ABSTRACT
In this study, to solve the singularly perturbed delay convection–diffusion–reaction problem, we proposed a hybrid numerical scheme that converges uniformly. Parabolic right boundary layer outcomes from the presence of the small perturbation parameter. To grip this layer behaviour, the problem is solved by Bakhvalov–Shishkin mesh for spatial domain discretization and uniform mesh for temporal domain discretization. A hybrid scheme consisting of a non-polynomial spline scheme for fine mesh and a midpoint upwind scheme for coarse mesh is used to discretize the spatial derivative, while an implicit Euler scheme is used to discretize the time derivative. To make computed solutions more accurate and increase rate of convergence of the scheme, we applied Richardson extrapolation technique. The stability and convergence of the scheme are established. The scheme has a second order of convergence in the discrete supreme norm and is parametric uniformly convergent. The scheme's application is demonstrated through two test problems.
1. Introduction
Differential equations that have their highest-order derivatives multiplied by a perturbation parameter ε are said to be singularly perturbed equations. The perturbation parameter takes an arbitrary value in which forms interior and/or boundary layers, that is, narrow subdomains on which the solution has a steep gradient. In these subdomains, when ε approaches zero, the derivatives of the solution increase rapidly. Singularly perturbed delay partial differential equations are singularly perturbed problems that involve one or more delay terms. The modelling of many real-world phenomena, including physical, chemical, and biological systems, which are characterized by variables that are both temporal and spatial, uses these equations [Citation1–4]. Particularly, in a better way to comprehend how singularly perturbed delay parabolic differential equations are applied, one can see the example given in [Citation3] , which is being used to model a furnace that processes metal sheet. In the example,the controller's finite speed is the cause of the delay.
Because of the boundary layer's existence in the singularly perturbed problem solution, the classical finite difference methods provide an unsatisfactory numerical result. So various numerical approaches have been developed for solving these problems, which need the so-called uniformly convergent method regardless of the perturbation parameter's value [Citation5,Citation6]. Fitted mesh and fitted operator are the two main numerical methods for constructing a uniformly convergent scheme. Fitted mesh methods use layer-adapted non-uniform mesh, which is dense in the layer(s), while the fitted operator uses an exponentially fitted scheme. There are also other parameter-uniform numerical approximations approaches for singularly perturbed problems, such as domain decomposition [Citation7,Citation8] and grid equidistribution [Citation9]. One can read literature on singularly perturbed delay problems in [Citation10–15]. Richardson extrapolation technique is a numerical method used to improve the accuracy of numerical solutions of differential equations. In the context of singularly perturbed problems, the Richardson extrapolation technique has been used to improve the accuracy of numerical schemes. The reader can get an exposition of this approach in [Citation16–19].
Using spline approximation methods, many authors developed uniformly convergent numerical methods for singularly perturbed delay convection–diffusion–reaction boundary value problems with Dirichlet boundary conditions. To name a few, a second-order exponentially fitted spline and B-spline collocation non-standard method on uniform mesh by a Negero [Citation20,Citation21], a second-order exponentially fitted tension spline on a uniform mesh, a reaction–diffusion version by Megiso et al. [Citation22], and essentially second-order tension spline on shishkin mesh but a reaction-diffusion version by Murali [Citation23]. A singularly perturbed problem with a large delay is one where the delay parameter exceeds the perturbation parameter; otherwise, it is said to be a singularly perturbed problem with small delay [Citation24,Citation25].To the best of our knowledge, the problem under consideration has not been solved using hybrid finite difference, which comprises midpoint upwind and the non-polynomial spline method on the Bakhvalov–Shishkin (B-S) mesh, till date. Motivated by articles [Citation23,Citation26], we developed a uniformly convergent numerical method for solving singularly perturbed parabolic convection–diffusion boundary value problem with a large time delay. The present work has novelties in terms of the existing work. The proposed method is free from convergency spoiled by a logarithmic factor, unlike using the method on Shishkin method. Additionally, the integration of Bakhvalov mesh and the non-polynomial spline method in the layer region potentially provide stability and an accurate numerical solution for singularly perturbed delay parabolic differential equations and allow for more flexibility in choosing the basis function used to approximate the function.
The rest of the paper is structured as follows: In Section 2 continuous problem formulation is described. Section 3 describes bounds on the continuous solution and its derivatives. In Section 4, the numerical scheme is implemented. Section 5 discusses the method's uniform convergence analysis. In Section 6, The technique of Richardson extrapolation has been explained. In Section 7, numerical experiments are illustrated in order to verify the theoretical result. The main conclusions are summarized in the last section.
