Abstract

Quenching characteristics based on the two-dimensional (2D) nonlinear unsteady convection-reaction-diffusion equation are creatively researched. The study develops a 2D compact finite difference scheme constructed by using the first and the second central difference operator to approximate the first-order and the second-order spatial derivative, Taylor series expansion rule, and the reminder-correction method to approximate the three-order and the four-order spatial derivative, respectively, and the forward difference scheme to discretize temporal derivative, which brings the accuracy resulted meanwhile. Influences of degenerate parameter, convection parameter, and the length of the rectangle definition domain on quenching behaviors and performances of special quenching cases are discussed and evaluated by using the proposed scheme on the adaptive grid. It is feasible for the paper to offer potential support for further research on quenching problem.

1. Introduction

As a common class of thermodynamic problems, quenching phenomena research has been widely applied in engineering area. There exist some typical researches including the flow, thermal properties, and its working mechanism of an object in heat associated with quenching phenomena [1, 2], the forced convection quench of hot sheets in super-cooled liquid [3], the quench process speed-up, and its quench characteristics in porous interface processing [4, 5]. In recent years, more and more scholars concern quenching problem built on parabolic equations [6–10], in which the authors depended on the different parabolic equations to analyze corresponding quench features. Additionally, quenching phenomena based on parabolic equation systems have attracted researchers’ attentions [11–13]. Nonlinear degeneration singularity reaction-diffusion equations or convection-reaction-diffusion equations, as a branch of parabolic equations, have been usually employed to handle quenching problems. References [14–16] focused on the quenching phenomena of the nonlinear degeneration singularity reaction-diffusion equations whereas Refs. [17, 18] focused on the quenching phenomena of the nonlinear degeneration singularity convection-reaction-diffusion equations. Selcuk discussed quench performances of solution and its time derivative and estimation of quenching time under special conditions [14]. Ge et al. devoted to an adaptive compact difference scheme to solve quench problems [15]. Beauregard analyzed quenching properties of the fractional Kawarada equation by using a new numerical method and proved the proposed method monotonic, nonnegative, and linearly stable [16]. Zhou et al. theoretically investigated quenching characteristics of the nonlinear degeneration singularity convection-reaction-diffusion equation [17]. Zhu and Rui constructed a high-accuracy method with adaptation mechanism for studying quenching behaviors and analyzed the influence of important elements on quenching [18].

References [6–18] concentrated on the quenching phenomena of the 1D heat equations. The quenching phenomena of the 2D heat equations are described as follows. References [19–21] used numerical method with adaptive algorithm to research degeneration singularity problem of the two-dimensional nonlinear reaction-diffusion equations. A Peaceman-Rachford finite difference scheme based on semiadaptive grids with stability and convergence under certain condition was posed to analyze the quenching behaviors of the 2D reaction-diffusion equation in Ref. [19]. Numerical analyses showed that the method is very efficient and reliable. An adaptive Peaceman-Rachford-Strang splitting method on exponentially evolving meshes was constructed to resolve the 2D problem of quenching type in Ref. [20]. Computational experiments stated that the method is of the satisfactory effectiveness, efficiency, and numerical stability. An exponential splitting method with spatial semidiscretization on arbitrary nonuniform meshes was investigated to explore the 2D degenerate stochastic Kawarada equation in Ref. [21]. Zhu and Ge developed an adaptive ADI strategy to simulate quenching phenomena based on 2D convection-reaction-diffusion equation [22]. Numerical cases illustrated that the method is of the positivity, monotonicity, and numerical stability. The low-accuracy strategies were adopted in the most aforementioned researches [6–15, 19–21]. There are only Refs. [15, 18, 22] delivering the high-accuracy strategies, in which the first two refer to the one-dimensional problem and the last one refers to the two-dimensional problem. It is well known that the development of high-order finite difference scheme of the 2D nonlinear convection-reaction-diffusion equations is maturing [23–25]. Reference [23] developed a new high-order difference approach for the 2D convection-reaction-diffusion equation with a small diffusivity, which firstly posed the 1D high-accuracy difference scheme and then extended it to the 2D case a nine-point stencil by using alternating direction algorithm and last resolve the 2D steady incompressible Navier-Stokes equations. In order to simulate groundwater pollution problems, Li et al. put forward a fourth-order compact difference scheme with unconditional stability and the second-order temporal accuracy, the fourth-order spatial accuracy built on the 2D convection-reaction-diffusion model, which was efficient through experiments [24]. Wu and Zhai combined exponential transformation and quadratic interpolation polynomial with Padé approximation to study the 2D time-fractional convection-dominated diffusion equation, which owned the higher accuracy and used alternating direction implicit algorithm to alleviate computational amount [25].

