Adaptive ADI Numerical Analysis of 2D Quenching-Type Reaction: Diffusion Equation with Convection Term

An adaptive high-order difference solution about a 2D nonlinear degenerate singular reaction-diffusion equation with a convection term is initially proposed in the paper. After the first and the second central difference operator approximating the firstorder and the second-order spatial derivative, respectively, the higher-order spatial derivatives are discretized by applying the Taylor series rule and the temporal derivative is discretized by using the Crank–Nicolson (CN) difference scheme. An alternating direction implicit (ADI) scheme with a nonuniform grid is built in this way. Meanwhile, accuracy analysis declares the second order in time and the fourth order in space under certain conditions. Sequentially, the high-order scheme is performed on an adaptive mesh to demonstrate quenching behaviors of the singular parabolic equation and analyse the influence of combustion chamber size on quenching. )e paper displays rationally that the proposed scheme is practicable for solving the 2D quenchingtype problem.


Introduction
Nonlinear reaction-diffusion equation with a singular or near-singular source term has been widely applied in ion conduct polarization theory [1], computational fluid dynamics [2], electromagnetism [3], material research [4], ecology [5], thermology [6,7], and so on. Different from the linear reaction-diffusion equation, it has a chance to produce a solution with singularity. Blow-up and quenching are considered as singularity of a solution. At a limited time, the former indicates that its solution tends to infinite, and the latter indicates that its temporal derivative tends to infinite, but its solution is restricted in a certain scope [8,9]. Researchers often use the nonlinear singular degenerate reaction-diffusion model to simulate ignition and burning and blow-up of the fuel, which can describe burning transformation from a stable status to an unstable status in a combustion chamber and from which quenching may arise [10]. To a large extent, it is to cognize features and change laws of quenching precisely that decides the design and optimization of the combustion chamber. We pay attention to the quenching problem of the degenerative singular reaction-diffusion equation with a convection term in this paper.
Let us recall the development of the singular reactiondiffusion equation of the quenching type. Since Kawarada concluded that the time differential quotient of the solution leans toward infinity when the solution approximates to one indefinitely for this equation [1], more and more scholars have focused on nonlinear singular problems. e academic system of researching the quenching phenomena of the nonlinear reaction-diffusion equation has been gradually established, which ranges from one dimension and two dimension to three dimension [11][12][13][14][15][16][17]. More one-dimensional problems are investigated rather than higher-dimensional ones. However, there are still some quenching principles of the two-dimensional degenerate singular reaction-diffusion equations discovered [6][7][8][9][10][12][13][14][15]. Specially, they qualitatively depicted quenching or nonquenching under certain conditions, asymptotic status of quenching time, the spatial definition domain size, i.e., critical length, resulting in quenching behaviors, and so on. Sheng et al. have been engaged in research in this field. Especially, Padgett and Sheng investigated the degenerate stochastic Kawarada equations from 1D to 2D and 3D by combining the finite difference schemes with arbitrary meshes [11,16,17]. e low-accuracy concepts are exploited in the above resolutions. Ge et al. firstly rendered a highorder compact difference scheme based on an adaptive mesh to study the quenching principle of the one-dimensional singular degenerate reaction-diffusion equation in 2018 [18].
Let us recall the development of the reaction-diffusion equation with a convection term of the quenching type again. Compared to the quenching equation mentioned in the prior paragraph, less attention is paid to the convectionreaction-diffusion equation with quenching singularity, especially to the 2D and 3D problems [19]. So far, it is difficult to find research on the 2D quenching problems. e authors in [19] discussed quenching phenomena under the conditions of different convection terms and singularity indexes: asymptotic status of quenching time, the spatial definition domain size, i.e., critical length, and so on, for the one-dimension singular equation. Sheng and Cheng studied quenching behaviors of the same type equation by using the fully adaptive difference scheme of low-order accuracy in [20]. Sheng and Khliq investigated the relationship between the critical value of the degenerative term that was not a function of spatial points and both quenching time and convergence time for the 1D singular degenerate problem in [21]. Zhou et al. theoretically depicted the critical length and the quenching spatial point for the convection-reactiondiffusion equation without the degenerate term [22]. Research studies above belong to the low-order difference strategy. Recently, a high-order adaptive difference scheme is firstly used to study the influence of degenerate function, convection function, nonlinear source function, and spatial definition length on quench behaviors for the one-dimensional convection-reaction-diffusion equation, respectively [23]. It provides the basis for adopting the high-precision algorithm to analyse the quenching states of the 2D convection-reaction-diffusion equation.
Depending on the previous research studies, we discover that there is an important significance for researching this two-dimensional convection-reaction-diffusion problem of the quenching type. It is well known that the finite element scheme and the finite difference scheme are popular numerical methods to solve the convection-reaction-diffusion equation [24][25][26][27][28][29]. e improved schemes based on the finite element are utilized to resolve the two-dimensional convection-reaction-diffusion problems [24,25]. In the past years, there have emerged many finite difference schemes for solving the two-dimensional convection-reaction-diffusion problem [26][27][28][29]. Ge and Cao built up the transformationfree high-order compact difference algorithm proposed by Kalita et al. [26] on the multigrid to solve the 2D steady convection-diffusion equation [27]. Huang proposed an ADI scheme to solve the 2D unsteady convection-reactiondiffusion equation, which is built by using the first and the second central difference operators to approximate the first-order and the second-order spatial derivative, respectively, and the Crank-Nicolson (CN) method to the discrete temporal derivative [28]. Li et al. represented a compact finite difference scheme of unconditional stability and the fourth order for solving the groundwater pollutant problem based on the 2D nonlinear unsteady convection-diffusion equation [29]. Of course, there exists other schemes for solving the 2D problem. A Fourier spectral discretization method was posted for explaining the high-dimensional fractional reaction-diffusion equations by using two separate mathematical techniques [30]. Pindza and Owolabi developed the Fourier spectral method in space and the exponential integrator scheme in time to solve the 2D and 3D fractional reaction-diffusion equations [31]. Nonlinear highdimensional convection-diffusion problems were resolved by a novel mesh-less local Petrov-Galerkin method proposed in [32]. Different from the above research studies, Owolabi and Atangana [33] studied the high-dimensional fractional reaction-diffusion systems of spatial interactions of three components' species.
e dynamical system was considered to be of asymptotical stability both in local and global style by analyzing the main equation theoretically. Furthermore, a theory certifying the existence and permanence of the three species was put forward.
In general, a high-order finite difference scheme has a greater advantage comparing with other methods. An adaptive ADI difference scheme based on the two-dimensional unsteady reaction-diffusion equation with convection function is used to analyze quenching phenomena of the 2D combustion model through serial cases and investigate relationship between burner shape and quenching features. Due to significant practical meaning of quenching behaviors, the important quenching information is beneficial to engineering applications such as fuel burning and industry manufacturing. e compact scheme with the adaptive algorithm is employed to compute a series of important information including quenching time and quenching location.
ere 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 proves adaptive mesh theory. Section 4 stimulates some typical numerical samples to explore and explain quenching problems. Section 5 draws the conclusion.