Throughout the paper, C,, and , have been taken as generic positive constants that are independent of mesh size, mesh point, and perturbation parameter.
In the following sections denote standard supremum norm and defined as defined on domain D.
2. Statement of the problem
The class of singularly perturbed delay parabolic convection–diffusion problems that we examine is as follows. (1) (1) with the subsequent interval and boundary conditions: (2) (2) where, Here and Also and are respectively, perturbation and delay parameters. The functions , and in , in , and and in Γ are assumed to be sufficiently smooth, bounded and independent of ε, that satisfy It is assumed that the terminal time T satisfies the condition , where k is positive integer. Under the aforementioned circumstances, the boundary layer of width along x = 1 is displayed in the boundary value problem Equations (Equation1(1) (1) )–(Equation2(2) (2) ) solution. The problem under consideration occurs in fluid dynamics. In this engineering field, the coefficient of the convective term represents the f mass transfer rate [Citation27]. If this transfer rate is influenced by external factors such as temperature gradient or fluid velocity variation, the convective flow is more spatially dependent, i.e. it is time-invariant . However, it is important to note that there are situations where there is both significant temporal and spatial variation occurs, so the coefficient of the convection term is In such cases, the convergence analysis is general and is covered in [Citation28,Citation29].
3. Bounds on the solution and its derivatives
If the data are sufficiently smooth in their domain of definition and meet the following compatibility conditions at the corner points ,,, and , then the existence and uniqueness of the Equations (Equation1(1) (1) )–(Equation2(2) (2) ) solution can be guaranteed. (3) (3)
Lemma 3.1
Maximum Principle
Assume , such that , for all and for all Then , for all
Proof.
Let such that , assume , clearly and
From Calculus we have, and then from Equations (Equation1(1) (1) )–(Equation2(2) (2) ) we have which contradicts to Hence , for all
We divide the time interval using the delay term τ, such as , and so forth, based on the available method of steps [Citation30–33]. On the interval , in (Equation1(1) (1) )–(Equation2(2) (2) ) becomes the expression becomes ,which is independent of ε in Hence, for , (Equation1(1) (1) )–(Equation2(2) (2) ) becomes (4) (4) Now, we decompose the solution in to its singular and regular components as The two terms asymptotic expansion for is , where satisfies (5) (5) and satisfies (6) (6) Thus, the regular component satisfies (7) (7) The singular component satisfy the homogeneous problem: (8) (8) Now, for , (Equation1(1) (1) )–(Equation2(2) (2) ) becomes (9) (9) Again, we decompose the solution in to its singular and regular components as The two terms asymptotic expansion for is , where satisfies the reduced problem (10) (10) and satisfies (11) (11) Clearly the regular component satisfies the following problem (12) (12) The singular component satisfy the homogeneous problem: (13) (13) In the same way, to get the decomposition on , we extend the decomposition approaches at every partition. This method of steps yields the existence and uniqueness results for all . For the interval , is known function,i.e. is independent of ε, and thus (Equation1(1) (1) )–(Equation2(2) (2) ) becomes a classical singularly perturbed parabolic problem, which can be solved the existing theories [Citation34,Citation35]. In addition, for the existence and uniqueness of a solution of (Equation1(1) (1) )–(Equation2(2) (2) ) is considered in [Citation30,Citation36–38].
Lemma 3.2
[Citation39],Lemma 3.2.
The solution of Equations (Equation1(1) (1) )–(Equation2(2) (2) ) satisfies the following bound:
where C is independent of ε.
Lemma 3.3
The solution of Equations (Equation1(1) (1) )–(Equation2(2) (2) ) satisfies the following estimate: ,for
Proof.
Since and from Lemma 3.2 we have , it follows that
Lemma 3.4
Uniform Stability Estimate
The ε-uniform bound on the solution of continuous problem equations (Equation1(1) (1) )–(Equation2(2) (2) ) satisfies
Proof.
For the barrier functions we have,on on on Furthermore, for all , Therefore, by using maximum principle stated in Lemma 3.1, we obtain for all . Accordingly, we get the required bound.
Theorem 3.1
The derivatives of exact solution of Equations (Equation1(1) (1) )–(Equation2(2) (2) ) satisfy where m and n are integers that are non-negative, such that
Proof.