In summary, it is easily found that there are much more low-order strategies than high-order strategies to solve degenerate singularity problems of the 2D reaction-diffusion equation. Especially, high-order difference schemes received few attentions for solving quenching problem of the two-dimensional convection-reaction-diffusion equation. It is the high-precision algorithms that have advantages in dealing with such problems because of its high accuracy and efficiency. So it is a meaningful try to explore the high-order compact difference schemes on adaptation mesh for analyzing quenching problems of the 2D degenerate singular reaction-diffusion equation with convection function. Going down this idea, we extend this study from Ref. [18] and construct a compact difference scheme on adaptive grid for solving the 2D unsteady convection-reaction-diffusion equation to explain the corresponding quenching phenomena. According to Ref. [18], we extend its 1D high-order compact finite difference scheme on adaptive mesh to a 2D strategy and use it to analyze the quenching behaviors of the 2D convection-reaction-diffusion combustion model. There exist three contributions of this paper. Firstly, we represent a new 2D high-order compact difference scheme for solving the corresponding unsteady convection-reaction-diffusion equation and give its accuracy performances. Secondly, we apply the scheme to explain the 2D reaction-diffusion of quenching type with and without convection function, respectively. Thirdly, we investigate a series of quenching characteristics including quenching time, quenching location, and so on, from which we can discover impacts of the parameters , , and () on quenching behaviors. There are five parts in the paper. Section 1 introduces the theme of this study. Section 2 describes carefully the proposed scheme of high-order accuracy. Section 3 introduces adaptive mesh algorithm. Section 4 stimulates some typical numerical samples to explore and explain quenching problems. Section 5 draws the conclusion.

2. 2D Problem and Difference Scheme

2.1. 2D Nonlinear Convection-Reaction-Diffusion Equation

The typical convection-reaction-diffusion equation of quenching type is written as

Its boundary conditions are

Its initial conditions is

This semilinear degenerate problem model involving two spatial dimensions is regarded as Eq. (1) with the boundary and initial conditions of Eqs. (2) and (3). The solution represents the temperature of the combustion chamber. refers to the smooth domain of the rectangle combustor in which and combustion chamber sizes and are the length of the definition area, , and is its boundary. and are the convection functions of and , and is the degeneracy function and degeneration parameter . The singularity source is strictly increasing for with, and singularity parameter is the power of the singular source term .

With the aid of the intermediate variables and , we replace and into Eq. (1) which can be defined as

For the convenience of expression, we use and instead of and . Then, the above formulation can be rearranged aswhere Correspondingly, the boundary conditions of Eq. (5) are

The initial conditions of Eq. (5) is

In the physical application, we rely on Eqs. (5)–(7) to compute and . Through observing a large number of values of and , we can capture quenching moment, i.e., quenching occurs when infinitely close to 1 and becomes so huge that its value blows up.

