Quenching for discretizations of a semilinear parabolic equation with nonlinear boundary outflux

Cyrille K. N’Dri,\(^1\) Ardjouma Ganon,\(^1\) Gozo Yoro,\(^2\) Kidjegbo A. Touré\(^1\)

April 11, 2023; accepted: September 28, 2023; published online: December 22, 2023.

\(^1\)UMRI Mathématiques et Nouvelles Technologies de l’Information, Institut National Polytechnique Félix HOUPHOUËT-BOIGNY, Yamoussoukro, Côte d’Ivoire, e-mail: C. K. N’Dri: ndri.pack@gmail.com
Ardjouma Ganon: ardjganon@gmail.com
Kidjegbo A. Touré: latoureci@gmail.com
\(^2\) Département Mathématiques et Informatique, Université Nangui Abrogoua d’Abobo Adjamé, Abidjan, Côte d’Ivoire, e-mail: yorocarol@yahoo.fr.

In this paper, we study numerical approximations of a semilinear parabolic problem in one-dimension, of which the nonlinearity appears both in source term and in Neumann boundary condition. By a semidiscretization using finite difference method, we obtain a system of ordinary differential equations which is an approximation of the original problem. We obtain conditions under which the positive solution of our system quenches in a finite time and estimate its semidiscrete quenching time. Convergence of the numerical quenching time to the theoretical one is established. Next, we show that the quenching rate of the numerical scheme is different from the continuous one. Finally, we give numerical results to illustrate our analysis.

MSC. 35K61, 65M06, 35B51

Keywords. Nonlinear parabolic equations, nonlinear boundary outflux, semidiscretizations, numerical quenching, quenching rate.

1 Introduction

Consider the following semilinear parabolic equation with nonlinear boundary outflux \( \left\{ \begin{array}{rl}& u_{t}(x,t)=u_{xx}(x,t)+f(x)(1-u(x,t))^{-p},\: 0{\lt}x{\lt}1,\: 0{\lt}t{\lt}\infty ,\\ & u_{x}(0,t)=u^{-q}(0,t),\quad 0{\lt}t{\lt}\infty ,\\ & u_{x}(1,t)=0 ,\quad 0{\lt}t{\lt}\infty ,\\ & u(x,0)=u_{0}(x),\quad 0\leq x\leq 1. \end{array} \right. \label{probleme} \)

Here \(p, q\) are positive constants, \(f\) is a non-negative function and initial datum \(u_{0}:[0,1]\rightarrow (0,1)\) is smooth enough and satisfies boundary conditions. We can regard this problem as a heat conduction model that incorporates the effects of nonlinear reaction (source) and nonlinear boundary outflux (emission).

Our problem derives from the general the problem \((1.1)\) of [ 18 ] , where the authors proved the existence and uniqueness of the classical solution. 

We say that the classical solution \(u(x,t)\) of 1 quenches in a finite time if there exists a finite time \(T_{q}\) such that:

\begin{equation*} \displaystyle \lim _{t\rightarrow T^{-}_{q}} \min _{0\leq x\leq 1}u(x,t)=0,\quad or \quad \displaystyle \lim _{t\rightarrow T^{-}_{q}} \max _{0\leq x\leq 1}u(x,t)=1. \end{equation*}

From now, we denote the quenching time with \(T_{q}\).

The quenching problem has been the subject of investigations of many authors since several decades (see [ 7 , 13 , 19 , 20 , 22 , 23 , 6 , 17 ] and the references cited therein). It was in \(1975\) that Kawarada [ 13 ] introduced the first concept of quenching, he found sufficient conditions under which the solutions of the following problem quenches in finite time,

\begin{align*} & u_{t}=u_{xx}+\tfrac {1}{1-u},\quad t{\gt}0,\quad x\in (0,l),\\ & u(t,0)=u(t,l)=0,\quad t{\gt}0,\\ & u(0,x)=0,\quad x\in (0,l), \end{align*}

where \(l\) is positive.

In our problem we have two singular heat sources, namely, a source term \(f(x)(1-u)^{-p}\) and the boundary outflux term \(u^{-q}\). Some authors studied such quenching problems with the nonlinearity both in source and in boundary conditions [ 7 , 20 , 22 , 23 , 17 ] . In [ 20 ] , Selcuk and Ozalp studied the quenching behavior of the solution of a semilinear heat equation with a singular boundary outflux. They showed that quenching occurs on the boundary under certain conditions and that the time derivative blows up at a quenching point.
Concerning problem 1, Zhi [ 22 ] showed that if the initial datum satisfies

\begin{equation*} u^{''}_{0}(x)+f(x)(1-u_{0}(x))^{-p}\leq 0\text{ and not equals} \ 0 \text{ identically},\: x\in (0,1), \end{equation*}

then the classical solution \(u\) of 1 quenches in a finite time \(T_{q}\) with the following estimate \(u(0,t)\sim (T_{q}-t)^{\frac{1}{2(q+1)}}\). This condition excludes the formation of a singularity for the source term, thus it is only sufficient to consider the case of quenching formation on the boundary for the solution of problem 1. He also asserted that the quenching can only occur on the point \(x=0\) if the given initial datum is monotone. A related problem was studied earlier in [ 23 ] where the author prescribed \(f(x)=1\), similar results have been also obtained.