Refer in [Citation40]
Bounds on the derivatives of Equations (Equation1(1) (1) )–(Equation2(2) (2) ) solution given Theorem in 3.1 are not adequate for proofing ε-uniform convergence. Strong bounds on the solution are therefore required.
Theorem 3.2
For non-negative integers n, m providing , with suitable compatibility conditions at the corners the regular and singular component satisfies the following estimate (14) (14) (15) (15)
Proof.
The details of the proof are found in [Citation26,Citation41].
4. Numerical scheme formulation
Since the model problem equation (Equation1(1) (1) )–(Equation2(2) (2) ) exhibit boundary layer at x = 1, we have to made more mesh points there. We use uniform mesh for time domain and Bakhvalov–Shishkin mesh for space domain.
4.1. Time discretization
The time domain is discretize with time step as where M denote mesh interval numbers in the direction on and satisfies for some positive integer q. If is the set of all mesh points from to 0: For the time derivative, we employ the implicit Euler method to obtain semi-discretized problem for Equation (Equation1(1) (1) )–(Equation2(2) (2) ) (16) (16) under the subsequent boundary and interval conditions (17) (17) where is the approximate solution of at th time level. Equations (Equation16(16) (16) )–(Equation17(17) (17) ) can be written as (18) (18) where The operator satisfy the following maximum principle.
Lemma 4.1
Semi-discrete maximum principle
Let be sufficiently smooth function on Ω, such that and . Then for all implies for all .
Proof.
Let ζ be such that and assume that . From the assumption made,clearly . From calculus, we have , and . Then which contradicts to the assumption . Therefore for all .
The local truncation error of the time semi-discretized problem equation (Equation18(18) (18) ) is given by .
Lemma 4.2
If for all and , then the local truncation error associated to Equation (Equation18(18) (18) ) satisfies .
Proof.
Applying Taylor's series expansion on ; Thus, (19) (19) At , we have (20) (20) substituting Equation (Equation20(20) (20) ) in to Equation (Equation19(19) (19) ) we get (21) (21) From Equation (Equation18(18) (18) ), satisfies (22) (22) From Equations (Equation21(21) (21) ) and (Equation22(22) (22) ), clearly the local truncation error is the solution of boundary value problem of type Applying the semi-discrete maximum principle now yields .
Local truncation error measures the contribution of each time step to the global error of the time semidiscretization, which is defined at , given as .
Theorem 4.1
Global error estimate,denoted by , of Equation (Equation18(18) (18) ) satisfy
Proof.
C is a positive constant independent of ε and .
Lemma 4.3
The semi-discretized solution of Equation (Equation18(18) (18) ) satisfy
Proof.
In Theorem 3.1 make m = 0 and
The strong bounds can obtained by decomposing the solution of semi-disctrized problem (Equation18(18) (18) ) in to regular and singular component as with the following properties and condition (23) (23) (24) (24)
Lemma 4.4
[Citation26]
The regular and singular components of and their derivatives satisfy the following bounds: (25) (25)
4.2. Full discretization
On the spatial domain , we choose B-S mesh which divides it in two intervals and . Each interval is divided in to N/2 equal sub intervals, where . B-S mesh in one of the Shishkin type mesh, where the boundary layer function in inverted on so that the mesh is graded. On the mesh is coarse and uniform. The transition parameter σ is defined as We shall assume otherwise N is exponentially smaller than ε. The mesh is given by [Citation26,Citation42] (26) (26) For we set and for . Given a function defined on , define the forward and backward difference operator as (27) (27) The approximation of second-order derivative at is defined as (28) (28)
4.2.1. Hybrid numerical scheme
Midpoint upwind method [Citation43,Citation44] Rewriting Equation (Equation18(18) (18) ), we get (29) (29) where The midpoint upwind scheme for Equation (Equation29(29) (29) ) is (30) (30) where Substituting Equations ((Equation27(27) (27) )–(Equation28(28) (28) )) in to Equation (Equation30(30) (30) ), we obtain (31a) (31a) where (31b) (31b) Non-polynomial spline scheme There have been literatures that propose the use of spline-based approaches for solving singularly perturbed problems with unform convergence analysis (e.g. [Citation45]). In this work we use the non-polynomial spline. For each segment , the non-polynomial spline function which interpolate has the following form (32) (32) where are the unknown constant coefficients, that are to be determined and is free parameter such that reduces to ordinary cubic spline as . To obtain these coefficients, the necessary conditions are : should satisfy interpolation conditions at and , and first derivative continuity at common nodes. Therefor to derive the coefficients in Equation (Equation32(32) (32) ) in terms of we define (33) (33) From Equation (Equation32(32) (32) ) we obtain (34) (34) From Equations (Equation32(32) (32) )–(Equation34(34) (34) ), we obtain (35) (35) where . From the first derivative continuity of ,we have . That is (36) (36) Substituting Equation (Equation35(35) (35) ) in to Equation (Equation36(36) (36) ) we obtain the following relation at i + 1 time level (37) (37) where . We have by equating the coefficients of from Equation (Equation37(37) (37) ). When .
