2, are smooth in their respective domains.

In the study of numerical methods for nonlinear singularly perturbed problems, the two major points to be developed are: (1) constructing robust difference schemes (this means that unlike classical schemes, the error does not increase to infinity, but rather remains bounded, as the small parameters approach zero); (2) obtaining reliable and efficient computing algorithms for solving nonlinear discrete problems. For solving these nonlinear discrete systems, the iterative approach presented in this paper is based on the method of upper and lower solutions and associated monotone iterates. The basic idea of the method of upper and lower solutions is the construction of two monotone sequences which converge monotonically from above and below to a solution of the problem. The monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration. An initial iteration in the monotone iterative method is either an upper or lower solution, which can be constructed directly from the difference equation, this method simplifies the search for the initial iteration as is often required in Newton’s method.

In [5], uniformly convergent numerical methods for solving linear singularly perturbed systems of type (1) were constructed. These uniform numerical methods are based on the piecewise uniform meshes of Shishkin-type [6].

In [2], we investigated uniform convergence properties of the monotone iterative method for solving scalar nonlinear singularly perturbed problems of type (1). In this paper, we extend our investigation to the case of the nonlinear singularly perturbed system (1).

The structure of the paper as follows. In Sect. 2, we introduce a nonlinear difference scheme for solving (1). The monotone iterative method is presented in Sect. 3. An analysis of the uniform convergence of the monotone iterates to the solution of the nonlinear difference scheme and to the solution of (1) is given in Sect. 4. The final Sect. 5 presents the results of numerical experiments with a gas-liquid interaction model.

2 The Nonlinear Difference Scheme

On \(\overline{\omega } = [0,1]\) and [0, T], we introduce meshes \(\overline{\omega }^{h}\) and \(\overline{\omega }^{\tau }\):

$$\displaystyle{\overline{\omega }^{h} =\{ x_{ m},0 \leq m \leq M_{x};x_{0} = 0,x_{M_{x}} = 1;h_{m} = x_{m+1} - x_{m}\},}$$
$$\displaystyle{\overline{\omega }^{\tau } =\{ t_{k},0 \leq k \leq N_{\tau };t_{0} = 0,t_{N_{\tau }} = T;\tau _{k} = t_{k} - t_{k-1}\},}$$

and consider the nonlinear implicit difference scheme

$$\displaystyle{ \mathcal{L}_{i}U_{i}(x_{m},t_{k}) + f_{i}(x_{m},t_{k},U) -\tau _{k}^{-1}U_{ i}(x_{m},t_{k-1}) = 0,\quad (x_{m},t_{k}) \in \omega ^{h} \times \omega ^{\tau }, }$$
(2)
$$\displaystyle{\mathcal{L}_{i}U_{i}(x_{m},t_{k}) \equiv -\varepsilon _{i}\mathcal{L}_{i}^{h}U_{ i}(x_{m},t_{k}) +\tau _{ k}^{-1}U_{ i}(x_{m},t_{k}).}$$
$$\displaystyle{U_{i}(x_{0},t_{k}) = U_{i}(x_{M_{x}},t_{k}) = 0,\quad U_{i}(x_{m},0) =\psi _{i}(x_{m}),\quad x_{m} \in \overline{\omega }^{h},\quad i = 1,2,}$$

where U ≡ (U 1, U 2), and the difference operators \(\mathcal{L}_{i}^{h}\), i = 1, 2, are defined by

$$\displaystyle{\mathcal{L}_{i}^{h}U_{ i}(x_{m},t_{k}) = \left [\frac{U_{i}(x_{m+1},t_{k}) - U_{i}(x_{m},t_{k})} {\hslash _{m}h_{m}} -\frac{U_{i}(x_{m},t_{k}) - U_{i}(x_{m-1},t_{k})} {\hslash _{m}h_{m-1}} \right ],}$$
$$\displaystyle{\hslash _{m} = (h_{m} + h_{m-1})/2,\quad i = 1,2.}$$

On each time level t k , k ≥ 1, we introduce the linear problems

$$\displaystyle{ (\mathcal{L}_{i} + c_{i})W_{i}(x_{m},t_{k}) =\varPhi _{i}(x_{m},t_{k}),\quad W_{i}(x_{0},t_{k}) = W_{i}(x_{M_{x}},t_{k}) = 0, }$$
(3)
$$\displaystyle{c_{i}(x_{m},t_{k}) \geq 0,\quad x_{m} \in \omega ^{h},\quad i = 1,2.}$$

In the following lemma, we state the maximum principle and we give estimates on solutions of (3) from [8].

Lemma 1

  1. (i)

    If mesh functions W i (x m , t k ), i = 1, 2, satisfy the conditions

    $$\displaystyle{(\mathcal{L}_{i} + c_{i})W_{i}(x_{m},t_{k}) \geq 0\ (\leq 0),\quad x_{m} \in \omega ^{h},}$$
    $$\displaystyle{W_{i}(x_{0},t_{k}) \geq 0\ (\leq 0),\quad W_{i}(x_{M_{x}},t_{k}) \geq 0\ (\leq 0),}$$

    then W i (x m , t k ) ≥ 0 ( ≤ 0) in \(\overline{\omega }^{h}\) , i = 1, 2.

  2. (ii)

    The following estimates on the solutions of ( 3 ) hold true

    $$\displaystyle{ \|W_{i}(\cdot,t_{k})\|_{\overline{\omega }^{h}} \leq \max _{x_{m}\in \omega ^{h}}\left \{ \frac{\vert \varPhi _{i}(x_{m},t_{k})\vert } {c_{i}(x_{m},t_{k}) +\tau _{ k}^{-1}}\right \},\quad i = 1,2, }$$
    (4)

    where \(\|W_{i}(\cdot,t_{k})\|_{\overline{\omega }^{h}} =\max _{x_{m}\in \overline{\omega }^{h}}\vert W_{i}(x_{m},t_{k})\vert \) .