Here we are interested in the numerical study of the phenomenon of quenching (For other works on numerical approximations of quenching solutions we refer to [ 15 , 5 , 14 , 17 ] ). We give assumptions under which the solution of a semidiscrete form of 1 quenches in a finite time and estimate its semidiscrete quenching time. We also prove that, under suitable assumptions on the initial datum, the semidiscrete quenching time converges to the theoretical one when the mesh size goes to zero. Our work was motived by the papers in [ 9 , 1 , 16 , 2 , 10 , 4 ] , where the authors have used semidiscrete forms for some parabolic equations to study the phenomenon of blow-up (we say that a solution blows up in a finite time if it reaches the value infinity in a finite time).

The paper is written in the following manner. In the next Section, we present a semidiscrete scheme of 1. In section 3, we give properties concerning our semidiscrete scheme. In section 4, under appropriate conditions, we prove that the solution of the semidiscrete form quenches in a finite time, estimate its semidiscrete quenching time and give results on the numerical quenching rate. In section 5, we study the convergence of semidiscrete quenching time. Finally, in the last section, we give numerical experiments.

2 The Semidiscrete Problem

Let \(I\) be a positive integer, we set \(h=\frac{1}{I}\), and we define the grid \(x_{i}=ih\), \(i=0,\ldots ,I\). We approximate the solution \(u\) of the problem 1 by the solution \(U_{h}=(U_{0},U_{1},\ldots ,U_{I})^{T}\) and the initial datum \(u_{0}\) by \(\varphi _{h}=(\varphi _{0},\varphi _{1},\ldots ,\varphi _{I})^{T}\) of the semidiscrete equations

\begin{align} & \tfrac {dU_{i}(t)}{dt}=\delta ^{2}U_{i}(t)+ f_{i}(1-U_{i}(t))^{-p},\quad i=1,\ldots ,I-1, \: t\in [0,T_{q}^{h}), \label{semidiscret1}\\ & \tfrac {dU_{0}(t)}{dt}=\delta ^{2}U_{0}(t)-\tfrac {2}{h}U^{-q}_{0}(t)+ f_{0}(1-U_{0}(t))^{-p},\quad [0,T_{q}^{h}), \label{semidiscret2}\\ & \tfrac {dU_{I}(t)}{dt}=\delta ^{2}U_{I}(t)+f_{I}(1-U_{I}(t))^{-p},\quad [0,T_{q}^{h}), \label{semidiscret3}\\ & U_{i}(0)=\varphi _{i}{\gt}0, \quad i=0,\ldots ,I, \label{semidiscret4} \intertext {where} & f_{i}\simeq f(x_{i}),\quad i=0,\ldots ,I,\notag \\ & \delta ^{2}U_{i}(t)=\displaystyle \tfrac {U_{i+1}(t)-2U_{i}(t)+U_{i-1}(t)}{h^{2}},\quad i=1,\ldots ,I-1,\notag \\ & \delta ^{2}U_{0}(t)=\displaystyle \tfrac {2U_{1}(t)-2U_{0}(t)}{h^{2}},\quad \delta ^{2}U_{I}(t)=\displaystyle \tfrac {2U_{I-1}(t)-2U_{I}(t)}{h^{2}}\notag . \end{align}

Here \([0,T_{q}^{h})\) is the maximal time interval on which \(\| U_{h}(t)\| _{\inf }{\gt}0\), where \(\| U_{h}(t)\| _{\inf }=\displaystyle \min _{0\leq i \leq I}\lvert U_{i}(t)\rvert \). The time \(T^{h}_{q}\) can be finite or infinite. When \(T_{q}^{h}\) is finite, we say that \(U_{h}\) quenches in finite time and \(T^{h}_{q}\) is called the quenching time of the solution \(U_{h}\). Whereas when \(T_{q}^{h}\) is infinite we say that \(U_{h}\) quenches globally.

In the following, we give important results which will be used later. Here, \(V_{i}\) and \(W_{i}\) denote the respective \((i+1)\)-th components of \(V_{h}\) and \(W_{h}\).

Definition 1

A function \(V_{h}\in \mathcal{C}^{1}([0,T],\mathbb {R}^{I+1})\) is an upper solution of 1–?? if

\begin{flalign*} & \displaystyle \tfrac {dV_{i}(t)}{dt}-\delta ^{2}V_{i}(t) \geq f_{i}(1-V_{i}(t))^{-p},\quad i=1,\ldots ,I, \: t\in (0,T],\\ & \displaystyle \tfrac {dV_{0}(t)}{dt}-\delta ^{2}V_{0}(t)+\tfrac {2}{h}V^{-q}_{0}(t)\geq f_{0}(1-V_{0}(t))^{-p},\: t\in (0,T],\\ & V_{i}(0)\geq \varphi _{i}, \quad i=0,\ldots ,I . \end{flalign*}

On the other hand, we say that \(V_{h}\in \mathcal{C}^{1}([0,T],\mathbb {R}^{I+1})\) is a lower solution of 1–?? if these inequalities are reversed.