Using Taylor series,let us derive first-order approximation (38) (38) Then (39) (39) (40) (40) Then (41) (41) Taking and as variables, then solving them from Equations (Equation39(39) (39) ) and (Equation41(41) (41) ), we get (42) (42) and (43) (43) Differentiation Equations (Equation39(39) (39) ) and (Equation41(41) (41) ) both sides, we get (44) (44) (45) (45) After substituting Equations (Equation42(42) (42) ) and (Equation43(43) (43) ) in to Equations (Equation44(44) (44) ) and (Equation45(45) (45) ), we obtain (46) (46) (47) (47) From Equation (Equation29(29) (29) ) we can have (48) (48) Now substitute Equation (Equation48(48) (48) ) in to Equation (Equation37(37) (37) ) to get (49) (49) By substituting Equations (Equation42(42) (42) ),(Equation46(46) (46) ) and (Equation47(47) (47) ) in to (Equation49(49) (49) ) at i + 1 time level we obtain (50a) (50a) where (50b) (50b) Combining Equation (Equation31a(31a) (31a) ) and (Equation50a(50a) (50a) ), we obtain fully discretized scheme (51) (51) Equation (Equation51(51) (51) ) gives system of equations as (52) (52) where In , The matrix representation of Equation (Equation52(52) (52) ) is (53) (53) The matrix E is tridiagonal matrix and ; diagonally dominant matrix. In addition, and . Thus at each time level the coefficient matrix E is irreducibly diagonally dominant M-matrix [Citation46] which assures the invertibility of the matrix.
Lemma 4.5
Discrete Maximum Principle
Let be any mesh on . Assume the discrete function satisfy . Then for implies for .
Proof.
Follow the same technique used in Lemma 4.1.
5. Convergence analysis of the method
In this section, the convergence is examined by induction on the subintervals , for positive integer κ.
Lemma 5.1
Discrete uniform stability estimate
The solution of Equation (Equation51(51) (51) ) satisfy
Proof.
Let . Define barrier functions such that . By applying Lemma 4.5, we can obtain the required result.
Lemma 5.2
[Citation47] Discrete comparison principle
If and are two mesh functions that satisfy ,, and , then for .
Lemma 5.3
Let for . Then there exists a positive constant C such that for
Proof.
For For , use the technique as done for .
Lemma 5.4
Let be a smooth function defined on and on . The following estimate holds true
(a) | For | ||||
(b) | For |
Proof.
One may refer [Citation35,Citation48] for the details.
Lemma 5.5
For , define a mesh function with convention that if . Then,for , there exist a positive constant C such that
Proof.
Refer [Citation6].
The error in numerical solution can be decomposed in to singular and regular component as Next we estimate the error of each component in the intervals and .
Theorem 5.1
The error in the regular component satisfy the following bound
Proof.
First consider . Let . Then we can have . Using Lemmas 4.4 and 5.4(a),as the derivatives of are bounded For the interval , by applying Lemmas 4.4 and 5.4(b) for the same technique we get the same result for . Now set for , where C chosen so that is barrier function for . By applying Lemma 5.3, we have given that C is a large enough constant. Clearly and . So Lemma 5.2 can be applied and we get for all j.
Theorem 5.2
The error in the singular component satisfy the following bound
Proof.
As done for regular component first let us consider . Recall we assumed ,which yield From Lemma 4.4, we have (54) (54) Now we want to show for a constant C. Let . From Taylor series , so for the mesh function (55) (55) Let for . Then by using definition of and Lemma 5.5 (56) (56) For sufficiently large value of constant C, we have (57) (57) (58) (58) From Equations (Equation56(56) (56) )–(Equation58(58) (58) ), by discrete comparison principle ,Lemma 5.2, we have (59) (59) But for (60) (60) After some manipulation of applying the Taylor series to the right end expression of Equation (Equation60(60) (60) ), we obtain (61) (61) Combining Equations (Equation54(54) (54) ) and (Equation61(61) (61) ), gives the required result for .