3 The Monotone Iterative Method

We say that the mesh functions

$$\displaystyle{\widetilde{U}(x_{m},t_{k}) = (\widetilde{U}_{1}(x_{m},t_{k}),\widetilde{U}_{2}(x_{m},t_{k})),\quad \widehat{U}(x_{m},t_{k}) = (\widehat{U}_{1}(x_{m},t_{k}),\widehat{U}_{2}(x_{m},t_{k}))}$$

are ordered upper and lower solutions if they satisfy the following inequalities:

$$\displaystyle{\widetilde{U}(x_{m},t_{k}) \geq \widehat{ U}(x_{m},t_{k}),\quad (x_{m},t_{k}) \in \overline{\omega }^{h} \times \omega ^{\tau },}$$
$$\displaystyle{\mathcal{L}_{i}\widetilde{U}_{i}(x_{m},t_{k}) + f_{i}(x_{m},t_{k},\widetilde{U}) -\tau _{k}^{-1}\widetilde{U}_{ i}(x_{m},t_{k-1}) \geq 0,\quad (x_{m},t_{k}) \in \omega ^{h} \times \omega ^{\tau },}$$
$$\displaystyle{\mathcal{L}_{i}\widehat{U}_{i}(x_{m},t_{k}) + f_{i}(x_{m},t_{k},\widehat{U}) -\tau _{k}^{-1}\widehat{U}_{ i}(x_{m},t_{k-1}) \leq 0,\quad (x_{m},t_{k}) \in \omega ^{h} \times \omega ^{\tau },}$$
$$\displaystyle{\widehat{U}_{i}(x_{{\ast}},t_{k}) \leq 0 \leq \widetilde{ U}_{i}(x_{{\ast}},t_{k}),\quad x_{{\ast}} = x_{0},x_{M_{x}},}$$
$$\displaystyle{\widehat{U}_{i}(x_{m},0) \leq \psi _{i}(x_{m}) \leq \widetilde{ U}_{i}(x_{m},0),\quad x_{m} \in \overline{\omega }^{h},\quad i = 1,2.}$$

We introduce the notation

$$\displaystyle{\langle \widehat{U}(t_{k}),\widetilde{U}(t_{k})\rangle =\{ U(x_{m},t_{k}):\widehat{ U}(x_{m},t_{k}) \leq U(x_{m},t_{k}) \leq \widetilde{ U}(x_{m},t_{k}),\ x_{m} \in \overline{\omega }^{h}\},}$$

and we assume that on each time level t k , k ≥ 1, the reaction functions satisfy the assumptions

$$\displaystyle{ 0 \leq \frac{\partial f_{i}} {\partial u_{i}}(x_{m},t_{k},U) \leq c_{i}(x_{m},t_{k}),\quad \mbox{ on}\ \langle \widehat{U}(t_{k}),\widetilde{U}(t_{k})\rangle, }$$
(5)
$$\displaystyle{0 \leq -\frac{\partial f_{i}} {\partial u_{i^{{\prime}}}}(x_{m},t_{k},U) \leq q_{i}(x_{m},t_{k}),\quad \mbox{ on}\ \langle \widehat{U}(t_{k}),\widetilde{U}(t_{k})\rangle,\quad i^{{\prime}}\neq i,}$$

where c i (x m , t k ) and q i (x m , t k ), i = 1, 2, are nonnegative bounded functions in \(\overline{\omega }^{h}\).

On each time level t k , k ≥ 1, the iterative method is given in the form

$$\displaystyle{ (\mathcal{L}_{i} + c_{i})Z_{i}^{(n)}(x_{ m},t_{k}) = -\mathcal{R}_{i}(x_{m},t_{k},U^{(n-1)}),\quad x_{ m} \in \omega ^{h}, }$$
(6)
$$\displaystyle{\mathcal{R}_{i}(x_{m},t_{k},U^{(n-1)}) \equiv \mathcal{L}_{ i}U_{i}^{(n-1)}(x_{ m},t_{k})+f_{i}(x_{m},t_{k},U^{(n-1)})-\tau _{ k}^{-1}U_{ i}(x_{m},t_{k-1}),}$$
$$\displaystyle{Z_{i}^{(n)}(x_{ {\ast}},t_{k}) = 0,\quad n \geq 1,\quad x_{{\ast}} = x_{0},x_{M_{x}},}$$
$$\displaystyle{Z_{i}^{(n)}(x_{ m},t_{k}) \equiv U_{i}^{(n)}(x_{ m},t_{k}) - U_{i}^{(n-1)}(x_{ m},t_{k}),}$$
$$\displaystyle{U_{i}(x_{m},0) =\psi _{i}(x_{m}),\quad x_{m} \in \overline{\omega }^{h},\quad i = 1,2,}$$

where c i , i = 1, 2, are defined in (5). For upper sequence, we have \(\overline{U}_{i}(x_{m},0) =\psi _{i}(x_{m})\), \(\overline{U}^{(0)}(x_{m},t_{k}) =\widetilde{ U}_{i}(x_{m},t_{k})\) and \(\overline{U}_{i}(x_{m},t_{k}) = \overline{U}_{i}^{(n_{k})}(x_{ m},t_{k})\), i = 1, 2, \(x_{m} \in \overline{\omega }^{h}\), where \(\overline{U}_{i}(x_{m},t_{k})\), i = 1, 2, are approximations of the exact solutions on time level t k and n k is a number of iterative steps on time level t k . For lower sequence, we have \(\underline{U}_{i}(x_{m},0) =\psi _{i}(x_{m})\), \(\underline{U}^{(0)}(x_{m},t_{k}) =\widehat{ U}_{i}(x_{m},t_{k})\) and \(\underline{U}(x_{m},t_{k}) =\underline{ U}^{(n_{k})}(x_{ m},t_{k})\), i = 1, 2, \(x_{m} \in \overline{\omega }^{h}\).