The following results are semidiscrete forms of a Maximum Principle and will be an important tool in the study of the semidiscrete problem 1–??. Their proofs are standard and will be omitted.

Lemma 2

Let \(a_{h}\in C^{0}([0,T],\mathbb {R}^{I+1})\) and \(V_{h}\in C^{1}([0,T],\mathbb {R}^{I+1})\) such that

\begin{align} & \displaystyle \tfrac {d}{dt}V_{i}-\delta ^{2}V_{i}+a_{i}V_{i}\geq 0,\quad i=0,\ldots ,I,\: t\in [0,T], \label{lemma1-1}\\ & V_{i}(0)\geq 0,\quad i=0,\ldots ,I, \label{lemma1-2} \end{align}

then we have \(V_{i}(t)\geq 0,\quad i=0,\ldots ,I,\: t\in [0,T]\).

Another form of the Maximum Principle for semidiscrete equations are the following comparison lemma.

Lemma 3

Let \(g\in C(\mathbb {R},\mathbb {R})\) and \(W_{h},V_{h}\in C^{1}([0,T],\mathbb {R}^{I+1})\) such that

\begin{align} & \tfrac {d}{dt}V_{i}-\delta ^{2}V_{i}-g(V_{i})\leq \tfrac {d}{dt}W_{i}-\delta ^{2}W_{i}-g(W_{i}),\: i=1,\ldots ,I,\: t\in (0,T],\label{C1}\\ & \tfrac {d}{dt}V_{0}+\tfrac {2}{h}V^{-q}_{0}-\delta ^{2}V_{0}-g(V_{0})\leq \tfrac {d}{dt}V_{0}+\frac{2}{h}W^{-q}_{0}-\delta ^{2}W_{0}-g(W_{0}),t\in (0,T],\label{C2}\\ & V_{i}(0)\leq W_{i}(0),\: i=0,\ldots ,I,\label{C3} \end{align}
then \(V_{i}(t)\leq W_{i}(t),\quad i=0,\ldots ,I,\: t\in [0,T]\).

3 Quenching in the semidiscrete problem

In this section, under appropriate assumptions, we show that solution \(U_{h}\) of the semidiscrete problem 1–?? quenches in a finite time \(T^{h}_{q}\) and we estimate its semidiscrete quenching time.

The following result gives a property of the operator \(\delta ^{2}\).

Lemma 4

Let \(U_{h}\in \mathbb {R}^{I+1}\) be such that \(U_{h}{\gt}0\). Then, we have

\begin{align*} & \delta ^{2}(U^{-q}_{i})\geq -qU^{-q-1}_{i}\delta ^{2}U_{i},\: i=0,\ldots ,I. \end{align*}

Proof â–¼
Let us introduce function \(f(s)=s^{-q}\). Using Taylor’s expansion we get

\begin{align*} & \delta ^{2}f(U_{0})=f^{'}(U_{0})\delta ^{2}U_{0}+\tfrac {(U_{1}-U_{0})^{2}}{h^{2}}f^{''}(\zeta _{0}),\\ & \delta ^{2}f(U_{I})=f^{'}(U_{I})\delta ^{2}U_{I}+\tfrac {(U_{I-1}-U_{I})^{2}}{h^{2}}f^{''}(\zeta _{I}),\\ & \delta ^{2}f(U_{i})=f^{'}(U_{i})\delta ^{2}U_{i}+\tfrac {(U_{i+1}-U_{i})^{2}}{2h^{2}}f^{''}(\eta _{i})+\tfrac {(U_{i-1}-U_{i})^{2}}{2h^{2}}f^{''}(\zeta _{i}),\: i=1,\ldots ,I-1. \end{align*}

where \(\eta _{i}\) is an intermediate value between \(U_{i}\) and \(U_{i+1}\) and \(\zeta _{i}\) the one between \(U_{i}\) and \(U_{i-1}\).
The result follows taking into account the fact that \(U_{h}\) is nonnegative.

Proof â–¼

In the rest of this paper, we assume that these conditions are satisfied

\begin{align} & 0\leq f_{i}\leq f_{i+1},\quad i=0,\ldots ,I-1, \label{croissance_f} \\ & 0{\lt}\varphi _{i}{\lt}\varphi _{i+1},\quad i=0,\ldots ,I-1. \label{croissance_donnee_initiale} \end{align}

Lemma 5

Let \(U_{h}\) be the solution of 1–??, then, we have

  1. \(U_{i}(t)\geq \varphi _{i}\) for \(i=0,\ldots ,I\), \(t\in [0,T]\);

  2. \(U_{i}(t){\lt}U_{i+1}(t)\) for \(i=0,\ldots ,I-1\),  \(t\in [0,T]\).