2.2. The Proposed Compact Difference Scheme

According to the idea given in [18], we employ the first and the second central difference operator to approximate the first-order and the second-order spatial derivatives of direction and direction, respectively, and the forward difference operator to discrete temporal derivative. The proposed high-accuracy finite difference scheme of Eq. (5) is deduced below. After the first-order and the second-order spatial derivatives of direction and direction are discretized on the nonuniform mesh, respectively, a scheme approximating the 2D unsteady convection-diffusion equation dispersed at point can be written aswhere

We use Taylor series to obtain the derivative expansions of direction: and , and the derivative expansions of direction: and from Eq. (5). The four expressions are substituted into the corresponding terms in Eq. (8). , which is the truncation error of Eq. (8). Omitting the truncation errors, we consider the situation of point for Eq. (8)where

The second-order derivative of time from Eq. (8) is . By using the forward difference scheme, we can get

Relying on the second-order backward Euler difference scheme, we can get

Provided , Eq. (14) can be derived as

The first-order and the second-order derivate of with regard to space variables and are discretized by the central difference scheme. The first-order derivate of with regard to time variable is discretized by the forward difference scheme. Subsequently, after the difference approximations are carried out to , , , , , , , and , a linear system is formed aswhere

Through the prior deducing procedure, it can be seen that the truncation error of the Eq. (15) is. When and , it owns spatial accuracy of fourth order and temporal accuracy of second order.

3. Adaptive Grid Structure

3.1. Adaptive Grid Structure in Time

The adaptation mesh technique in the temporal and the spatial direction is employed for solving the 2D problem of quenching type. Obviously, the 2D adaptive mesh algorithm can be derived from the 1D case. The arc-length monitor function of the temporal derivative function resulting from equal distribution principle is used to design the temporal and the spatial moving mesh algorithm, respectively. From Ref. [19, 20], a self-adaptive grid in the time direction is given as follows. A maximized ratio equation for computing the adaptive temporal interval is

The standard uniform central difference formula is substituted for the in Eq. (17). Of course, when the prior temporal step is given, the current temporal step can be calculated through Eq. (17).

3.2. Adaptive Grid Structure in Space

Let a spatial adaptation algorithm be deduced as follows. is taken as the monitor function in direction and is taken as the monitor function in direction.

and point to the set of 2D original spatial mesh points in direction and direction. The 2D computational area is transformed as in direction and in direction through mesh movement. The continuous function and in the definition area equally distributes over the 2D mesh refreshed. In fact, the 1D adaptive mesh algorithm individually carries out in each spatial direction. According to [19, 20], we can get the semi-implicit scheme in direction

Similarly, we can get the semi-implicit scheme in direction

3.3. Iterative Adaption Algorithm

This paragraph shows the iterative process of the presented method in the paper.

Step 1. Depending on the spatial mesh , at the th time layer, we can obtain the corresponding monitor functions and via Eqs. (18) and (19).

Step 2. The point set refreshed is extracted from the prior grid via Eq. (20) whereas refreshed is extracted from by Eq. (21). We use the new point sets to get the replacement for the old point sets iteratively once , , which takes 10-5. The last value of takes .

Step 3. By virtue of the last grid point sets , , and area ratio algorithm, the solutions in the th time line are all estimated. Combining with obtained from Eq. (17), in the th time line can be calculated by Eq. (15).

Step 4. Repeat steps from the first to the third till quenching occurs, or the solution converges to a steady solution.

4. Simulation Demonstration

4.1. Numerical Case

This is a common numerical sample utilized to evaluate the performance of Eq. (15) measure by Eq. (22)