The following theorem gives the monotone property of the iterative method (6).

Theorem 1

Let \(\widetilde{U}\) and \(\widehat{U}\) be ordered upper and lower solutions, and assumption ( 5 ) be satisfied. On each time level t k , k ≥ 1, the sequences \(\{\overline{U}^{(n)}\}\) , \(\{\underline{U}^{(n)}\}\) with \(\overline{U}^{(0)} =\widetilde{ U}\) and \(\underline{U}^{(0)} =\widehat{ U}\) , generated by the iterative method ( 6 ), converge monotonically

$$\displaystyle{ \underline{U}^{(n-1)}(x_{ m},t_{k}) \leq \underline{ U}^{(n)}(x_{ m},t_{k}) \leq \overline{U}^{(n)}(x_{ m},t_{k}) \leq \overline{U}^{(n-1)}(x_{ m},t_{k}),\quad x_{m} \in \overline{\omega }^{h}, }$$
(7)

Proof

Since \(\overline{U}^{(0)} =\widetilde{ U}\) and \(\underline{U}^{(0)} =\widehat{ U}\), then from (6) we conclude that

$$\displaystyle{(\mathcal{L}_{i} + c_{i})\overline{Z}_{i}^{(1)}(x_{ m},t_{1}) \leq 0,\quad (\mathcal{L}_{i} + c_{i})\underline{Z}_{i}^{(1)}(x_{ m},t_{1}) \geq 0,\quad x_{m} \in \omega ^{h},}$$
$$\displaystyle{\overline{Z}_{i}^{(1)}(x_{ {\ast}},t_{1}) \leq 0,\quad \underline{Z}_{i}^{(1)}(x_{ {\ast}},t_{1}) \geq 0,\quad x_{{\ast}} = x_{0},x_{M_{x}},\quad i = 1,2.}$$

From Lemma 1, it follows that

$$\displaystyle{ \overline{Z}_{i}^{(1)}(x_{ m},t_{1}) \leq 0,\quad \underline{Z}_{i}^{(1)}(x_{ m},t_{1}) \geq 0\quad x_{m} \in \overline{\omega }^{h},\quad i = 1,2. }$$
(8)

We now prove (7) for n = 1 and k = 1. From (6), in the notation \(W_{i}^{(n)} = \overline{U}_{i}^{(n)} -\underline{ U}_{i}^{(n)}\), n ≥ 0, i = 1, 2, we conclude that

$$\displaystyle{ (\mathcal{L}_{i} + c_{i})W_{i}^{(1)}(x_{ m},t_{1}) = F_{i}(x_{m},t_{1},\overline{U}^{(0)}) - F_{ i}(x_{m},t_{1},\underline{U}^{(0)}),\quad x_{ m} \in \omega ^{h}, }$$
$$\displaystyle{W_{i}^{(1)}(x_{ {\ast}},t_{1}) = 0,\quad x_{{\ast}} = x_{0},x_{M_{x}},\quad i = 1,2,}$$

where F i (x k , t k , U) = c i (x m , t k )U i (x m , t k ) − f i (x m , t k , U). Since \(\overline{U}^{(0)}(x_{m},t_{1}) \geq \underline{ U}^{(0)}(x_{m},t_{1})\), by Lemma 2 from [1], we conclude that the right hand sides in the difference equations are nonnegative. From Lemma 1, it follows W i (1)( p, t 1) ≥ 0, i = 1, 2, and this leads to (7) for n = 1, k = 1.

Using the mean-value theorem, from (6) we obtain

$$\displaystyle{ \mathcal{R}_{i}(x_{m},t_{1},\overline{U}^{(1)}) = -\left (c_{ i} -\frac{\partial f_{i}} {\partial u_{i}}\right )\overline{Z}_{i}^{(1)}(x_{ m},t_{1}) + \frac{\partial f_{i}} {\partial u_{i^{{\prime}}}}\overline{Z}_{i^{{\prime}}}^{(1)}(x_{ m},t_{1}),\quad i^{{\prime}}\neq i, }$$
(9)

where the partial derivatives are calculated at intermediate points which lie in the sector \(\langle \overline{U}^{(1)}(t_{1}),\overline{U}^{(0)}(t_{1})\rangle\). From (5) and (8), we conclude that

$$\displaystyle{\mathcal{R}_{i}(x_{m},t_{1},\overline{U}^{(1)}) \geq 0,\quad x_{ m} \in \omega ^{h},\quad \overline{U}_{ i}^{(1)}(x_{ {\ast}},t_{1}) = 0,\quad x_{{\ast}} = x_{0},x_{M_{x}},\ i = 1,2.}$$

Thus, \(\overline{U}^{(1)}(x_{m},t_{1})\) is an upper solution. Similarly, we prove that \(\underline{U}^{(1)}(x_{m},t_{1})\) is a lower solution. By induction on n, we can prove that \(\{\overline{U}^{(n)}(x_{m},t_{1})\}\) and \(\{\underline{U}^{(n)}(\,p,t_{1})\}\) are, respectively monotonically decreasing and monotonically increasing sequences.

From (7) with t 1, it follows that for i = 1, 2,

$$\displaystyle{ \widehat{U}_{i}(x_{m},t_{1}) \leq \underline{ U}_{i}^{(n_{1})}(x_{ m},t_{1}) \leq \overline{U}_{i}^{(n_{1})}(x_{ m},t_{1}) \leq \widetilde{ U}_{i}(x_{m},t_{1}),\quad x_{m} \in \overline{\omega }^{h}. }$$
(10)

From here and by the assumption of the theorem that \(\widetilde{U}(\,p,t_{2})\) and \(\widehat{U}(\,p,t_{2})\) are, respectively, upper and lower solutions, we conclude that \(\widetilde{U}(x_{m},t_{2})\) and \(\widehat{U}(x_{m},t_{2})\) are upper and lower solutions with respect to \(\overline{U}^{(n_{1})}(x_{m},t_{1})\) and \(\underline{U}^{(n_{1})}(x_{m},t_{1})\).