Proof â–¼

  1. Using lemma 3, we obtain \(U_{i}(t)\geq \varphi _{i}{\gt}0,\quad i=0,\ldots ,I,\: t\in [0,T]\).

  2. For \(i=0,\ldots ,I-1\), \(t\in [0,T]\), introduce \(Z_{i}\) such that

    \(Z_{i}(t)=U_{i+1}(t)-U_{i}(t)\). Let \(t_{0}\) be the first \(t{\gt}0\) such that \(Z_{i}(t){\gt}0\) for \(t\in [0,t_{0})\), but \(Z_{i_{0}}(t_{0})=0\) for a certain \(i_{0}\in \{ 0,\ldots ,I-1\} \). Without loss of generality, we suppose that \(i_{0}\) is the smallest integer checking the equality above. We observe that

    \begin{align*} & \tfrac {d}{dt}Z_{i_{0}}(t_{0})=\displaystyle \lim _{k\rightarrow 0}\tfrac {Z_{i_{0}}(t_{0})-Z_{i_{0}}(t_{0}-k)}{k}\leq 0,\quad i_{0}=0,\ldots ,I-1,\\ & \delta ^{2}Z_{i_{0}}(t_{0})=\tfrac {Z_{i_{0}-1}(t_{0})-2Z_{i_{0}}(t_{0})+Z_{i_{0}+1}(t_{0})}{h^{2}}{\gt}0,\quad i_{0}=1,\ldots ,I-2,\\ & \delta ^{2}Z_{0}(t_{0})=\tfrac {Z_{1}(t_{0})-3Z_{0}(t_{0})}{h^{2}}{\gt}0,\\ & \delta ^{2}Z_{I-1}(t_{0})=\tfrac {Z_{I-2}(t_{0})-3Z_{I-1}(t_{0})}{h^{2}}{\gt}0. \end{align*}

    By a computation, we get

    \begin{align*} & \tfrac {d}{dt}Z_{0}(t_{0})-\delta ^{2}Z_{0}(t_{0})-f_{1}(1-U_{1}(t_{0}))^{-p}+f_{0}(1-U_{0}(t_{0}))^{-p}-\tfrac {2}{h}U^{-q}_{0}(t_{0}){\lt}0,\\ & \tfrac {d}{dt}Z_{i_{0}}(t_{0})-\delta ^{2}Z_{i_{0}}(t_{0})-f_{i_{0}+1}(1-U_{i_{0}+1}(t_{0}))^{-p}+f_{i_{0}}(1-U_{i_{0}}(t_{0}))^{-p}{\lt}0,\: i_{0}=1,\ldots ,I-1. \end{align*}

    But these inequalities contradict 13 and the proof is complete.

Proof â–¼

Theorem 6

Let \(U_{h}\) be the solution of 1–??. Assume that the initial data at ?? verifies

\begin{align} & \delta ^{2}\varphi _{i}+f_{i}(1-\varphi _{i})^{-p}\leq -\epsilon \varphi _{i}^{-q},\quad i=1,\ldots ,I,\label{condition theo quench1}\\ & \delta ^{2}\varphi _{0}+f_{0}(1-\varphi _{0})^{-p}-\frac{2}{h}\varphi ^{-q}_{0}\leq - \epsilon \varphi ^{-q}_{0}\label{condition theo quench2}, \intertext {for a certain constant $\epsilon \in (0,1]$.} \intertext {Then, the solution $U_{h}$ quenches in a finite time $T_{q}^{h}$ and we have the following estimate} & T_{q}^{h}\leq \frac{\| \varphi _{h}\| _{\inf }^{q+1}}{\epsilon (q+1)}.\notag \end{align}

Proof â–¼
Since \((0,T_{q}^{h})\) is the maximal time interval which on which \(\| U_{h}\| _{\inf }{\gt}0\). We want to show that \(T_{q}^{h}\) is finite and satisfies the above inequality. Introduce the vector \(J_{h}(t)\) defined as follows

\begin{flalign} & J_{i}(t)=\tfrac {dU_{i}(t)}{dt}+\epsilon U^{-q}_{i}(t),\quad i=0,\ldots ,I,\label{fonctionnel theo quench1} \intertext {by a straightforward computation, we get} & \tfrac {d}{dt}J_{i}-\delta ^{2}J_{i}=\tfrac {d}{dt}(\tfrac {dU_{i}}{dt}-\delta ^{2}U_{i})-\epsilon (qU^{-q-1}_{i}\tfrac {dU_{i}}{dt}+\delta ^{2}U^{-q}_{i}),\quad i=0,\ldots ,I.\notag \end{flalign}

Using lemma 4 and equalities 13 and 12, the above equalities give

\begin{flalign} & \tfrac {d}{dt}J_{i}-\delta ^{2}J_{i}\leq pf_{i}(1-U_{i})^{-p-1}J_{i},\quad i=1,\ldots ,I,\\ & \tfrac {d}{dt}J_{0}-\delta ^{2}J_{0}\leq (\tfrac {2q}{h}U^{-q-1}_{0}+pf_{0}(1-U_{0})^{-p-1})J_{0}. \end{flalign}

We observe from 11–?? that

\begin{flalign*} & J_{i}(0)=\delta ^{2}\varphi _{i}+f_{i}(1-\varphi _{i})^{-p}+\epsilon \varphi ^{-q}_{i}\leq 0,\quad i=1,\ldots ,I,\\ & J_{0}(0)= \delta ^{2}\varphi _{0}+f_{0}(1-\varphi _{0})^{-p}+(\epsilon -\tfrac {2}{h})\varphi ^{-q}_{0}\leq 0. \end{flalign*}