Semilinear Reaction-Diffusion Equation with Convection Term.
is semilinear degenerate problem model involving two spatial dimensions is regarded as equation (1) with the boundary and initialization conditions of equation (2) and (3). e solution u(x, y, t) represents the temperature of the combustion chamber. Ω ∈ (0, α) × (0, β) refers to the smooth rectangle domain of the combustor, and zΩ is its boundary. c 1 (x, y) and c 2 (x, y) are the convection functions of x and y, and σ(x, y) � (x 2 + y 2 ) q/2 is the degeneracy function and q ≥ 0. e singularity source f(u) reaction-diffusion equation of the quenching type is written as (1)

Its boundary conditions are
(2)

Its initialization condition is
With the aid of the intermediate variables x and y, we replace x � x/α and y � y/β in equation (1) which can be defined as For the convenience of expression, we use x and y instead of x and y. en, the above formulation can be rearranged as where σ(x, y) � (α 2 x 2 + β 2 y 2 ) q/2 and Ω ∈ (0, 1) × (0, 1). Correspondingly, the boundary conditions of equation (5) are e initialization condition of equation (5) is In the physical application, we rely on equations (5) and (7) to compute u and u t . rough observing a large number of values of u and u t , we can capture the quenching moment, i.e., quenching occurs when u is infinitely close to 1 and u t and becomes so huge that its value blows up.