From (6), we conclude that W (1)(x m , t 2) satisfies

$$\displaystyle\begin{array}{rcl} (\mathcal{L}_{i} + c_{i})W_{i}^{(1)}(x_{ m},t_{2})& =& F_{i}(x_{m},t_{2},\overline{U}^{(0)}) - F_{ i}(x_{m},t_{2},\underline{U}^{(0)}) + {}\\ & & \tau _{2}^{-1}[\overline{U}_{ i}^{(n_{1})}(x_{ m},t_{1}) -\underline{ U}_{i}^{(n_{1})}(x_{ m},t_{1})], {}\\ \end{array}$$
$$\displaystyle{x_{m} \in \omega ^{h},\quad W_{ i}^{(1)}(x_{ {\ast}},t_{2}) = 0,\quad x_{{\ast}} = x_{0},x_{M_{x}},\quad i = 1,2.}$$

Since \(\overline{U}^{(0)}(x_{m},t_{2}) \geq \underline{ U}^{(0)}(x_{m},t_{2})\) and taking into account (10), by Lemma 2 from [1], we conclude that the right hand sides in the difference equations are nonnegative. From Lemma 1, we have W i (1)( p, t 2) ≥ 0, i = 1, 2, that is,

$$\displaystyle{ \underline{U}_{i}^{(1)}(\,p,t_{ 2}) \leq \overline{U}_{i}^{(1)}(\,p,t_{ 2}),\quad p \in \overline{\omega }^{h},\quad i = 1,2. }$$

The proof that \(\overline{U}_{i}^{(1)}(x_{m},t_{2})\) and \(\underline{U}_{i}^{(1)}(x_{m},t_{2})\), i = 1, 2, are, respectively, upper and lower solutions is similar to the proof on the time level t 1. By induction on n, we can prove that \(\{\overline{U}^{(n)}(x_{m},t_{2})\}\) and \(\{\underline{U}^{(n)}(x_{m},t_{2})\}\) are, respectively, monotonically decreasing and monotonically increasing sequences.

By induction on k, k ≥ 1, we prove that \(\{\overline{U}^{(n)}(x_{m},t_{k})\}\) and \(\{\underline{U}^{(n)}(\,p,t_{k})\}\) are, respectively, monotonically decreasing and monotonically increasing sequences, which satisfy (7).

3.1 Convergence on [0, T]

We now choose the stopping criterion of the iterative method (6) in the form

$$\displaystyle{ \max _{i}\|\mathcal{R}_{i}(\cdot,t_{k},U^{(n)})\|_{\omega ^{ h}} \leq \delta, }$$
(11)

where δ is a prescribed accuracy, and \(U(x_{m},t_{k}) = U^{(n_{k})}(x_{ m},t_{k})\), \(x_{m} \in \overline{\omega }^{h}\), where n k is minimal subject to the stopping test.

Instead of (5), we now impose the two-sided constraints on f i , i = 1, 2, in the form

$$\displaystyle{ \rho _{k} \leq \frac{\partial f_{i}} {\partial u_{i}}(x_{m},t_{k},U) \leq c_{i}(x_{m},t_{k}),\quad \mbox{ on}\ \langle \widehat{U}(t_{k}),\widetilde{U}(t_{k})\rangle, }$$
(12)
$$\displaystyle{0 \leq -\frac{\partial f_{i}} {\partial u_{i^{{\prime}}}}(x_{m},t,U) \leq q_{i}(x_{m},t_{k}),\quad \mbox{ on}\ \langle \widehat{U}(t_{k}),\widetilde{U}(t_{k})\rangle,\quad i\neq i^{{\prime}},}$$

where ρ k , k ≥ 1, are defined in (13).

Remark 1

We mention that the assumption ∂f i ∕∂u i ≥ ρ k , i = 1, 2, in (12) can always be obtained via a change of variables. Indeed, introduce the following functions u i (x, t) = exp(λt)z i (x, t), i = 1, 2, where λ is a constant. Now, z i (x, t), i = 1, 2, satisfy (1) with

$$\displaystyle{\varphi _{i} =\lambda z_{i} +\exp (-\lambda t)f_{i}(x,t,\exp (\lambda t)z_{1},\exp (\lambda t)z_{2}),}$$

instead of f i , i = 1, 2, and we have

$$\displaystyle{ \frac{\partial \varphi _{i}} {\partial z_{i}} =\lambda +\frac{\partial f_{i}} {\partial u_{i}},\quad \frac{\partial \varphi _{i}} {\partial z_{i^{{\prime}}}} = \frac{\partial f_{i}} {\partial u_{i^{{\prime}}}},\quad i^{{\prime}}\neq i,\quad i = 1,2.}$$

Thus, if λ ≥ max k ≥ 1 ρ k , from here, we conclude that ∂φ i ∕∂z i and \(\partial \varphi _{i}/\partial z_{i^{{\prime}}}\) satisfy (12)

We impose the constraint on τ k

$$\displaystyle{ \tau _{k} < \frac{1} {\rho _{k}},\quad \rho _{k} =\max _{i}\{\max _{x_{m}\in \overline{\omega }^{h}}[q_{i}(x_{m},t_{k})]\}. }$$
(13)

If assumptions (12) and (13) hold, then the nonlinear difference scheme (2) has a unique solution (see Lemmas 3 and 4 in [1] for details).

We prove the following convergence result for the iterative method (6), (11).

