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

Zhu, Xiaoliang; Ge, Yongbin

doi:https://doi.org/10.1155/2020/8161804

Mathematical Problems in Engineering

On this page

Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 8161804 | https://doi.org/10.1155/2020/8161804

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

Xiaoliang Zhu^1,2and Yongbin Ge¹

Academic Editor: Ivan Giorgio

Received16 Feb 2020

Accepted29 Apr 2020

Published15 Jun 2020

Abstract

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 first-order 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. The paper displays rationally that the proposed scheme is practicable for solving the 2D quenching-type problem.

1. 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 reaction-diffusion 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. The 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 [11–17]. 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 [6–10, 12–15]. 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]. The low-accuracy concepts are exploited in the above resolutions. Ge et al. firstly rendered a high-order 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 convection-reaction-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. The 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-reaction-diffusion 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 [24–29]. The 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 [26–29]. Ge and Cao built up the transformation-free 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-reaction-diffusion 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 high-dimensional 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. The 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. The compact scheme with the adaptive algorithm is employed to compute a series of important information including quenching time and quenching location. 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 proves adaptive mesh theory. Section 4 stimulates some typical numerical samples to explore and explain quenching problems. Section 5 draws the conclusion.

2. Problem and Difference Discretization

2.1. Semilinear Reaction-Diffusion Equation with Convection Term

This semilinear degenerate problem model involving two spatial dimensions is regarded as equation (1) with the boundary and initialization conditions of equation (2) and (3). The solution represents the temperature of the combustion chamber. refers to the smooth rectangle domain of the combustor, and is its boundary. and are the convection functions of and , and is the degeneracy function and . The singularity source is strictly increasing for with . The typical convection-reaction-diffusion equation of the quenching type is written as

Its boundary conditions are

Its initialization condition is

With the aid of the intermediate variables and , we replace and in equation (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 and Correspondingly, the boundary conditions of equation (5) are

The initialization condition of equation (5) is

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

2.2. Nonuniform ADI Difference Scheme

According to Taylor formula [28], the high-accuracy ADI difference scheme of equation (5) is deduced below. The steady convection-diffusion equation derived from the 1D unsteady convection-diffusion equation in the direction is considered as where is a function of . By virtue of the Taylors’ series expansions, we get expressions of and , including the central difference operators of the first and the second spatial derivatives, discretized on the nonuniform mesh, respectively. The two expressions are substituted into the above steady convection-diffusion equation. So, a 1D steady convection-diffusion equation dispersed at point can be written aswhere

After discretizing and in the spatial direction, equation (8) is reorganized aswhere

Similarly, from the one-dimensional steady convection-diffusion equation in the direction: , and the equation in the direction can be derived aswhere

Adding the two hands of equation (10) and equation (12), we can get a 2D steady convection-diffusion equation:

We can mark the truncation error as and get

Subsequently, after substituting for in equation (14), the situation of point for equation (15) is considered as

The Crank–Nicolson method is used to discretize the temporal term of equation (16), and an equation of point can be formed aswhere is the time step and is the truncation error. When is added to the left side of equation (18) and 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 , equation (18) is reorganized as

After equation (19) is computed and arranged again, a linear system is formed as

In equation (21),

The elements of are

The element of iswhere

Similarly, from equation (20), another linear system is written aswherein which

It can be observed that each on the th time layer is obtained, according to 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 , which represents the direction. Subsequently, we depend on and equation (28) to gain , which carries information in the direction and so on, until the solutions on the last time axis are worked out. The 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. The scheme of uniform owns spatial accuracy of the fourth order, whereas its nonuniform scheme owns spatial accuracy of the third order.

3. Adaptive Mesh Method

3.1. Temporal Adaptive Mesh Method

Comparing with solution , the temporal derivative is more sensible. Therefore, 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 . We can extend the 1D adaptive temporal grid schemes in Reference [23] to the 2D case:

After in equation (32) is discretized through the central difference formulas to calculate the maximum ratio, the adaptive time step is easily obtained by equation (32) and in the prior time layer.

3.2. Spatial Adaptive Mesh Method

Similarly, we can utilize the arc-length monitor function in the space direction based on and extend from the 1D adaptive spatial algorithm to the 2D problem by relying on Reference [23]. Therefore, representing the monitor function in the direction and representing the monitor function in the direction can be written as

The two spatial monitor functions are positive in their definition area. and 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 in the direction and in the direction. It is easily observed that the 1D adaptive grid scheme acts on in the direction and direction, respectively, to produce new grid points.