In the paper, . The initial and boundary values can be computed through the exact solution. The tested results display in Table 1, in which there are four criteria containing Max. error, Aver. error, CPU time, and Conv. rate. Max. error means the maximal absolute error between analysis solution and difference solution; Aver. error means the average absolute error; CPU time means the running time of system; Conv. rate means convergence rate between the two interfacing mesh numbers. The nonuniform mesh follows the rule: in direction and in direction, where point to intervals of spatial direction and , whereas and mean telescopic transformation coefficient and . There exist six difference schemes that are compared on another for the example in Table 1, which are the schemes in Ref. [26] on the uniform and the nonuniform grid, the schemes in Ref. [27] on the uniform and the nonuniform grid, and the proposed schemes on the uniform and the nonuniform grid. The schemes in Ref. [27] and the proposed schemes all chose the five spatial intervals 16, 32, 64, 128, 256, and and the four aforementioned criteria. Only the schemes in Ref. [26] choose the five spatial intervals 16, 32, 64, 128, and and the three aforementioned criteria with the exception of Aver. error.

Table 1 
The parameter values for common example.

With regard to Max. error and Aver. error, the schemes on the uniform grid are inferior to the schemes on the nonuniform grid; the proposed schemes are superior to the other schemes, and the schemes (nonuniform) in Ref. [27] are superior to the scheme (nonuniform) in Ref. [26], and the tendency becomes more obviously as rises. When is 128, the maximal absolute error of the scheme in Ref. [27] (uniform) is , and the maximal absolute error of the scheme in Ref. [27] (nonuniform) is ; the maximal absolute error of the scheme in Ref. [26] (uniform) is , and the maximal absolute error of the scheme in Ref. [26] (nonuniform) is ; the maximal absolute error of the proposed scheme (uniform) is , and the maximal absolute error of the proposed scheme (nonuniform) is . When is 256, the average error of the scheme in Ref. [27] (uniform) is , and the average error of the scheme in Ref. [27] (nonuniform) is ; the average error of the proposed scheme (uniform) is , and the average error of the proposed scheme (nonuniform) is . In terms of CPU time, the scheme (nonuniform) in Ref. [27] spends the most time among the schemes when is 32, 64, 128, and 256, and the tendency becomes more obviously as rises; the schemes in Ref. [26] spend much more time than the proposed schemes do. Typically, the running time of the schemes on the uniform grid is less than that of the schemes on the nonuniform grid. It just means that the latter complies with a time-for-space rule, which refers to it may improve the spatial accuracy of an algorithm by adding its running time.

4.2. Quenching Case without Convection Term

When and , with the initial and boundary conditions Eqs. (6) and (7), the original Eq. (5) can be described as

For the situation of and , we take advantage of Eq. (15) to approximate Eq. (24) and gain quenching case without convection term. We set the three parameters , , and to the case which had been illustrated in Refs [19, 20]. In this case, the initial temporal step is configured as , and the initial space step is configured as .

Table 2 offers quenching performances of the three different schemes, which are from Refs [19, 20] besides the proposed method on adaptive mesh, for , , and without convection term. The criteria in Table 2 are explained as follows. refers to the point of quenching location, refers to the quenching time, refers to the maximum temperature value immediately before quenching happens, and refers to the maximum variation of temperature with respect to time immediately before quenching happens. For the three schemes, their quenching locations are all . There are subtle differences among the three schemes for quenching time and . of the proposed scheme is 0.5990174307488624, and of the proposed scheme is 0.9901182463562488. But there are greater differences among the three schemes for . of the proposed scheme is 83.20947696438735 while those of the schemes in Refs [19, 20] are 148.887767 and 1249.917563. There is no effect of the differences on quenching research. Figure 1 offers the three-dimensional scenes of and with regard to spatial variables , when the time is 0.5990174307488624, respectively. From the two 3D surfaces at the time that is immediately before quenching occurs, it is found that and at the quenching location and the quenching time .