Nonuniform ADI Difference Scheme.
According to Taylor formula [28], the high-accuracy ADI difference scheme of equation (5) is deduced below. e steady convection-diffusion equation derived from the 1D unsteady convection-diffusion equation in the x− direction is con- where g is a function of x. By virtue of the Taylors' series expansions, we get expressions of (u x ) k and (u xx ) k , including the central difference operators of the first and the second spatial derivatives, discretized on the nonuniform mesh, respectively. e two expressions are substituted into the above steady convection-diffusion equation. So, a 1D steady convection-diffusion equation dispersed at point x k can be written as where After discretizing (u x 3 ) k and (u x 4 ) k in the spatial direction, equation (8) is reorganized as where Mathematical Problems in Engineering Similarly, from the one-dimensional steady convectiondiffusion equation in the y− direction: (1/β 2 σ(y))u yy + (c 2 (y)/βσ(y))u y � g(y), and the equation in the y− direction can be derived as where P y � P 1y δ y + P 2y δ 2 y , P 1y � 1 βσ j + P 9y δ y − βψ 2y δ 2 y c 2j , P 9y � − βψ 1y + β 2 ψ 2y c 2j , C y � C 0y + C 1y δ y + C 2y δ 2 y , Adding the two hands of equation (10) and equation (12), we can get a 2D steady convection-diffusion equation: Subsequently, after substituting (u t − (f(u)/σ(x, y))) k,j for g k,j in equation (14), the situation of point (n + 1/2) for equation (15) is considered as e Crank-Nicolson method is used to discretize the temporal term of equation (16), and an equation of point (n + 1) can be formed as where τ n is the time step and O(τ n Δ) + O(τ 3 n ) is the truncation error. When (τ 2 n /4)P y P x u n+1 k,j is added to the left side of equation (18) and (τ 2 n /4)P y P x u n+1 k,j is added to the right and the truncation error is deleted, a group of linear system is organized as By virtue of the intermediate variable u * k,j , equation (18) is reorganized as After equation (19) is computed and arranged again, a linear system is formed as In equation (21), where Mathematical Problems in Engineering 5 where Similarly, from equation (20), another linear system is written as where in which It can be observed that each u n+1 k,j on the n + 1th time layer is obtained, according to u n k,j on the prior one, by computing the tridiagonal linear equations twice. We adopt the catch-up method to solve the linear equations. We rely on equation (21) to get the medial variable u * k,j , which represents the x− direction. Subsequently, we depend on u * k,j and equation (28) to gain u n+1 k,j , which carries information in the y− direction and so on, until the solutions on the last time axis are worked out. e accuracy analysis of equation (18) is performed, and its truncation error is Note that equations (19) and (20) have the second-order accuracy in time and the third-or fourth-order accuracy in space. e scheme of uniform owns spatial accuracy of the fourth order, whereas its nonuniform scheme owns spatial accuracy of the third order.

Temporal Adaptive Mesh Method.
Comparing with solution u, the temporal derivative u t is more sensible. erefore, like the 1D adaptive method, the 2D adaptive mesh method in the time direction is based on the arc-length monitor function of the temporal derivative function u t . We can extend the 1D adaptive temporal grid schemes in Reference [23] to the 2D case: After u tt in equation (32) is discretized through the central difference formulas to calculate the maximum ratio, the adaptive time step τ n is easily obtained by equation (32) and τ n− 1 in the prior time layer.

Spatial Adaptive Mesh Method.
Similarly, we can utilize the arc-length monitor function in the space direction based on u t and extend from the 1D adaptive spatial algorithm to the 2D problem by relying on Reference [23]. erefore, W x (x, t) representing the monitor function in the x− direction and W y (y, t) representing the monitor function in the y− direction can be written as e two spatial monitor functions are positive in their definition area. ξ k (x k , t) K k�0 and ζ j (y j , t) J j�0 represent the set of 2D original spatial mesh points. After an adaptive improvement is complemented, the grid in the 2D computational area is updated as x k (ξ k , t) K k�0 in the x− direction and y j (ζ j , t) J j�0 in the y− direction. It is easily observed that the 1D adaptive grid scheme acts on in the x− direction and y− direction, respectively, to produce new grid points.
We take the equidistributing mesh in the y− direction as an example below. From [23], we know that the subareas of W y (y, t) in intervals of [y j− 1 (t), y j (t)](j � 1, 2, . . . , J) are equal. Obviously, when W y ≡ 1, the scope [0, 1] is uniform. It can be expressed as 6 Mathematical Problems in Engineering where ς y (t) is the average arc-length. Equation (35) is equal to After the second derivation of the aforementioned formula, the following equation can be obtained: where m represents the iterative number, y (m) j J j�0 represents the spatial grid node set before iteration, and y (m+1) j J j�0 represents the space grid node set after iteration. Equation (37) can be solved by the difference scheme: which approximates to the former. Similarly, the semiimplicit scheme in the x− direction is defined as