Theorem 2

Assume that the mesh \(\overline{\omega }^{\tau }\) satisfies ( 13 ), and f i ( p, t, U), i = 1, 2, satisfy ( 12 ), where \(\widetilde{U}\) and \(\widehat{U}\) are ordered upper and lower solutions of ( 2 ). Then for the sequences \(\{\overline{U}^{(n)}\}\) , \(\{\underline{U}^{(n)}\}\) , generated by ( 6 ), ( 11 ) with, respectively, \(\overline{U}^{(0)} =\widetilde{ U}\) and \(\underline{U}^{(0)} =\widehat{ U}\) , the following uniform in ɛ estimate holds

$$\displaystyle{ \max _{i}\left [\max _{t_{k}\in \overline{\omega }^{\tau }}\|U_{i}(\cdot,t_{k}) - U_{i}^{{\ast}}(\cdot,t_{ k})\|_{\overline{\omega }^{h}}\right ] \leq T\delta, }$$
(14)

where U i ∗( p, t k ), i = 1, 2, is the unique solution to ( 2 ).

Proof

The difference problem for \(U(x_{m},t_{k}) = U^{(n_{k})}(x_{ m},t_{k})\), k ≥ 1, can be represented in the form

$$\displaystyle{\mathcal{L}_{i}U_{i}(x_{m},t_{k}) + f_{i}(x_{m},t_{k},U) -\tau _{k}^{-1}U_{ i}(x_{m},t_{k-1}) = \mathcal{R}_{i}(x_{m},t_{k},U^{(n_{k})}),\quad x_{ m} \in \omega ^{h},}$$
$$\displaystyle{U_{i}(x_{{\ast}},t_{k}) = 0,\quad x_{{\ast}} = x_{0},x_{M_{x}},\quad i = 1,2.}$$

From here, (2) and using the mean-value theorem, we get the difference problem for W i (x m , t k ) = U i (x m , t k ) − U i ∗(x m , t k )

$$\displaystyle{ \left (\mathcal{L}_{i} + \frac{\partial f_{i}} {\partial u_{i}}\right )W_{i}(x_{m},t_{k}) = \mathcal{R}_{i}(x_{m},t_{k},U) + \frac{1} {\tau _{k}}W_{i}(x_{m},t_{k-1}) - \frac{\partial f_{i}} {\partial u_{i^{{\prime}}}}W_{i^{{\prime}}}(x_{m},t_{k}),\ }$$
(15)
$$\displaystyle{x_{m} \in \omega ^{h},\quad W_{ i}(x_{{\ast}},t_{k}) = 0,\quad x_{{\ast}} = x_{0},x_{M_{x}}\quad i^{{\prime}}\neq i,\quad i = 1,2,}$$

where the partial derivatives are calculated at intermediate points E i , i = 1, 2, such that \(U_{i}^{{\ast}}\leq E_{i} \leq \overline{U}_{i}^{(0)}\), i = 1, 2, in the case of upper solutions and \(\underline{U}_{i}^{(0)} \leq E_{i} \leq U_{i}^{{\ast}}\), i = 1, 2, in the case of lower solutions. Thus, the partial derivatives satisfy (12). From here, (12), using (4) and taking into account that according to Theorem 1 the stopping criterion (11) can always be satisfied, in the notation \(w_{k} =\max _{i}\|W_{i}(\cdot,t_{k})\|_{\overline{\omega }^{h}}\) we have

$$\displaystyle{w_{k} \leq \frac{1} {\rho _{k} +\tau _{ k}^{-1}}\left [\delta +\tau _{k}^{-1}w_{ k-1} +\rho _{k}w_{k}\right ].}$$

Solving the last inequality for w k and taking into account that τ k −1∕(ρ k + τ k −1) > 0, we have

$$\displaystyle{w_{k} \leq \delta \tau _{k} + w_{k-1}.}$$

Since w 0 = 0, by induction on k, we conclude (14)

$$\displaystyle{w_{k} \leq \delta \sum _{l=1}^{k}\tau _{ l} \leq T\delta,\quad k \geq 1.}$$

3.2 Construction of Initial Upper and Lower Solutions

Here, we give some conditions on functions f i and ψ i , i = 1, 2, to guarantee the existence of upper \(\widetilde{U}\) and lower \(\widehat{U}\) solutions, which are used as the initial iterations in the monotone iterative method (6).

Bounded Reactions Functions

Assume that f i , ψ i , i = 1, 2, from (1) satisfy the conditions

$$\displaystyle{ -\sigma _{i} \leq f_{i}(x,t,0) \leq 0,\quad \psi _{i}(x) \geq 0,\quad u_{i}(x,t) \geq 0,\quad x \in \overline{\omega }, }$$

where σ i , i = 1, 2, are positive constants. Then

$$\displaystyle{ \widehat{U}_{i}(x_{m},t_{k}) = \left \{\begin{array}{ll} \psi _{i}(x_{m}),&k = 0, \\ 0, &k \geq 1, \end{array} \right.\quad x_{m} \in \overline{\omega }^{h},\quad i = 1,2, }$$

are lower solutions to (2). The solutions of the following linear problems:

$$\displaystyle{ \mathcal{L}_{i}(x_{m},t_{k})\widetilde{U}_{i}(x_{m},t_{k}) =\tau _{ k}^{-1}\widetilde{U}_{ i}(x_{m},t_{k-1}) +\sigma _{i},\quad x_{m} \in \omega ^{h},\quad k \geq 1, }$$
$$\displaystyle{\widetilde{U}_{i}(x_{{\ast}},t_{k}) = 0,\quad x_{{\ast}} = x_{0},x_{M_{x}},\quad k \geq 1,\quad \widetilde{U}_{i}(x_{m},0) =\psi _{i}(x_{m}),\quad x_{m} \in \overline{\omega }^{h},}$$

are upper solutions to (2).

Constant Upper and Lower Solutions

Assume that functions f i , ψ i , i = 1, 2, from (1) satisfy the conditions

$$\displaystyle{ f_{i}(x,t,0) \leq 0,\quad f_{i}(x,t,L) \geq 0,\quad \psi _{i}(x) \geq 0,\quad u_{i}(x,t) \geq 0,\quad x \in \overline{\omega }, }$$
(16)

where L = const > 0. The functions

$$\displaystyle{ \widehat{U}_{i}(x_{m},t_{k}) = \left \{\begin{array}{ll} \psi _{i}(x_{m}),&k = 0, \\ 0, &k \geq 1, \end{array} \right.\qquad \widetilde{U}_{i}(x_{m},t_{k}) = L,\quad x_{m} \in \overline{\omega }^{h}, }$$
(17)

are, respectively, lower and upper solutions.