We take the equidistributing mesh in the direction as an example below. From [23], we know that the subareas of in intervals of are equal. Obviously, when , the scope is uniform. It can be expressed aswhere 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 represents the iterative number, represents the spatial grid node set before iteration, and represents the space grid node set after iteration. Equation (37) can be solved by the difference scheme:which approximates to the former. Similarly, the semi-implicit scheme in the direction is defined as

3.3. Iterative Adaption Method

The adaptive iterative procedure of the presented method is carried forward as follows. Firstly, the spatial monitor functions and can be obtained through equations (33) and (34) based on the grids and in the th time layer. Secondly, depending on equations (38) and (39), we can gain derived from in the direction and derived from in the direction. The new points substitute for the old points iteratively when and are reached for . is the terminal value of . Thirdly, from the value built on the last grid point sets and , is computed by using the method of area ratio. Thus, we can get in the th time layer depending on equation (18). Lastly, repeat the aforementioned produce till quenching occurs or the solution converges to a steady solution.

4. Numerical Illustration

4.1. 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:

The initial and boundary value conditions are derived from the exact solution. There are four metrics for the data test, which are maximum error, average error, CPU time, and convergence rate. We adopt these parameters to fulfill the two difference schemes: the proposed scheme based on the uniform mesh and the proposed scheme based on the nonuniform mesh. and represent the nonuniform grid, where point to intervals of the spatial direction and , whereas point to the mesh transformation coefficient and .

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 impact these precision criteria. In the paper, takes 0.7. The 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 . 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. The tendency of the conv. rate from the former scheme is similar to that from the latter scheme. We take the latter as the example. The four values are positive, which means the accuracy increases as rises when . The conv. rate value is maximum when transfers from 16 to 32, which means the accuracy promotes greatly in the stage. The latter three values are close, which means the accuracy promotes gently in the three stages.

4.2. Quenching Case without Convection Term

This section will describe a quenching status without the convection term, i.e., and . The 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 that has been described in [13, 14]. In Table 2 and Figure 1, , , and which are the same for the three approaches, but the initial space step and the initial temporal step are different among them, where we take and . In Table 2, the five parameters represent the same senses, in which represents the point of quenching location, represents the quenching time, represents the maximum temperature value immediately before quenching happens, and means the maximum variation of temperature with respect to time immediately before quenching happens. From Table 2, we can see that the quenching location of the referred scheme is identical with that of the scheme in [14], and and of the first one are close to those of the last two. Obviously, is different among the various methods. From Figure 1, it is found that and at the location and the quenching time .

(a)

(b)

4.3. Quenching Case with Convection Term

The 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]. The first convection term can be written as two functions and , where and take the constants. The four groups of variables: , and , and , and in equations (5)–(7) affect the quenching characteristics of the 2D problem, whose meaning has been stated in Section 2. There follows a special case for . The initial time step is considered as , and the initial space step is considered as , in which the -direction is identical with the -direction. The four groups of parameters are configured as which is defined as Case 7.0. This will lead to a quenching phenomenon. Quenching location is ; the quenching time is 4.487776579638927; is 0.9992409498317957; is 1539.153786381783. When 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 , , , , and before quenching occurs of Case 7.0. and refer to maximal and maximal in each time layer before quenching occurs; and refer to the coordinate point of synchronously producing and in each time layer before quenching occurs; refers to the moment at which each time layer is located before quenching occurs. first appears at the point when is 4.487088240486234 which is the first row in Table 3 and maintains at this point until quenching occurs. When is which is the immediate time before quenching happens, we can obtain important characteristics. When is 4.487777138788559, blows up and reaches 1.051076326841211 × 10⁹.

Figures 2–7 provide seven groups of graphs to illustrate more particular quenching statuses. Figure 2 shows the 2D plots of and changing as rises from the initial moment until quenching occurs. Figure 2(a) corresponds to , and Figure 2(b) corresponds to . Figure 3 sketches the 2D contours of and regarding immediately before quenching happens. Figure 3(a) corresponds to , and Figure 3(b) corresponds to . Figure 4 draws the 2D outlines of and regarding immediately before quenching happens. Figure 4(a) corresponds to , and Figure 4(b) corresponds to . The three 2D plots in Figure 5 describe the adaptive mechanism before quenching occurs. The first one reflects spatial adaptation, and the last two reflect temporal adaptation. Figure 5(a) describes the trend 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. The red sign on the blue curve in Figure 5(a) denotes the initial temporal adaptive moment (i.e., 4.486155872847072). The temporal step is uniform before the adaptive moment but becomes nonuniform when varies from 4.486155872847072 to 4.487777138788559. The green sign on the blue curve in Figure 5(a) denotes the initial moment (i.e., 4.487088240486234) of producing the quenching location . The green lines in Figures 5(b) and 5(c) describe the uniform spatial steps in the direction and the direction, respectively. The spatial adaptation occurs at 4.487777138788559 after which the spatial step becomes nonuniform. The red curves with blue spots in Figures 5(b) and 5(c) describe, respectively, the nonuniform spatial steps in the direction and the direction at . Although the two curves are very similar, there is still subtle difference between their data. Figures 6–7 provide the 3D surfaces of and with respect to spatial variables and at certain special time. Figure 6(a) records the 3D surfaces about , , and , and Figure 6(b) records the 3D surfaces about , , and when . Figure 7(a) records the 3D surfaces about , , and , and Figure 7(b) records the 3D surfaces about , , and when .