Iterative Adaption
Method. e adaptive iterative procedure of the presented method is carried forward as follows. Firstly, the spatial monitor functions W x (x, t) and W y (y, t) can be obtained through equations (33) and (34) irdly, from the value built on the last grid point sets x in the n + 1th time layer depending on equation (18). Lastly, repeat the aforementioned produce till quenching occurs or the solution converges to a steady solution.

Common Example.
It is not a quenching case. It is just for demonstrating the accuracy and convergence of the proposed difference scheme. We have solved the following unsteady convection-diffusion equation with coefficient whose exact solution is offered: Mathematical Problems in Engineering e initial and boundary value conditions are derived from the exact solution. ere are four metrics for the data test, which are maximum error, average error, CPU time, and convergence rate. We adopt these parameters T � 0.5, τ n � 0.01, and p � 100 to fulfill the two difference schemes: the proposed scheme based on the uniform mesh and the proposed scheme based on the nonuniform mesh. k/K + (λ x /π)sin(πk/K) K k�0 and j/J + (λ y /π)sin(πj/J) J j�0 represent the nonuniform grid, where K and J point to intervals of the spatial direction and K � J, whereas λ x and λ y point to the mesh transformation coefficient and λ x � λ y . Table 1 offers these measuring data. It displays the accuracies and running time of our scheme (uniform and nonuniform). We select five spatial intervals: 16, 32, 64, 128, and 256 as test cases, which are given to each of the above two meshes (i.e., uniform and nonuniform). For the nonuniform mesh, the different values of λ x impact these precision criteria. In the paper, λ x takes 0.7. e maximum errors of the first scheme are higher than those of the second one, and the average errors of the latter are superior to those of the former. Furthermore, the larger the number of spatial intervals is, the higher the accuracies of the scheme based on the nonuniform grid are. Its accuracies arrive at 1.3364 × 10 − 7 in the maximum error and 6.4492 × 10 − 9 in the average error when K � J � 256. In terms of computing time of the two schemes whose measurement unit is millisecond, the first scheme is lower than the second one. Obviously, by comparing with the uniform mesh, the difference scheme based on the nonuniform mesh spends more time to run. Lastly, the conv. rate values are observed. e tendency of the conv. rate from the former scheme is similar to that from the latter scheme. We take the latter as the example. e four values are positive, which means the accuracy increases as K rises when K ∈ [16,256]. e conv. rate value is maximum when K transfers from 16 to 32, which means the accuracy promotes greatly in the stage. e latter three values are close, which means the accuracy promotes gently in the three stages.

Quenching Case without Convection Term.
is section will describe a quenching status without the convection term, i.e., c 1 (x, y) � 0 and c 2 (x, y) � 0. e mathematical model can be expressed as Its initial and boundary conditions are just as in equations (6) and (7).
After equations (19) and (20) are used to approximate equation (41), we can get some quenching behaviors. For the 2D nonlinear degeneration singularity equation without the convection term, we demonstrate the quenching case of q � 0 that has been described in [13,14]. In Table 2 and Figure 1, , and θ � 1 which are the same for the three approaches, but the initial space step h 0 and the initial temporal step τ 0 are different among them, where we take h 0 � 1/50 and τ 0 � 4.166677 × 10 − 3 . In Table 2, the five parameters represent the same senses, in which (x max , y max ) represents the point of quenching location, t max represents the quenching time, max urepresents the maximum temperature value immediately before quenching happens, and max u t means the maximum variation of temperature with respect to time immediately before quenching happens. From Table 2, we can see that the quenching location (0.5, 0.5) of the referred scheme is identical with that of the scheme in [14], and t max � 0.5958348109999995 and max u � 0.998562875689732 of the first one are close to those of the last two. Obviously, max u t is different among the various methods. From Figure 1, it is found that u ⟶ 1 − and u t ⟶ 690 − at the location (x max , y max ) � (0.5, 0.5) and the quenching time t max � 0.5958348109999995.