We deduce from lemma 2 that \(J_{h}(t)\leq 0,\: \text{for}\: t\in (0,T_{q}^{h})\), which implies that

\begin{flalign} & \tfrac {dU_{i}}{dt}+\epsilon U^{-q}_{i}\leq 0\quad \text{for}\quad t\in (0,T_{q}^{h}),\: i=0,\ldots ,I. \end{flalign}

By the above inequality we obtain the following form \(U^{q}_{i}dU_{i}\leq -\epsilon dt\: \text{for}\: t\in (0,T_{q}^{h})\) and \(i=0,\ldots ,I\). Integrating this inequality over \([0,T_{q}^{h})\), we obtain

\begin{flalign} & T_{q}^{h}-t\leq \tfrac {U^{q+1}_{i}(t)}{\epsilon (q+1)}.\label{estimation temps1} \end{flalign}

From lemma 5, we have \(U_{0}(0)=\| \varphi _{h}\| _{\inf }\) and taking \(t=0\) in 15 we get the desired result.

Proof â–¼

The following result concerns the lower bound for the quenching rate.

Remark 7

Using the inequality 15 we obtain \(T_{q}^{h}-t_{0}\leq \tfrac {U^{q+1}_{0}(t_{0})}{\epsilon (q+1)}\quad \text{for}\quad t_{0}\in (0,T_{q}^{h})\), which implies that \(U_{0}(t)\geq (\epsilon (q+1))^{\frac{1}{q+1}}(T_{q}^{h}-t)^{\frac{1}{q+1}}\quad \text{for}\quad t\in (0,T_{q}^{h})\).

Theorem 8

Assume that 11–?? remains true. Then, near the quenching time \(T_{q}^{h}\), the solution \(U_{h}\) to problem 1–?? has following quenching rate estimate

\( U_{0}(t)\sim (T_{q}^{h}-t)^{\frac{1}{(q+1)}}\), in the sense that there exist two positive constants \(C_{1}\), \(C_{2}\) such that

\begin{flalign*} \displaystyle C_{1}(T_{q}^{h}-t)^{\frac{1}{q+1}}\leq U_{0}(t)\leq C_{2}(T_{q}^{h}-t)^{\frac{1}{q+1}},\quad \text{for}\quad t\in (0,T_{q}^{h}). \end{flalign*}

Proof â–¼
remark 7 ensures the term of left hand side.

Let \(i_{0}\) be such that \(U_{i_{0}}(t)=\displaystyle \min _{0\leq i\leq I} U_{i}(t)\), \(t\in (0,T_{q}^{h})\). From lemma 5 we obtain

\begin{align*} & \delta ^{2}U_{i_{0}}(t)=\displaystyle \tfrac {U_{i_{0}+1}(t)-2U_{i_{0}}(t)+U_{i_{0}-1}(t)}{h^{2}}\geq 0,\quad i_{0}=1,\ldots ,I-1,\\ & \delta ^{2}U_{0}(t)=\displaystyle \tfrac {2U_{1}(t)-2U_{0}(t)}{h^{2}}\geq 0,\\ & \delta ^{2}U_{I}(t)=\displaystyle \tfrac {2U_{I-1}(t)-2U_{I}(t)}{h^{2}}\geq 0. \end{align*}

Which leads to \(\displaystyle \tfrac {dU_{0}}{dt}\geq -\tfrac {2}{h}U^{-q}_{0}\), integrating over \((t,T_{q}^{h})\) we have

\[ U_{0}\leq (\tfrac {2(q+1)}{h})^{\frac{1}{q+1}}(T_{q}^{h}-t)^{\frac{1}{q+1}}. \]

Hence, we have \(U_{0}\leq C_{2 }(T_{q}^{h}-t)^{\frac{1}{q+1}}\) and the proof is completed.

Proof â–¼

Remark 9

Let us point out that the quenching rate for the numerical scheme,

\((T_{q}^{h}-t)^{\frac{1}{q+1}}\), is different from the continuous one, \((T_{q}-t)^{\frac{1}{2(q+1)}}\) [ 22 ] .

4 Convergence of the semidiscrete Quenching time

In this section, with suitable assumptions, we establish the convergence of the quenching time of the approximate semidiscrete solution to the quenching time of the theoretical solution.
The next theorem establishes that, for each fixed time interval \([0,T]\), \((T{\lt} T_{q})\) where \(u\) is defined, the solution of the semidiscrete problem 1-?? approximates \(u\), as \(h\rightarrow 0\) .