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(c)

(a)

(b)

(a)

(b)

4.4. Quenching Case with Convection Term

The second convection term can be written as the two convection functions: and , where and take the constants. Comparing with the first convection term , it is hard to conduct quenching behavior for the second convection term . We make a following configuration to gain a quenching status. The initial step (step) is , and the initial step is . The four groups of important parameters are set as which is defined as Case 8.0. The 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. The 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 is . 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. The latter quenching time is , is 0.9835341519876587, and is 73.15674241637485 immediately before quenching occurs, which is the sixth row in Table 4. gets the value 6.637517818582074 × 10¹¹ and blows up when is 5.604272047535358, which is the last row in Table 4.

This paragraph supplies five sets of figures (i.e., Figures 8–12) to more vividly depict the quenching information of the case. The 2D curves in Figure 8 describe the relationship between and and the relationship between and . The 2D curves in Figure 9 describe the relationship between and and the relationship between and . The 2D curves in Figure 10 describe the relationship between and and the relationship between and . In Figures 8–10, (a) corresponds to and (b) corresponds to . Figure 11(a) draws the tendency of the temporal step varying with temporal variable ; Figure 11(b) draws the tendency of the initial spatial step and the adaptive step varying with spatial variable ; and Figure 11(c) draws the tendency of the initial spatial step and the adaptive step varying with spatial variable . 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). in the time step becomes 1.185615675885128 × 10¹² when is 5.603982012140827. The temporal step is uniform before the adaptive moment but becomes nonuniform since then. The green spot on the blue curve expresses the initial moment of producing the quenching location , 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 and when . Figure 12(a) denotes the relationship between , , and , and Figure 12(b) denotes the relationship between , , and . Figure 12(a) records the 3D surfaces about , , and , and Figure 12(b) records the 3D surfaces about , , and when . There is a narrowband projecting the rear plane of the 3D axis box in Figure 12(b). This is because some larger values of appear at the positions, in which the axis and the axis are simultaneously near the origin of the coordinates. From the data source of the 3D surface, we can observe some larger varying from 1 to 27, whereas at other positions are positive numbers less than 1, which can be seen via Figures 9(b) and 10(b). Obviously, the larger are all less than 73.15674241637485, which is reasonable for a quenching case.

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(c)

(a)

(b)

4.5. The Effect of Burner Size on Quenching

The 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 , we also consider serial values of taken in besides corresponding to Case 7.0 in Table 5 because it does not produce quenching behavior when 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. Specially, Case 7.0.1: , Case 7.0.2: , Case 7.0.3: , Case 7.0.4: , Case 7.0.5: , Case 7.0.6: , Case 7.0.7: , Case 7.0.8: , Case 7.0.9: , and Case 7.0.10: . The quenching data associated with different based on , , and for the convection term are recorded in Table 5. From Table 5, we can discover that quenching time that is 4.517239999869761 is the largest when and quenching location that is 0.03636363636363636 when .

Similarly, we investigate quenching behavior variation with the changes of for convection function . From a number of tests, we select eleven meaningful cases to be given in Table 6. Case 8.0: , Case 8.0.1: , Case 8.0.2: , Case 8.0.3: , Case 8.0.4: , Case 8.0.5: , Case 8.0.6: , Case 8.0.7: , Case 8.0.8: , Case 8.0.9: , and Case 8.0.10: . The initial step is , and the initial step (step) is . , , , and are set as c. It is easily seen that quenching occurs, and quenching location is 0.025 when and quenching time is the smallest when from Table 6.