Quenching Case with Convection Term b/(1 + αx + βy).
e standard 2D degenerate semilinear quenching problem with the initial boundary condition lists in equations (5)-(7) is extended from the equation in one-space dimension [23]. e first convection term b/(1 + αx + βy) can be written as two functions c 1 (x, y) � b x /(1 + αx + βy) and c 2 (x, y) � b y /(1 + αx + βy), where b x and b y take the constants. e four groups of variables: q, b x and b y , α and β, and θ in equations (5)-(7) affect the quenching characteristics of the 2D problem, whose meaning has been stated in Section 2. ere follows a special case for b/(1 + αx + βy). e initial time step is considered as τ 0 � 0.0003, and the initial space step is considered as h 0 � 1/55, in which the x-direction is identical with the y-direction. e four groups of parameters are configured as q � 0.5, α � β � ��� 200 √ , b x � b y � 10, and θ � 1 which is defined as Case 7.0. is will lead to a quenching phenomenon. Quenching location (x * 7 , y * 7 ) is (0.03636363636363636, 0.03636363636363636); the quenching time T * 7 is 4.487776579638927; max x,y u is 0.9992409498317957; max x,y (u t ) is 1539.153786381783. When t is 4.486155872847072, the temporal step starts to become smaller and smaller until quenching occurs. We choose six rows of representative data given in Table 3 which poins to features relative to x, y, t, u, and u t before quenching occurs of Case 7.0. max x,y u and max x,y (u t ) refer to maximal u and maximal u t in each time layer before quenching occurs; x and y refer to the coordinate point of synchronously producing max x,y u and max x,y (u t ) in each time layer before quenching occurs; t refers to the moment at which each time layer is located before quenching occurs. (x, y) first appears at the point (x * 7 , y * 7 ) when t is 4.487088240486234 which is the first row in Table 3 and maintains at this point until quenching occurs. When t is T * 7 which is the immediate time before quenching happens, we can obtain important characteristics. When t is 4.487777138788559, max x,y (u t ) blows up and reaches 1.051076326841211 × 10 9 . Figures 2-7 provide seven groups of graphs to illustrate more particular quenching statuses. Figure 2 shows the 2D 8 Mathematical Problems in Engineering plots of max x,y u and max x,y (u t ) changing as t rises from the initial moment until quenching occurs. Figure 2(a) corresponds to u, and Figure 2(b) corresponds to u t . Figure 3 sketches the 2D contours of u and u t regarding x immediately before quenching happens. Figure 3(a) corresponds to u, and Figure 3(b) corresponds to u t . Figure 4 draws the 2D outlines of u and u t regarding y immediately before quenching happens. Figure 4(a) corresponds to u, and Figure 4(b) corresponds to u t . e three 2D plots in Figure 5 describe the adaptive mechanism before quenching occurs. e first one reflects spatial adaptation, and the last two reflect temporal adaptation. Figure 5(a) describes the trend   [13] 0.58712499993751 --0.990432 148.887767 e scheme in [14] 0  Figure 1: (a) e 3D plots of u immediately before quenching occurs and (b) the 3D plots of u t immediately before quenching occurs. e parameters are q � 0, a � β � �� 10 √ , and θ � 1 without the convection term. of the temporal step over time. To distinguish more clearly the variation, we intercept the curves of the period from 4.470000000001407 to 4.487777138788559. e red sign on the blue curve in Figure 5(a) denotes the initial temporal adaptive moment (i.e., 4.486155872847072). e temporal step is uniform before the adaptive moment but becomes nonuniform when t varies from 4.486155872847072 to 4.487777138788559. e green sign on the blue curve in Figure 5(a) denotes the initial moment (i.e., 4.487088240486234) of producing the quenching location (x * 7 , y * 7 ). e green lines in Figures 5(b) and 5(c) describe the uniform spatial steps in the x− direction and the y− direction, respectively. e spatial adaptation occurs at 4.487777138788559 after which the spatial step becomes nonuniform. e red curves with blue spots in Figures 5(b) and 5(c) describe, respectively, the nonuniform spatial steps in the x− direction and the y− direction at T * 7 . Although the two curves are very similar, there is still subtle difference between their data. Figures 6-7 provide the 3D surfaces of u and u t with respect to spatial variables x and y at certain special time. Figure 6 and u t when t � 4.487770702255042. Figure 7(a) records the 3D surfaces about x, y, and u, and Figure 7(b) records the 3D surfaces about x, y, and u t when T * 7 .