4 Uniform Convergence of the Monotone Iterates

We assume that 0 < ɛ 1 ≤ ɛ 2 ≤ 1.

In the notation u = (u 1, u 2), ɛ = (ɛ 1, ɛ 2) and f = ( f 1, f 2), the following linear system is considered in [5]:

$$\displaystyle{\frac{\partial u} {\partial t} -\varepsilon \frac{\partial ^{2}u} {\partial x^{2}}+A(x,t)u = f(x,t),\quad A(x,t) = \left [\begin{array}{cc} a_{11}(x,t)&a_{12}(x,t) \\ a_{21}(x,t)&a_{22}(x,t) \end{array} \right ],}$$

where the matrix A(x, t) satisfies the assumptions

$$\displaystyle{\quad a_{ii}(x,t) > 0,\quad \quad a_{ii^{{\prime}}}(x,t) \leq 0,\quad a_{ii}(x,t) + a_{ii^{{\prime}}}(x,t) \geq \alpha = \mbox{ const} > 0,}$$
$$\displaystyle{i\neq i^{{\prime}},\quad i = 1,2,\quad (x,t) \in \overline{\omega }\times [0,T].}$$

From [5], we write down the bounds on ∂u i ∕∂x, i = 1, 2 in the form

$$\displaystyle{ \left \vert \frac{\partial u_{1}} {\partial x} (x,t)\right \vert \leq C\left [1 +\mu _{ 1}^{-1}\pi _{ \mu _{1}}(x) +\mu _{ 2}^{-1}\pi _{ \mu _{2}}(x)\right ], }$$
(18)
$$\displaystyle{ \left \vert \frac{\partial u_{2}} {\partial x} (x,t)\right \vert \leq C\left [1 +\mu _{ 2}^{-1}\pi _{ \mu _{2}}(x)\right ],\quad \pi _{\gamma }(x) \equiv \exp (-\gamma ^{-1}x) +\exp (-\gamma ^{-1}(1 - x)), }$$

where \(\mu _{i} = \sqrt{\varepsilon _{i}}\), i = 1, 2, and γ is a positive constant. These bounds show that there are two overlapping boundary layers at x = 0 and x = 1.

By using the mean-value theorem, we write f i , i = 1, 2, from (1) in the form

$$\displaystyle{f_{i}(x,t,u) = f_{i}(x,t,0) + \frac{\partial f_{i}} {\partial u_{i}}(x,t,v)u_{i} + \frac{\partial f_{i}} {\partial u_{i^{{\prime}}}}(x,t,v)u_{i^{{\prime}}},\quad i^{{\prime}}\neq i,\quad i,i^{{\prime}} = 1,2,}$$

where v lies between 0 and u. We suppose that ∂f i ∕∂u i and \(\partial f_{i}/\partial u_{i^{{\prime}}}\), i ′ ≠ i, i, i ′ = 1, 2, for \((x,t,v) \in \overline{\omega }\times [0,T] \times (-\infty,\infty )\) satisfy the following assumptions:

$$\displaystyle{ \frac{\partial f_{i}} {\partial u_{i}}(x,t,v) > 0,\quad \frac{\partial f_{i}} {\partial u_{i^{{\prime}}}}(x,t,v) \leq 0,\quad \quad i^{{\prime}}\neq i,\quad i,i^{{\prime}} = 1,2, }$$
(19)
$$\displaystyle{\min _{-\infty \leq v\leq \infty }\left [\frac{\partial f_{i}} {\partial u_{i}}(x,t,v) + \frac{\partial f_{i}} {\partial u_{i^{{\prime}}}}(x,t,v)\right ] >\alpha = \mbox{ const} > 0.}$$

Remark 2

If assumptions (19) hold, then Theorem 3.1, Chap. 8 in [7] guarantees existence and uniqueness of the solution to problem (1).

We may now consider (1) as a linear problem and use bounds (18) on the exact solutions. We introduce the piecewise uniform mesh \(\overline{\omega }^{h}\) of Shishkin-type from [5], where the boundary layer thicknesses \(\varsigma _{\varepsilon _{ i}}\), i = 1, 2, and mesh spacings \(h_{\varepsilon _{i}}\), i = 1, 2, h are defined by

$$\displaystyle{ \varsigma _{\varepsilon _{2}} =\min \left \{1/4,2\sqrt{\varepsilon _{2}}\ln M_{x}\right \},\quad \varsigma _{\varepsilon _{1}} =\min \left \{\varsigma _{\varepsilon _{2}}/2,2\sqrt{\varepsilon _{1}}\ln M_{x}\right \}, }$$
(20)
$$\displaystyle{h_{\varepsilon _{1}} = 8\varsigma _{\varepsilon _{1}}/M_{x},\quad h_{\varepsilon _{2}} = 8(\varsigma _{\varepsilon _{2}} -\varsigma _{\varepsilon _{1}})/M_{x}\quad h = 2(1 - 2\varsigma _{\varepsilon _{2}})/M_{x}.}$$

The mesh \(\overline{\omega }^{h}\) is constructed thus: in each of the subintervals \([0,\varsigma _{\varepsilon _{1}}]\), \([\varsigma _{\varepsilon _{1}},\varsigma _{\varepsilon _{2}}]\), \([\varsigma _{\varepsilon _{2}},1 -\varsigma _{\varepsilon _{2}}]\), \([1 -\varsigma _{\varepsilon _{2}},1 -\varsigma _{\varepsilon _{1}}]\) and \([1 -\varsigma _{\varepsilon _{1}},1]\), mesh points are distributed uniformly with M x ∕8 + 1, M x ∕8 + 1, M x ∕2 + 1, M x ∕8 + 1 and M x ∕8 + 1 mesh points, respectively. The mesh spacings \(h_{\varepsilon _{1}}\), \(h_{\varepsilon _{2}}\) and h are in use, respectively, in the first and last, in the second and fourth, in the third domains.