Figures 13 and 14 display the relationships between , , , and , respectively, in which, the red curves in Figures 13(a) and 14(a) reflect quenching time variations as rises; the blue curves in Figures 13(b) and 14(b) reflect quenching location variations as rises; the green curves in Figures 13(b) and 14(b) reflect quenching location variations as rises. Figure 13 corresponds to Table 5, and Figure 14 corresponds to Table 6. The 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 drops monotonously as increases when , and it promotes monotonously as increases when . The 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 is 0.95. The contour shows the fluctuation of the points up and down. For the relationship between and , we rely on the tendencies of the red dots on the blue curves in Figures 13(b) and 14(b), respectively, to see that 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 keeps a smaller value than when and becomes the same value as when for Figure 13(b) and is equal to 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).

(a)

(b)

(a)

(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 and when for convection function . But, it can result in quenching phenomena to properly take the ratio of and when for the convection term . 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 takes a value not less than 0.7. There 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., ), the horizontal coordinate of quenching location is equivalent to its vertical coordinate.

5. 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. The 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. The 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

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

References

H. Kawarada, On Solutions of Initial-Boundary Value Problem for u_t = u_xx + 1(1 − u), vol. 10, Publications of The Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan, 1975.
T. Hou, “Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and Navier-Stokes equations,” Acta Numerica, vol. 18, pp. 277–346, 2009.
View at: Publisher Site | Google Scholar
P. Chang and H. A. Levine, “The quenching of solutions of semi-linear hyper-bolic equations,” SIAM Journal of Mathematical Analysis, vol. 12, pp. 893–903, 1982.
View at: Google Scholar
A. Acker and B. Kawohl, “Remarks on quenching,” Nonlinear Analysis: Theory, Methods & Applications, vol. 13, no. 1, pp. 53–61, 1989.
View at: Publisher Site | Google Scholar
C. Chen, “The method of fundamental solutions for nonlinear thermal explosions,” Communications in Numerical Methods in Engineering, vol. 11, no. 8, pp. 675–681, 1995.
View at: Publisher Site | Google Scholar
K. W. Liang, P. Lin, M. T. Ong, and R. C. E. Tan, “A splitting moving mesh method for reaction–diffusion equations of quenching type,” Journal of Computational Physics, vol. 215, no. 2, pp. 757–777, 2006.
View at: Publisher Site | Google Scholar
Q. Sheng, “Adaptive decomposition finite difference methods for solving singular problems—a review,” Frontiers of Mathematics in China, vol. 4, no. 4, pp. 599–626, 2009.
View at: Publisher Site | Google Scholar
C. Y. Chan and L. Ke, “Parabolic quenching for nonsmooth convex domains,” Journal of Mathematical Analysis and Applications, vol. 186, no. 1, pp. 52–65, 1994.
View at: Publisher Site | Google Scholar
H. Cheng, P. Lin, Q. Sheng, and R. C. E. Tan, “Solving degenerate reaction-diffusion equations via variable step peaceman-rachford splitting,” SIAM Journal on Scientific Computing, vol. 25, no. 4, pp. 1273–1292, 2003.
View at: Publisher Site | Google Scholar
M. A. Beauregard and Q. Sheng, “An adaptive splitting approach for the quenching solution of reaction-diffusion equations over non-uniform grids,” Journal of Computational and Applied Mathematics, vol. 241, pp. 30–44, 2013.
View at: Publisher Site | Google Scholar
J. L. Padgett and Q. Sheng, “Non uniform crank-nicolson scheme for solving the stochastic Kawarada equation via arbitrary grids,” Numerical Methods for Partial Differential Equations, vol. 33, no. 4, pp. 1305–1328, 2017.
View at: Publisher Site | Google Scholar
A. Q. M. Khaliq and Q. Sheng, “On the monotonicity of an adaptive splitting scheme for two-dimensional singular reaction-diffusion,” International Journal of Computer Mathematics, vol. 84, no. 6, pp. 795–806, 2007.
View at: Publisher Site | Google Scholar
M. A. Beauregard and Q. Sheng, “A semi-adaptive compact splitting method for the numerical solution of 2-dimensional quenching problems,” Applied Mathematics and Computation, vol. 218, no. 22, pp. 11240–11254, 2012.
View at: Publisher Site | Google Scholar
M. A. Beauregard and Q. Sheng, “Solving degenerate quenching-combustion equations by an adaptive splitting method on evolving grids,” Computers and Structures, vol. 122, pp. 33–43, 2013.