Quenching Case with Convection Term b/(αx + βy).
e second convection term b/(αx + βy) can be written as the two convection functions: c 1 (x, y) � b x /(αx + βy) and c 2 (x, y) � b y /(αx + βy), where b x and b y take the constants. Comparing with the first convection term b/(1 + αx + βy), it is hard to conduct quenching behavior for the second convection term b/(αx + βy). We make a following configuration to gain a quenching status.
e initial x− step (y− step) is h 0 � 1/40, and the initial t− step is τ 0 � 0.0005. e four groups of important parameters are set as q � 1.1, α � β � 20, b x � b y � 3, and θ � 1 which is defined as Case 8.0. e parameter statuses of the case are recorded in Table 4. For the case, the initial time of producing the quenching location is earlier than the temporal adaptive time. e former time is 5.596499999999618, which is the first row in Table 4, the latter one is 5.603982012140827, which is the fifth row in Table 4, and the quenching location (x * 8 , y * 8 ) is (0.025, 0.025). Moreover, it is so transient from the front time of quenching location appearing to the terminative time of quenching location appearing that there are 18 time layers between the two moments. e latter quenching time is (T * 8 � 5.604262757909247), max x,y u is 0.9835341519876587, and max x,y (u t ) is 73.15674241637485 immediately before quenching occurs, which is the sixth row in Table  4. max x,y (u t ) gets the value 6.637517818582074 × 10 11 and blows up when t is 5.604272047535358, which is the last row in Table 4.
is paragraph supplies five sets of figures (i.e., Figures 8-12) to more vividly depict the quenching information of the case. e 2D curves in Figure 8 describe the relationship between max x,y u and t and the relationship between max x,y (u t ) and t. e 2D curves in Figure 9 describe the relationship between u and x and the relationship between u t and x. e 2D curves in Figure 10 describe the relationship between u and y and the relationship between u t and y. In Figures 8-10, (a) corresponds to u and (b) corresponds to u t . Figure 11(a) draws the tendency of the temporal step varying with temporal variable t; Figure 11(b) draws the tendency of the initial spatial step and the adaptive step varying with spatial variable x; and Figure 11(c) draws the tendency of the initial spatial step and the adaptive step varying with spatial variable y. For Figure 11(a), to distinguish more clearly the variation, we intercept the curves of the period from 5.580499999999627 to 5.604272047535358; and the red spot on the blue curve expresses the earliest temporal adaptive time (i.e., 5.603982012140827). max x,y (u t ) in the time step becomes 1.185615675885128 × 10 12 when t is 5.603982012140827. e temporal step is uniform before the adaptive moment but becomes nonuniform since then. e green spot on the blue curve expresses the initial moment of producing the quenching location (x * 8 , y * 8 ), which is 5.604262757909247. In Figures 11(b) and 11(c), the green lines indicate the uniform initial spatial steps while the red curves with blue signs indicate the nonuniform adaptive spatial steps. Lastly, we provide two 3D surfaces in Figure 12 to state quenching statuses of u and u t when t � T * 8 . Figure 12(a) denotes the relationship between u, x, and y, and Figure 12(b) denotes the relationship between u t , x, and y. Figure 12(a) records the 3D surfaces about x, y, and u, and Figure 12(b) records the 3D surfaces about x, y, and u t when t � T * 8 . ere is a narrowband projecting the rear plane of the 3D axis box in Figure 12(b).
is is because some larger values of max x,y (u t ) appear at the positions, in which the x− axis and the y− axis are simultaneously near the origin of the coordinates. From the data source of the 3D surface, we can observe some larger max x,y (u t ) varying from 1 to 27,    whereas max x,y (u t ) at other positions are positive numbers less than 1, which can be seen via Figures 9(b) and 10(b). Obviously, the larger max x,y (u t ) are all less than 73.15674241637485, which is reasonable for a quenching case.