Table 2 
Quenching performances of different schemes for , , and without convection term.
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Figure 1 
(a) The 3D plots of immediately before quenching occurs, and (b) the 3D plots of immediately before quenching occurs. The parameters are , , and without convection term.
4.3. Quenching Case with Convection Term

We will investigate what role degeneration parameter , convection parameter , and combustion chamber size () in Eqs. (5)–(7) play during the quenching process and enumerate the representative Case 5.0 to show special quenching features when and , where and take the constants and convection parameter . When singularity parameter is larger than 1, the quenching situation is complex. Therefore, we only consider quenching cases of . In numerical samples, and range from -200 to 200; range from 15 to 300000. These are the same to Section 4.4. To better illustrate the problems, we choose some representative data from lots of tests to discuss quenching phenomena in the next paragraphs. For the convection model , we set as the initial () step and as the initial step. Tables 3–6 offer some typical quenching cases to demonstrate the relationship between the three groups of parameters and quenching phenomena. Case 5.0 recorded as a reference in Table 3, i.e., , is compared with other cases in Tables 3–5. Table 6 displays the specific quenching data of Case 5.0. Figures 1–3 show the quenching statuses of Case 5.0 relative to , , , , and .

Table 3 
Quenching data of based on convection term .
Table 4 
Quenching data of and based on convection term .
Table 5 
Quenching data of and based on convection term .
Table 6 
The maximal values and locations of and for based on convection term .

Except Case 5.0, there are ten quenching cases chosen in Table 3. The ten items list as follows: Case 5.1.1 is ; Case 5.1.2 is ; Case 5.1.3 is ; Case 5.1.4 is ; Case 5.1.5 is ; Case 5.1.6 is ; Case1.1.7 is ; Case1.1.8 is ; Case 5.1.9 is ; Case 5.1.10 is . Although the program may run when , it is hard to form quenching behaviors. If belongs to the definition domain , the quenching phenomena will occur. From serial numerical cases, we can easily see the performances of quenching location and time. The quenching location declines as evolves and finally converges to . There is an intermediate point dividing the definition domain as and . The quenching location and do decline as evolves in the first subdomain and converge to in the second subdomain. The quenching time reaches the maximum 9.686902288384093 when . and are in positive proportion when in .

We choose ten items written in Table 4 to display the quenching characteristics related to the parameters and . Case 5.2.1 is ; Case 5.2.2 is ; Case 5.2.3 is; Case 5.2.4 is ; Case 5.2.5 is ; Case 5.2.6 is ; Case 5.2.7 is ; Case 5.2.8 is ; Case 5.2.9 is ; Case 5.2.10 is ; and in Eq. (5) are theoretically equivalent, i.e., . A valid scope of is for these cases. Because quenching occurs fastest when takes 4, is thought as a special point for quenching time. Quenching time is inversely proportional to when and positively proportional to when . Quenching location rises as increases when and takes when .

Table 5 offers seventeen referencing cases. The first ten items are Case 5.3.1 is ; Case 5.3.2 is ; Case 5.3.3 is ; Case 5.3.4 is ; Case 5.3.5 is ; Case 5.3.6 is ; Case 5.3.7 is ; Case 5.3.8 is ; Case 5.3.9 is ; Case 5.3.10 is . We configure and for the prior ten items. Experiments show even if reaches , quenching phenomena still happen. Furthermore, if falls in the interval , then quenching phenomena will occur normally, and quenching location and do not decrease monotonously as declines. When lies in the area of , always keeps . In terms of quenching time, there is no obvious linear relationship between and in a small range of the definition field of . If the measurement scale is enlarged in the domain of , especially when reaches 2900, will rise with the increase of , i.e., is 3.376127343223672 when is 2900 and is 39.49651984658089 when is .

We take Case 5.0 to demonstrate specific quenching phenomena. In this case, quenching occurs at , , and with the parameters . The next context declares more details about quenching. Let us observe the performances of the solution , the derivative function , and their time and space locations of five stages relative to quenching for Case 5.0, which show in Table 6 and Figures 2–4. The first four items reflect the four states before quenching occurs in Table 6. The last item represents the quenching state in which is more than 1 and reaches 17.07446290189856, blows up and reaches , the position of is (, 0.2902651144597507), and the position of is (, ) when . Figures 2(a) and 2(b) denote the profiles of both solution and derivative function . In the aforementioned figures, the maximum values of and grow from 0 to . As soon as time variable arrives at quenching spot and quenching time , we have and . By locating the quenching time , we can draw the plots of and . Figures 3(a)–3(d) depict the function relationships between and , between and , between and , and between and , respectively, when is. Owing to the identity of and in the original equation, Figures 3(a) and 3(c) have the same shape, and Figures 3(b) and 3(d) have the same shape.