View at: Publisher Site | Google Scholar
M. A. Beauregard, “Numerical solutions to singular reaction diffusion equation over elliptical domains,” Applied Mathematics and Computation, vol. 254, pp. 75–91, 2015.
View at: Publisher Site | Google Scholar
J. L. Padgett and Q. Sheng, “Numerical solution of degenerate stochastic Kawarada equations via a semi-discretized approach,” Applied Mathematics and Computation, vol. 325, pp. 210–226, 2018.
View at: Publisher Site | Google Scholar
J. L. Padgett and Q. Sheng, “On the stability of a variable step exponential splitting method for solving multidimensional quenching-combustion equations,” in Modern Mathematical Methods and High Performance Computing in Science and Technology, vol. 171, pp. 155–167, Springer, Berlin, Germany, 2016, in Mathematics & Statistics.
View at: Publisher Site | Google Scholar
Y. Ge, Z. Cai, and Q. Sheng, “A compact adaptive approach for degenerate singular reaction -diffusion equations,” Numerical Methods for Partial Differential Equations, vol. 34, no. 4, pp. 1166–1187, 2018.
View at: Publisher Site | Google Scholar
Q. Sheng and A. Q. M. Khaliq, “A compound adaptive approach to degenerate nonlinear quenching problems,” Numerical Methods for Partial Differential Equations, vol. 15, no. 1, pp. 29–47, 1999.
View at: Publisher Site | Google Scholar
Q. Sheng and H. Cheng, “An adaptive grid method for degenerate semi-linear quenching problems,” Computers and Mathematics with Applications, vol. 39, no. 9–10, pp. 57–71, 2000.
View at: Publisher Site | Google Scholar
Q. Sheng and A. Q. M. Khliq, “Adaptive algorithms of convection-diffusion-reaction equations of quenching type, dynamic continuous,” Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical and Analysis, vol. 8, pp. 129–148, 2001.
View at: Google Scholar
Q. Zhou, Y. Nie, X. Zhou, and W. Guo, “Quenching of a semi-linear diffusion equation with convection and reaction,” Electronic Journal of Differential Equations, vol. 2015, no. 1, pp. 1–7, 2015.
View at: Google Scholar
X. Zhu and R. Hong, “High-order compact difference scheme of 1D nonlinear degenerate convection- reaction–diffusion equation with adaptive algorithm,” Numerical Heat Transfer, Part B: Fundamentals, vol. 75, no. 1, pp. 43–66, 2019.
View at: Publisher Site | Google Scholar
P. Sun, L. Chen, and J. Xu, “Numerical studies of adaptive finite element methods for two dimensional convection-dominated problems,” Journal of Scientific Computing, vol. 43, no. 1, pp. 24–43, 2010.
View at: Publisher Site | Google Scholar
V. Joshi and R. Jaiman, “A positivity preserving variational method for multi-dimensional convection-diffusion-reaction equation,” Journal of Computational Physics, vol. 339, pp. 247–284, 2017.
View at: Publisher Site | Google Scholar
J. C. Kalita, A. K. Dass, and D. C. Dalal, “A transformation-free HOC scheme for steady convection–diffusion on non-uniform grids,” International Journal for Numerical Methods in Fluids, vol. 44, pp. 33–53, 2004.
View at: Publisher Site | Google Scholar
Y. Ge and F. Cao, “Multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems,” Journal of Computational Physics, vol. 230, no. 10, pp. 4051–4070, 2011.
View at: Publisher Site | Google Scholar
X. Huang and Y. Ge, A High-Order Compact ADI Method on Non-uniform Grid for Unsteady Convection-Diffusion Equations, Ningxia University, Yinchuan, China, 2013, in Chinese.
L. Li, Z. Jiang, and Z. Yin, “Fourth-order compact finite difference method for solving two-dimensional convection–diffusion equation,” Advances in Differential Equations, vol. 234, pp. 1–24, 2018.
View at: Publisher Site | Google Scholar
K. M. Owolabi, “Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems,” Chaos, Solitons and Fractals, vol. 93, pp. 89–98, 2016.
View at: Publisher Site | Google Scholar
E. Pindza and K. M. Owolabi, “Fourier spectral method for higher order space fractional reaction–diffusion equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 40, pp. 112–128, 2016.
View at: Publisher Site | Google Scholar
J. Li, Y. Qiao, S. Zhai, and X. Feng, “Meshless local Petrov Galerkin method for 2D 3D nonlinear convection diffusion equations based on LS RBF PUM,” Advances in Differential Equations, vol. 74, no. 1, pp. 450–464, 2018.
View at: Publisher Site | Google Scholar
K. M. Owolabi and A. Atangana, “Numerical simulation of noninteger order system in subdiffusive, diffusive, and superdiffusive scenarios,” Journal Computational and Nonlinear Dynamics, vol. 12, no. 3, 2017.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 Xiaoliang Zhu and Yongbin Ge. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

675

Downloads

603

Citations