e Effect of Burner Size on Quenching.
e relationship between combustion the chamber shape and quenching features is discussed in this paragraph, which shows how the ratio of length to width of the burner affects the quenching behavior. In terms of c(x, y) � b/(1 + αx + βy), we also consider serial values of α/β taken in [0.7, 0.97] besides α � β corresponding to Case 7.0 in Table 5 because it does not produce quenching behavior when α/β is smaller than 0.7. We choose ten values: 0.7, 0.72, 0.82, 0.85, 0.88, 0.89, 0.9, 0.92, 0.95, and 0.97 for Case 7.0.1-Case 7.0.10 in Table 5.  Table 5. From Table 5, we can discover that quenching time t max that is 4.517239999869761 is the largest when α/β � 0.85 and quenching location x max � y max that is 0.03636363636363636 when 0.89 ≤ α/β ≤ 1. Similarly, we investigate quenching behavior variation with the changes of α/β for convection function b/(αx + βy). From a number of tests, we select eleven meaningful cases to be given in Table 6 1/40. q, α, b x (b y ), and θ are set as cq � 1.1, α � 20, b x � b y � 3, and θ � 1. It is easily seen that quenching occurs, and quenching location x max � y max is 0.025 when α/β ≥ 0.15 and quenching time t max is the smallest when α/β � 0.375 from Table 6. Figures 13 and 14 display the relationships between t max , x max , y max , and α/β, respectively, in which, the red curves in Figures 13(a) and 14(a) reflect quenching time t max variations as α/β rises; the blue curves in Figures 13(b) and 14(b) reflect quenching location x max variations as α/β rises; the green curves in Figures 13(b) and 14(b) reflect quenching location y max variations as α/β rises. Figure 13 corresponds to Table 5, and Figure 14 corresponds to Table 6. e highest value marked by the green sign of the red contour in Figure 13(a) is the quenching time 4.517239999869761 of Case 7.0.4. Quenching time t max drops monotonously as α/β increases when 0.85 < α/β ≤ 1.0, and it promotes monotonously as α/β increases when 0.7 ≤ α/β ≤ 0.85. e shape of the red contour in Figure 14(a) is different from that of the red contour in Figure 13(a). Its lowest green point is just the quenching time 5.160277808704819 of Case 8.0.10, and α/β is 0.95. e contour shows the fluctuation of the points up and down. For the relationship between x max and α/β, we rely on the tendencies of the red dots on the blue curves in Figures 13(b) and 14(b), respectively, to see that x max keeps a same value in each case. Depending on the trend of the purple dots on the green curves in Figures 13(b)  and 14(b), respectively, we can find that y max keeps a smaller value than x max when 0.7 ≤ α/β ≤ 0.89 and becomes the same value as x max when 0.89 ≤ α/β ≤ 1 for Figure 13(b) and y max is equal to x max for Figure 14(b). Meanwhile, it is clearly observed that there is only the blue contour with ten red points extending to the end in Figure 14(b).
Finally, we can conclude that quenching problems based on different convection functions require different combustion chamber sizes. It can lead to quenching behaviors to properly take the ratio of α and β when α/β ≥ 0.7 for convection function b/(1 + αx + βy). But, it can result in quenching phenomena to properly take the ratio of α and β when α/β ≥ 0.15 for the convection term b/(αx + βy). By comparing, it is easier to form quenching behaviors for the latter than the former. Generally, we can consider it will produce quenching phenomena when α/β takes a value not less than 0.7. ere must be a suitable ratio that will spend  the most quenching time. When the length and the width of the combustion chamber are close (i.e., 0.89 ≤ α/β ≤ 1), the horizontal coordinate of quenching location is equivalent to its vertical coordinate.

Conclusions
Nonlinear singular degenerate reaction-diffusion equation plays more and more important role in the quenching area such as diffusion of thermal and energy, combustion theory, aerospace engineering, and biomedical treatment. We put forward a compact difference scheme and built an adaptation algorithm to solve the 2D convection-reaction-diffusion equation of the quenching type. Depending on theoretical and numerical experiments above, we can get some conclusions as follows. As a valid and powerful mathematical tool, the high-order compact difference scheme combining with the proper adaptive procedure may offer elegant manners to resolve the semilinear singular degenerate partial differential equation. It is very efficient to employ the two-layer self-adaptive grid principle for solving the quenching problem. e self-adaptive temporal step makes time adaptation activated when matching the critical condition. It not only saves computational cost, but also impacts on specific moment of quenching. Similarly, it is important to choose spatial adaptation parameters, which may decide where the temporal derivative will blow-up, even whether quenching occurs or not. e study provides the feasibility of analyzing and resolving different engineering applications of the quenching problem in different finite difference schemes on an adaptive mesh. We will conduct further research in the future, which includes the 3D reaction-diffusion equations with the convection function of the quenching type.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.