Theorem 3

Assume that meshes \(\overline{\omega }^{\tau }\) and \(\overline{\omega }^{h}\) satisfy, respectively, ( 13 ) and ( 20 ), and f i (x, t, u), i = 1, 2, satisfy ( 19 ). Then the nonlinear difference scheme ( 2 ) converges ɛ-uniformly to the solution of ( 1 )

$$\displaystyle{ \max _{i}\left [\max _{t_{k}\in \overline{\omega }^{\tau }}\|U_{i}^{{\ast}}(\cdot,t_{ k}) - u_{i}^{{\ast}}(\cdot,t_{ k})\|_{\overline{\omega }^{h}}\right ] \leq C(M_{x}^{-1}\ln M_{ x}+\tau ),\quad \tau =\max _{k}\tau _{k}, }$$
(21)

where U i ∗ and u i ∗ , i = 1, 2, are, respectively, the exact solutions to ( 2 ) and ( 1 ), C is a generic constant which is independent of ɛ, M x and τ.

Proof

Since the proof of the theorem follows the proof of Theorem 1 from [3], then we only present the sketch of it.

The exact solutions u i ∗(x, t), i = 1, 2, can be presented on [x m−1, x m+1] in the integral-difference form (compare with (5) from [3])

$$\displaystyle{ \varepsilon _{i}\mathcal{L}_{i}^{h}u_{ i}^{{\ast}}(x_{ m},t_{k}) = \frac{\partial u_{i}^{{\ast}}} {\partial t} + f_{i}(x_{m},t_{k},u^{{\ast}}) + I_{ i}(x_{m},t_{k},u^{{\ast}}),\quad x_{ m},t_{k} \in \omega ^{h} \times \omega ^{\tau }, }$$

where u ∗ = (u 1 ∗, u 2 ∗), \(\mathcal{L}_{i}^{h}\), i = 1, 2, are defined in (2) and I i , i = 1, 2, are given in the form

$$\displaystyle\begin{array}{rcl} I_{i}(x_{m},t_{k},u^{{\ast}})& =& \frac{1} {\hslash _{m}}\int _{x_{m-1}}^{x_{m} }\phi _{2,m-1}(s)\left (\int _{x_{m}}^{s}\frac{d\psi _{i}(\xi,t_{k})} {d\xi } d\xi \right )ds {}\\ & & + \frac{1} {\hslash _{m}}\int _{x_{m}}^{x_{m+1} }\phi _{1,m}(s)\left (\int _{x_{m}}^{s}\frac{d\psi _{i}(\xi,t_{k})} {d\xi } d\xi \right )ds, {}\\ \end{array}$$
$$\displaystyle{\psi _{i}(x,t_{k}) = f_{i}(x,t_{k},u^{{\ast}}) + \frac{\partial u_{i}^{{\ast}}(x,t_{ k})} {\partial t},\quad x \in [x_{m-1},x_{m+1}],}$$
$$\displaystyle{\phi _{1,m}(x) = \frac{x_{m+1} - x} {\hslash _{m}},\quad \phi _{2,m}(x) = \frac{x - x_{m}} {\hslash _{m}},\quad }$$

The truncation errors T i (x m , t k ), i = 1, 2, can be represented in the form

$$\displaystyle{ T_{i}(x_{m},t_{k}) = T_{i,1}(x_{m},t_{k}) - I_{i}(x_{m},t_{k},u^{{\ast}}),\quad }$$
$$\displaystyle{T_{i,1}(x_{m},t_{k}) \equiv \frac{u_{i}^{{\ast}}(x_{m},t_{k}) - u_{i}^{{\ast}}(x_{m},t_{k-1})} {\tau _{k}} -\frac{\partial u_{i}^{{\ast}}(x_{m},t_{k})} {\partial t}.}$$

Using the Taylor expansion about (x m , t k ), we obtain

$$\displaystyle{ \|T_{i}(\cdot,t_{k})\|_{\omega ^{h}} \leq \frac{1} {2}\max _{(x,t)\in Q}\vert u_{i,tt}^{{\ast}}\vert \tau _{ k} +\| I_{i}(\cdot,t_{k})\|_{\omega ^{h}}. }$$
(22)

Thus, similar to [3], using bounds (18), the following estimates on dψ i ∕dx, i = 1, 2, hold true

$$\displaystyle{ \left \vert \frac{d\psi _{i}(x,t)} {dx} \right \vert \leq C\left [1 +\mu _{ 1}^{-1}\pi _{ \mu _{1}}(x) +\mu _{ 2}^{-1}\pi _{ \mu _{2}}(x)\right ],\quad i = 1,2. }$$

From here, using the properties of the piecewise uniform mesh of Shishkin-type and repeating the proof of Theorem 1 from [3], we prove the estimates

$$\displaystyle{\|I_{i}(\cdot,t_{k})\|_{\omega ^{h}} \leq C\left (M_{x}^{-1}\ln M_{ x}\right ),\quad i = 1,2.}$$

From here and (22), we obtain

$$\displaystyle{ \|T_{i}(\cdot,t_{k})\|_{\omega ^{h}} \leq C\left (M_{x}^{-1}\ln M_{ x}+\tau \right ),\quad i = 1,2. }$$

The difference problems for u i ∗, i = 1, 2, can be represented in the form

$$\displaystyle{\mathcal{L}_{i}u_{i}^{{\ast}}(x_{ m},t_{k}) + f_{i}(x_{m},t_{k},u^{{\ast}}) -\tau _{ k}^{-1}u_{ i}^{{\ast}}(x_{ m},t_{k-1}) = T_{i}(x_{m},t_{k}),\quad x_{m} \in \omega ^{h},}$$
$$\displaystyle{u_{i}^{{\ast}}(x_{ {\ast}},t_{k}) = 0,\quad x_{{\ast}} = x_{0},x_{M_{x}},\quad i = 1,2.}$$