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Figure 2 
The graphs of solution and : (a) as increases, and (b) as increases from the initial moment to the occurrence of quenching behavior with the parameters based on convection term .
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Figure 3 
The graphs of (a) as increases, (b) as increases, (c) as increases, and (d) as increases when is for convection term .
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Figure 4 
(a) The variation of spatial step as processes; (b) the variation of spatial step as processes; (c) the curve of the temporal steps as time increases with the parameters based on convection term .

Figure 4 depicts curve of the adaptive temporal steps as the time variable progresses and graphs of the adaptive spatial steps changing as the space variables change. Figures 4(a) and 4(b) portray the distributions of spatial steps. The plots in Figure 4(a) are similar to those in Figure 4(b), but there are some subtle differences in their specific data. The blue curve depicts the tendency of spatial steps varying before quenching occurs, which underlines (). The green line refers to spatial step distribution of uniform mesh with nonadaption, which is extracted from the initial temporal layer. Mesh adaptation is stimulated at the position of red square sign, and becomes gradually smaller and falls rapidly toward its floor when in Figure 4(c). The red square mark represents the moment and the temporal step when the process approximates to quenching time. The time adaption is triggered before the moment and lasts through the rest course of calculation. So this quenching phenomenon is caught with the features of and as reaches .

There are three groups of 3D profiles of the solutions and their temporal derivatives in Figure 5. Figures 5(a) and 5(b) denote the two three-dimensional plots of and when , respectively. Figures 5(c) and 5(d) denote the two three-dimensional plots of andwhen, respectively. Figures 5(e) and 5(f) denote the two three-dimensional plots of and when , respectively. Thought the 3D plots, we can get richer information. The first group is the three-dimensional views from both and at the first 500th temporal layers in the excursion, in which and when . The second group is the three-dimensional views from both and including the penultimate temporal steps before , in which and when . The third group is the three-dimensional views from both and immediately before quenching happens, in which and when . During the temporal layers’ moving forward, the solution changes smoothly and then almost arrives at the peak value while approximates to . Its peak is the maximal value but before quenching time and location. There is some subtle perturbation at the initial temporal axis of the left boundary for the change rate of the solution. Afterward, the temporal derivative varies smoothly, and their maximums of each time axis increase steadily. While approaches to the quenching time , the maximum also increases rapidly and infinitely and then blows up at the next time layer of quenching time.

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Figure 5 
(a) The 3D plot of when ; (b) the 3D plot of when ; (c) the 3D plot of when ; (d) the 3D plot of when; (e) the 3D plot of when ; (f) the 3D plot of when . All is for Case 5.0.
4.4. Quenching Example with Convection Term

Similarly, we need to only consider and conduct a series of simulation experiments to investigate the 2D quenching regularity for the convection term which is related to the three groups of elements: and and , , and and exemplify the quenching behaviors of the representative Case 6.0. The convection term should written as and , where and take the constants and convection parameter . For Case 6.0, the initial step is , and the initial step is . We choose some representative data to record in Tables 7–10 from the simulation results. Case 6.0 is regarded as the reference standard and defined as . Tables 7–10 and Figures 6–9 demonstrate specific quenching information of Case 6.0.