Theorem 10

Assume that the problem 1 has a solution \(u\in C^{4,1}([0,1]\times [0,T])\) such that \(\displaystyle \inf _{t\in [0,T]}\| u(.,t)\| _{\infty }=\alpha {\gt}0\) and the initial condition \(\varphi _{h}\) at ?? verifies

\begin{align} \| \varphi _{h}-u_{h}(0)\| _{\infty }=\circ (1), \quad as\quad h\rightarrow 0,\label{uh moins u tend vers 0} \end{align}

where \(u_{h}(t)=(u(x_{0},t),\ldots ,u(x_{I},t))^{T}\), \(t\in [0,T]\). Then, for \(h\) small enough, the semidiscrete problem 1–?? has a unique solution \(U_{h}\in C^{1}([0,T],\mathbb {R}^{I+1})\) such that

\begin{align*} \displaystyle \max _{t\in [0,T]}(\| U_{h}(t)-u_{h}(t)\| _{\infty })=O( \| \varphi _{h}-u_{h}(0)\| _{\infty }+h^{2})\quad as\quad h\rightarrow 0. \end{align*}
Proof â–¼
The semidiscrete problem (1)–(??) has for each \(h\), a unique solution \(U_{h}\in C^{1}([0,T_{q}^{h}),\mathbb {R}^{I+1})\). Let \(t(h)\) be the greatest value of \(t{\gt}0\) such that

\begin{align} \| U_{h}(t)-u_{h}(t)\| _{\infty }{\lt}\tfrac {\alpha }{2}\quad for\quad t\in (0,t(h)).\label{majorationUh-uh} \end{align}

The relation (16) implies \(t(h){\gt}0\) for \(h\) small enough.

Let \(t^{\star }(h)=\min \{ t(h),T\} \), using the triangle inequality we obtain

\(\| U_{h}(t)\| _{\inf }\geq \| u(.,t)\| _{\inf }-\| U_{h}(t)-u_{h}(t)\| _{\infty }\) for \(t\in (0,t^{\star }(h))\), which implies that

\begin{align} \| U_{h}(t)\| _{\inf }\geq \tfrac {\alpha }{2} \quad for \quad t\in (0,t^{\star }(h)). \label{minoration Uh} \end{align}

Let \(e_{h}(t)=U_{h}(t)-u_{h}(t)\) be the error discretization and the vector \(z_{h}(t)\) defined by \(z_{i}(t)=e^{(K+1)(t-x^{2}+2x)}(\| \varphi _{h}-u_{h}(0)\| _{\infty }+Lh^{2}),\quad 0\leq i\leq I,:\ t\in (0,t^{\star }(h))\). Using the lemma 2 we can prove that \(z_{i}(t){\gt}\lvert e_{i}(t)\rvert \) for \(t\in (0,t^{\star }(h)),\: i=0,\ldots ,I\), which implies that

\begin{align*} \| U_{h}(t)-u_{h}(t)\| _{\infty }\leq e^{(K+1)(t-x^{2}+2x)}(\| \varphi _{h}-u_{h}(0)\| _{\infty }+Lh^{2}),\quad t\in (0,t^{\star }(h)). \end{align*}

Suppose that \(T{\gt}t(h)\), from (17) we have \(\frac{\alpha }{2}=\| U_{h}(t(h))-u_{h}(t(h))\| _{\infty }\leq e^{(K+1)(T-x^{2}+2x)}(\| \varphi _{h}-u_{h}(0)\| _{\infty }+Lh^{2})\). Since the term on the right hand side of the above inequality goes to zero as \(h\) goes to zero, we deduce that \(\tfrac {\alpha }{2}\leq 0\), which is impossible.

Proof â–¼

Theorem 11

Let \(T_{q}\) be the quenching time of the solution \(u\) of 1 such that

\(u\in C^{4,1}([0,1]\times [0,T_{q}))\). We assume that the initial data at ?? satisfies

\(\| \varphi _{h}-u_{h}(0)\| _{\infty }=o(1),\: \text{as} \quad h\rightarrow 0\). Then under the hypothesis of theorem 6, the solution \(U_{h}\) of the problem 1–?? quenches in finite time \(T^{h}_{q}\) and we have

\begin{align*} \displaystyle \lim _{h\rightarrow 0}T^{h}_{q}=T_{q}. \end{align*}

Proof â–¼
Let \(\eta {\gt}0\), there exists a positive constant \(\alpha \) such that

\begin{align} \tfrac {1}{\epsilon }\frac{s^{q+1}}{q+1}\leq \tfrac {\eta }{2}\quad for\quad s\in [0,\alpha ].\label{inegalie convergence thq} \end{align}

Since \(u\) quenches in a finite time \(T_{q}\), there exists \(T_{0}\in (T_{q}-\tfrac {\eta }{2},T_{q})\) such that
\(0{\lt}\| u(.,t)\| _{\inf }\leq \tfrac {\alpha }{2}\) for \(t\in [T_{0},T_{q})\). From theorem 10, the problem 1-?? has a solution \(U_{h}(t)\) such that \(\| U_{h}(t)-u_{h}(t)\| _{\infty }\leq \tfrac {\alpha }{2}\) for \(t\in [0,T_{1}]\), where \(T_{1}=\tfrac {T_{0}+T_{q}}{2}\), which gives \(\| U_{h}(T_{1})-u_{h}(T_{1})\| _{\infty }\leq \tfrac {\alpha }{2}\). Using the triangle inequality, we get \(\| U_{h}(T_{1})\| _{\inf }\leq \| U_{h}(T_{1})-u_{h}(T_{1})\| _{\infty }+\| u_{h}(T_{1})\| _{\inf }\leq \alpha \). From theorem 6, \(U_{h}\) quenches in a finite time \(T_{q}^{h}\). We deduce from remark 7 and 19 that

\begin{equation*} \lvert T_{q}^{h}-T_{q}\rvert \leq \lvert T_{q}^{h}-T_{1}\rvert + \lvert T_{1}-T_{q}\rvert \leq \frac{1}{\epsilon }\tfrac {\| U_{h}(T_{1})\| ^{q+1}_{\inf }}{q+1}+\frac{\eta }{2}\leq \eta . \end{equation*}