From here, (2) and using the mean-value theorem, we get the difference problem for W i (x m , t k ) = U i (x m , t k ) − u i ∗(x m , t k ) in the form

$$\displaystyle{ \left (\mathcal{L}_{i} + \frac{\partial f_{i}} {\partial u_{i}}\right )W_{i}(x_{m},t_{k}) = -T_{i}(x_{m},t_{k}) + \frac{1} {\tau _{k}}W_{i}(x_{m},t_{k-1}) - \frac{\partial f_{i}} {\partial u_{i^{{\prime}}}}W_{i^{{\prime}}}(x_{m},t_{k}),\ }$$
$$\displaystyle{x_{m} \in \omega ^{h},\quad W_{ i}(x_{{\ast}},t_{k}) = 0,\quad x_{{\ast}} = x_{0},x_{M_{x}}\quad i^{{\prime}}\neq i,\quad i = 1,2.}$$

Now the proof of the theorem repeats the proof of Theorem 2 starting from (15), where − T i , i = 1, 2, are in use instead of \(\mathcal{R}_{i}\), i = 1, 2, in (15).

Theorem 4

Assume that all the assumptions in Theorem  3 are satisfied. Then for the sequences \(\{\overline{U}^{(n)}\}\) and \(\{\underline{U}^{(n)}\}\) , generated by ( 6 ), ( 11 ) with, respectively, \(\overline{U}^{(0)} =\widetilde{ U}\) and \(\underline{U}^{(0)} =\widehat{ U}\) , the uniform in ɛ estimate holds

$$\displaystyle{ \max _{i}\left [\max _{t_{k}\in \overline{\omega }^{\tau }}\|U_{i}(\cdot,t_{k}) - u_{i}^{{\ast}}(\cdot,t_{ k})\|_{\overline{\omega }^{h}}\right ] \leq C(\delta +M_{x}^{-1}\ln M_{ x}+\tau ), }$$

where \(U_{i}(\,p,t_{k}) = \overline{U}^{(n_{k})}(\,p,t_{ k})\) or \(U_{i}(\,p,t_{k}) =\underline{ U}^{(n_{k})}(\,p,t_{ k})\) and u i ∗ , i = 1, 2, are the exact solutions to ( 1 ).

Proof

The proof of the theorem follows from Theorems 2 and 3.

5 Gas-Liquid Interaction Model

The gas-liquid interaction model in the non-dimensional variables can be presented in the form (see [4] for details)

$$\displaystyle{ \frac{\partial u_{1}} {\partial t} -\frac{\partial u_{1}} {\partial x^{2}} -\kappa _{1}(1 - u_{1})u_{2} = 0,\quad (x,t) \in \omega \times (0,T], }$$
$$\displaystyle{\frac{\partial u_{2}} {\partial t} -\varepsilon \frac{\partial u_{2}} {\partial x^{2}} +\kappa _{2}(1 - u_{1})u_{2} = 0,\quad (x,t) \in \omega \times (0,T],}$$
$$\displaystyle{u_{1}(0,t) = u_{1}(1,t) = 0,\quad u_{2}(0,t) = u_{2}(1,t) = 1,}$$
$$\displaystyle{u_{1}(x,0) = 0,\quad u_{2}(x,0) =\sin (\pi x),\quad x \in \overline{\omega },}$$

where u 1 and u 2 are, respectively, concentrations of a dissolved gas and a dissolved reactant and κ i , i = 1, 2, are positive constants. The test problem, which corresponds to the case ɛ 1 = 1, ɛ 2 = ɛ, for small values of ɛ is singularly perturbed and u 2 has boundary layers of width \(\mathcal{O}(\sqrt{\varepsilon })\) near x = 0 and x = 1.

It is easy to verify that assumptions (16) with L i = 1, i = 1, 2, hold true. Thus, \(\widehat{U}_{i}\) and \(\widetilde{U}_{i}\), i = 1, 2, from (17) are, respectively, lower and upper solutions to the test problem. From here, it follows that the inequalities in (12) hold, and one can choose c i (x m , t k ) = κ i , i = 1, 2, in (5) The exact solution is not available, so we estimate the error of the numerical solutions \(U_{i}^{M_{x}}\), i = 1, 2, with respect to the reference solutions \(U_{i}^{2M_{x}}\), i = 1, 2,

$$\displaystyle{E_{M_{x}} =\max _{i=1,2}\|U_{i}^{M_{x} }(\cdot,t_{N_{\tau }}) - U_{i}^{2M_{x} }(\cdot,t_{N_{\tau }})\|_{\overline{\omega }^{h}},}$$

and assume that \(E_{M_{x}} = C(1/M_{x})^{p_{M_{x}}}\), where constant C is independent of M x , and \(p_{M_{x}}\) is the order of maximum numerical error. For each M x , we compute \(p_{M_{x}}\) from

$$\displaystyle{p_{M_{x}} =\log _{2} \frac{E_{M_{x}}} {E_{2M_{x}}}.}$$

We choose δ = 10−8 in the stopping test (11). In Table 1, for parameters κ i = 1, i = 1, 2, \(t_{N_{\tau }} = 0.5\), τ = 5 × 10−4 and different values of ɛ and M x , we present the maximum numerical error \(E_{M_{x}}\), the order of maximum numerical error \(p_{M_{x}}\) and the number of monotone iterations \(n_{M_{x}}\) on each time level. The data in the table show that for ɛ ≤ 10−4, the numerical solution converges uniformly in ɛ, has the first-order accuracy in the space variable, and the monotone sequences converge in few iterations.

Table 1 Numerical results
Full size table