Table 7 
Quenching data of based on convection term .
Table 8 
Quenching data of and based on convection term .
Table 9 
Quenching data of and based on convection term .
Table 10 
The maximal values and locations of and for based on convection term .
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Figure 6 
The graphs of solution and : (a) as increases, and (b) as increases from the initial time to the occurrence of quench behavior with based on convection term .
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Figure 7 
The graphs of (a) as increases, (b) as increases, (c) as increases, and (d) as increases when is for convection term .
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Figure 8 
(a) The variation of spatial step as space processes; (b) the variation of spatial step as space processes; (c) the curve of the temporal steps as time increases with the parameters based on convection term .
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Figure 9 
(a) The 3D plot of when and (b) the 3D plot of when for Case 6.0.

The effect of on quenching problem can be illustrated by continual experiments. A reasonable quenching behavior will occur when or takes a value in . When is 0.1, 0.2, 0.3, or , it does not produce quenching phenomena. There are quenching characteristics for : quenching location is not equal to when takes 0, 0.5 while is equal to in the other cases; is when takes 0.6, 0.7, and 0.8; is when is in . Quenching time increases as increases in the domain of . Table 7 gives some of these quenching cases, in which Case 6.1.1 is; Case 6.1.2 is ; Case 6.1.3 is ; Case 6.1.4 is ; Case 6.1.5 is ; Case 6.1.6 is ; Case 6.1.7 is ; Case 6.1.8 is ; Case 6.1.9 is ; Case 6.1.10 is ; Case 6.1.11 is ; Case 6.0 is between Case 6.1.5 and Case 6.1.6.

The paragraphs describe the relationship between () and quenching features. We write ten cases in Table 8. Case 6.2.1 is ; Case 6.2.2 is ; Case 6.2.3 is ; Case 6.2.4 is ; Case 6.2.5 is ; Case 6.2.6 is ; Case 6.2.7 is ; Case 6.2.8 is ; Case 6.2.9 is ; Case 6.2.10 is ; Case 6.0 is between Case 6.2.4 and Case 6.2.5. Quenching behavior can be formed when () takes the values in , in which the function curve of (,) is concave and is smallest when . When , , , , , or , quenching location is equal to whereas when takes other values in , is different from .

We describe the influence of and on quenching results just depending on quenching data in Table 9. Of course, we do a great quantity of experiments, from which some typical cases chosen in Table 9. We set and from 10 to for Cases 6.3.1-6.3.10. Specifically, Case 6.3.1 is ; Case 6.3.2 is ; Case 6.3.3 is ; Case 6.3.4 is ; Case 6.3.5 is ; Case 6.3.6 is ; Case 6.3.7 is ; Case 6.3.8 is ; Case 6.3.9 is ; Case 6.3.10 is ; Case 6.0 is between Case 6.3.8 and Case 6.3.9.

It does not form quenching status when or . After is defined as 1, is evaluated from 10 to . We find that it is more likely to produce quenching behaviors when is between 10 and . By means of continual tests, Cases 6.3.1-6.3.10 represent the ten special cases, and we can observe the quenching status as follows. As far as quenching spatial characteristic is concerned, quenching location does not monotonously increase as increases when is in the definition domain. For example, quenching location reaches the maximum when and quenching location keeps the coordination point of when is in . Next, quenching temporal characteristic is concerned. In fact, there does not exist strict linear relationship between quenching time and . Comparatively, quenching time is smaller in the former half than in the latter half of the domain of . When the measurement scale is enlarged in the domain of , especially when reaches 19991, increases with the increase of .