Proof â–¼

5 Numerical experiments

In the section, we present numerical approximations to the quenching time of problem (1) in the case where \(u_{0}(x) = -x^{\varepsilon }+\varepsilon x+\varepsilon ^{-1/q}\) with \(0{\lt}p\leq 1\), \(0{\lt}q\leq \frac{1}{2}\), \(\varepsilon =\tfrac {101}{100}\) and \(f(x)=\tfrac {1}{1000}\). We also discuss the quenching sets. To do this, we transform the semidiscrete scheme (1)–(??) into the following semidiscrete equations. Set us \(V_{i}(t)=\displaystyle \tfrac {1}{U_{i}(t)}\), we obtain

\begin{align} \tfrac {d}{dt}V_{i}(t)=\tfrac {1}{h^{2}}\Big(2V_{i}(t)-\tfrac {V^{2}_{i}(t)}{V_{i-1}(t)}-\tfrac {V^{2}_{i}(t)}{V_{i+1}(t)}\Big)-f_{i}V^{p+2}_{i}(t)(V_{i}(t)-1)^{-p},i=0,\ldots ,I-1,\label{transformat1}\\ \tfrac {d}{dt}V_{0}(t)=\tfrac {2}{h^{2}}\Big(V_{0}(t)-\tfrac {V^{2}_{0}(t)}{V_{1}(t)}\Big)+\tfrac {2}{h}V^{2+q}_{0}(t)-f_{0}V^{2+p}_{0}(t)(V_{0}(t)-1)^{-p},\label{transformat2}\\ \tfrac {d}{dt}V_{I}(t)=\tfrac {2}{h^{2}}\Big(V_{I}(t)-\tfrac {V^{2}_{I}(t)}{V_{I-1}(t)}\Big)-f_{I}V^{p+2}_{I}(t)(V_{I}(t)-1)^{-p}\label{transformat3},\\ V_{i}(0)=(\varphi _{i})^{-1},\quad i=0,\ldots ,I.\label{transformat4} \end{align}
Using the method presented by Hirota and Ozawa [ 12 ] , we transform the semidiscrete scheme (20)–(23) into a tractable form by the arc length transformation technique (see [ 21 ] , [S. Moriguti, C. Okuno, R. Suekane, M. Iri, K. Takeuchi, Ikiteiru Suugaku - Suuri Kougaku no Hatten (in Japanese), Baifukan, Tokyo, 1979.]) like this:

\begin{equation} \label{arc} \begin{cases} \dfrac {d}{d\eta }\begin{pmatrix} t \\ V_0 \\ \vdots \\ V_I \end{pmatrix} = \dfrac {1}{\sqrt{1+\sum _{i=0}^I f_i^2}}\begin{pmatrix} 1 \\ f_0 \\ \vdots \\ f_I \end{pmatrix}, \quad 0<\eta <\infty ,\\ t(0)=0, \quad V_i(0)= (\varphi _{i})^{-1}, \quad i=0,\ldots ,I, \end{cases} \end{equation}
24

where

\begin{align*} & f_{0}=\tfrac {2}{h^{2}}\Big(V_{0}(t)-\tfrac {V^{2}_{0}(t)}{V_{1}(t)}\Big)+\tfrac {2}{h}V^{2+q}_{0}(t)-f_{0}V^{2+p}_{0}(t)(V_{0}(t)-1)^{-p},\\ & f_{i}=\tfrac {1}{h^{2}}\Big(2V_{i}(t)-\tfrac {V^{2}_{i}(t)}{V_{i-1}(t)}-\tfrac {V^{2}_{i}(t)}{V_{i+1}(t)}\Big)-f_{i}V^{p+2}_{i}(t)(V_{i}(t)-1)^{-p},\quad i=1,\ldots ,I-1,\\ & f_{I}=\tfrac {2}{h^{2}}\Big(V_{I}(t)-\tfrac {V^{2}_{I}(t)}{V_{I-1}(t)}\Big)-f_{I}V^{p+2}_{I}(t)(V_{I}(t)-1)^{-p},\\ & V_{i}(0)=(\varphi _{i})^{-1},\quad i=0,\ldots ,I. \end{align*}