Quenching phenomena of Case 6.0 can be considered in the following text. Its quenching time appears at , and its quenching position appears at in this situation. Table 10 shows six stages around quenching for Case 6.0. The items from the first to the fifth describe the five stages of representation before the occurrence of quenching, and the last item just notes the quenching moment. is at (, ), and is at (, ) when quenching occurs. A comprehensive statement of quenching states with the parameters is recorded in the next paragraphs. Although there is slight perturbation in the left boundary of , it does not influences on distribution of . It is evident that Figures 6(a) and 6(b) give two pairs curves of both and . The first one is for the distribution of the solution as varies, and the second one is for the distribution of as increases. There is small perturbation near the start-up of the adaptive procedure for the left figure of . The four figures in Figure 7 give function relationship physical quantities between, , and spatial variables. Figure 7(a) paints the contour of (,), Figure 7(b) paints the contour of (,), Figure 7(c) paints the contour of (,), and Figure 7(d) paints the contour of (,). Similarly, Figures 7(a) and 7(c) have the same shape, and Figures 7(b) and 7(d) have the same shape.

There are three graphics in Figure 8, which represent three function relationships between spatial step in direction and , between spatial step in direction and , and between temporal step and , respectively. Figure 8(a) depicts two curves, in which one is between initial spatial steps and , and another is between quenching spatial steps and . Figure 8(b) also describes two curves, in which one is between initial spatial steps and , and another is between quenching spatial steps and . The green lines with gradient marks reflect the initial spatial step distribution that is uniform regardless of direction or direction. The red curves covered by blue squares reflect quenching spatial steps that are self-adaptive when quenching occurs. There is a blue curve with a red square marks in Figure 8(c), which portrays temporal stepvaries as changes from 0 to . keeps 0.001 when , but after becomes when which signed as a red square, sharply declines up to when .

The three-dimensional surfaces can render more reliable information for both the solution and the time derivative which display in Figure 9. The group of surfaces is the three-dimensional views of both and when is equal to . It is smooth and steady that the solution carries forward at the temporal axis in Figure 9(a). Moreover, it is almost at the middle position of each time axis that the case gives birth to the peak value of . When the position is at immediately before quenching occurs, the peak value of the solution approaches 1. The three-dimensional representation of Figure 9(b) reveals the evolution of the temporal derivative with regard to spatial variables. The temporal derivative of the solution varies smoothly and reaches its maximums at the quenching domain . At last, the temporal derivative becomes infinite when quenching occurs at another location (, ), which is not observed in Figure 9(b). It is obvious for Case 6.0 that quenching occurs based on and as reaches .

5. Conclusion

Relying on the analyses in this paper, we can sort up the relationship between the parameters , (), (), and quenching behaviors. There exist four aspects concluded as follows. First, the degenerate function acts within the normal range if the degeneracy parameter is larger than or equal to 0.4 but not more than 1.3, and quenching time increases as increases. Second, there is a special point dividing the definition domain of convection parameter as the left and the right subdomain, in which quenching phenomena occur normally and have a variety of features. Quenching time decreases as increases in the left subdomain, and quenching time increases as increases in the right subdomain. Quenching location either decreases or convergences upon a certain value as increases in the domain of . Third, under the condition of , the influence of on quenching result is researched. There exists still a special point categorizing the definition domain of as two subdomains, in which quenching phenomena occur normally and display different spatial effects. Specially, with the rise of , quenching location does not increases monotonously in the left hand of and tends to a fixed coordinates in the right hand of . Additionally, if we rely on a small scale to observe the domain of , then there is no linear relationship between quenching time and . We investigate the definition by virtue of a big scale to find that quenching time also increases when become larger, especially when reaches certain value; quenching time increases as rises. Fourth, it must be a good choice to set as 1. Through experiments, it is hard to produce quenching phenomena that can be formed when . From the result analyses, it can be discovered that it is meaningful to initially investigate the 2D singularity degeneracy problem of quenching type based on the unsteady convection-reaction-diffusion equation by using the high-order difference scheme, which will probably lead to the potentially support to research the next quenching problems.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (Grant nos. 11772165, 11961054), the National Natural Science Foundation of Ningxia (Grant no. 2018AAC02003), the Key Research and Development Program of Ningxia (Grant no. 2018BEE03007), and the Major Innovation Projects for Building First-class Universities in China’s Western Region (Grant no. ZKZD2017009).