“\(\eta \)” is the arc length and we have \(d\eta ^{2}=dt^{2}+\displaystyle \sum ^{I}_{i=0}dV^{2}_{i}\). Note that in the transformation below the variables \(t\) and \(V_{i}\) are functions of \(\eta \) such that \(\displaystyle \lim _{\eta \rightarrow \infty }t(\eta )=T^{h}\quad and \quad \displaystyle \lim _{\eta \rightarrow \infty }\| V_{h}(\eta )\| _{\infty }=\infty \). Now we introduce \(\{ \eta _{j}\} \) which is the sequence of the arc lengths and we apply an ODE solver to (24) for each value of \(\eta _{j}\) in order to generate a linearly convergent sequence to the blow-up time. The resulting sequence is accelerated by the Aitken \(\Delta ^{2}\) method [ 3 ] . We use the DOP54 [ 11 ] as the adaptive code for the integration of the ODEs. It has been written by Hairer et al. [ 11 ] based on explicit Runge-Kutta method of order \((4)5\) due to Dormand and Prince [ 8 ] . Let us define the sequence \(\eta _{j}\) by \(\eta _{j}=2^{12}.2^{j}\) \((j=0,\ldots ,10)\), and the parameters in the DOP54 are \(InitialStep=0\) and \(AbsTol=RelTol=1.e-15\). The parameters \(AbsTol\) and \(RelTol\) specify the tolerances of the absolute and relative errors, respectively, and \(InitialStep\) is used to choose the manner in which the errors are controlled. In the following tables, in rows, we present the numerical quenching times \(T^{h}\) of problem (1)-(??), the Step and the orders of the approximations corresponding to meshes of \(16\), \(32\), \(64\), \(128\), \(256\) and \(512\). The order \((s)\) of the method is computed from

\begin{equation*} \frac{\log ((T^{4h}-T^{2h})/(T^{2h}-T^{h}))}{\log (2)}. \end{equation*}

\(I\)

\(T^{h}\)

\(Steps\)

\(s\)

16

0.320236408

6304

-

32

0.318644976

12122

-

64

0.318150680

23261

1.69

128

0.318001392

44785

1.73

256

0.317957379

87402

1.76

512

0.317944663

178187

1.79

Table 1 Semidiscrete solution for \(p=1/4\), \(q=1/4\).

\(I\)

\(T^{h}\)

\(Steps\)

\(s\)

16

0.200847014

4490

-

32

0.198905750

8498

-

64

0.198304689

16154

1.69

128

0.198124110

30905

1.73

256

0.198071185

59982

1.77

512

0.198055981

120646

1.80

Table 2 Semidiscrete solution for \(p=1/4\), \(q=1/2\).

\(I\)

\(T^{h}\)

\(Steps\)

\(s\)

16

0.320475664

6306

-

32

0.318883760

12127

-

64

0.318389339

23270

1.69

128

0.318240017

44802

1.73

256

0.318195997

87437

1.76

512

0.318183278

178267

1.79

Table 3 Semidiscrete solution for \(p=1/2\), \(q=1/4\).

\(I\)

\(T^{h}\)

\(Steps\)

\(s\)

16

0.201026888

4492

-

32

0.199084652

8502

-

64

0.198483334

16161

1.69

128

0.198302688

30920

1.74

256

0.198249747

60011

1.77

512

0.198234539

120710

1.80

Table 4 Semidiscrete solution for \(p=1/2\), \(q=1/2\).

Remark 12

From the tables, we can see the convergence of \(T^{h}\) to the quenching time of the solution of 1, since the rate of convergence is near \(2\), which is just the accuracy of the difference approximation in space. The tables of our numerical results show that there is a relationship between the quenching time and the flow on the boundary on the one hand and the absorption in the interior of the domain on the other hand. Indeed, when the absorption in the interior of the domain is constant \((p=1/4)\) and that the flow on the boundary increases from \(1/4\) to \(1/2\), the quenching time decreases from \(0.318\) to \(0.198\) whereas when the flow on the boundary is constant \((q=1/4)\) and that the absorption in the interior of the domain increases from \(1/4\) to \(1/2\), the quenching time remains substantially the same at \(0.318\). The absorption in the interior of the domain has in fact no essential effect upon the quenching behavior of problem 1, whereas the flow on the boundary leads to the quenching, which is in agreement with the theoretical results [ 23 , 22 ] .

For other illustrations, in what follows, we present several graphs to illustrate our analysis. In Figures 14, we have used the case where \(I=64\) and \(p=1/4\). We can appreciate in 1 and 2 that the discrete solution quenches in a finite time at the first node, which is well known in a theoretical point of view [ 23 , 22 ] . In 3 and 4 we see that the approximation of \(u(x,T)\) increases and gives the value zero at the first node. The time \(T\) represents the quenching time of the solution \(u\).

\includegraphics[height=4in, width=5.0776in]{SOL2.jpg}
Figure 1 Evolution of the semidiscrete solution, \(q=1/4\).
\includegraphics[height=4in, width=5.0776in]{SOL1.jpg}
Figure 2 Evolution of the semidiscrete solution, \(q=1/2\).

\includegraphics[height=4in, width=5.0776in]{NODE1.jpg}
Figure 3 Profile of the approximation of \(u(x,T)\), \(q=1/4\).
\includegraphics[height=4in, width=5.0776in]{NODE2.jpg}
Figure 4 Profile of the approximation of \(u(x,T)\), \(q=1/2\).

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