Now consider the interval .
Let . For the bounds on in Lemma 4.4 an error analysis for is (62) (62) for sufficiently large value of , for all . Set for . Then by using Equation (Equation55(55) (55) ) and Lemma 5.5 (63) (63) (64) (64) (65) (65) By discrete comparison principle Lemma 5.2 and Equations (Equation62(62) (62) )–(Equation65(65) (65) ) we obtain .
Theorem 5.3
Error in the totally discrete scheme
Let be the solution of continuous problem given by Equations (Equation1(1) (1) )–(Equation2(2) (2) ) and be the approximation to the solution of the fully discretized problem equation (Equation52(52) (52) ). Then, the error estimate for the total discrete scheme is given
proof.
By combining the estimates provided in Theorems 4.1,5.1 and 5.2, the proof can be readily obtained.
6. Richardson extrapolation
In this section, we use the Richardson extrapolation technique to improve the scheme's rate of convergence and computed solution accuracy. Let us define a mesh with and mesh intervals in spatial and temporal directions respectively. Here is a Bakhvalov–Shishkin mesh possessing the same transition point as used in . The discrete spatial domain and temporal domain are obtained from bisecting of each intervals of and , respectively.
Let be the numerical solution of the fully disretized scheme Equation (Equation51(51) (51) ). Then from Theorem 5.3, one can write (66) (66) Similarly assume be the numerical solution of the fully disretized scheme Equation (Equation51(51) (51) ) on . Then (67) (67) The two remainders and are . From Equations (Equation66(66) (66) )–(Equation67(67) (67) ) using the approaches in [Citation49–51], we get (68) (68) Now, we define the extrapolation formula as (69) (69) which gives a better approximation of than or . The truncation error of spatial discretization when approximating (Equation69(69) (69) ) can be written as follows: (70) (70)
Theorem 6.1
Error After Richardson Extrapolation
Let be the exact solution of Equations (Equation1(1) (1) )–(Equation2(2) (2) ) and be the extrapolated solution as defined in Equation (Equation69(69) (69) ). Then
Proof.
Like the exact solution of Equations (Equation1(1) (1) )–(Equation2(2) (2) ) and the numerical solution of Equation (Equation51(51) (51) ), the numerical solution of Equation (Equation51(51) (51) ) on can be decomposed in to regular and singular components such that (71) (71) Therefore, at the point , we have (72) (72) The combination of Equations (Equation70(70) (70) ) and (Equation72(72) (72) ) gives (73) (73) where is a regular component, and is a singular component such that . First we consider the regular components in outer layer region and interior layer region, i.e. boundary layer region. From Theorem 5.1 and the techniques used in [Citation52,Citation53], on the mesh , we have (74) (74) where is and is . For , by using Theorem 5.1 and Equation (Equation74(74) (74) ), we have Thus (75) (75) For , using the same procedure employed above, we can get the same result (Equation75(75) (75) ). Now consider the singular components in outer layer region and interior layer region. First consider the estimate in the outer layer region . From Theorem 3.2, we have (76) (76) For a given time level , we consider the following barrier function in order to determine the bound of . By the same technique used in [Citation44], we obtain . Therefore (77) (77) In a similar approach, we can get (78) (78) From Equations (Equation77(77) (77) )–(Equation78(78) (78) ) for outer layer region, we obtain (79) (79) For interior layer region , by following the same procedure in [Citation17,Citation53,Citation54], we obtain Therefore, for inner layer region, we get (80) (80) Hence the combination of Equations (Equation75(75) (75) ),(Equation79(79) (79) ) and (Equation80(80) (80) ) gives the required result.
7. Numerical examples, results and discussion
In this section, we present the numerical results obtained by the proposed numerical scheme for the problems type Equations (Equation1(1) (1) )–(Equation2(2) (2) ). In all case, the numerical experiments performed by taking constant .
Example 7.1
[Citation55]
Consider singularly perturbed delay parabolic initial boundary value problem:
Example 7.2
[Citation56]
Now consider the following singularly perturbed time-delay parabolic convection–diffusion problem
Example 7.3
[Citation54]
Consider the following singularly perturbed delay parabolic initial boundary value problem: We choose the initial data and the source function to fit with the exact solution where and .
The numerical solution profile for Examples 7.1, 7.2 and 7.3 are shown in Figures , , and , respectively. These figures confirm existence of the boundary layer in the solution near x = 1 and show effects of the parameter ε on gradient of the boundary layer. The exact solutions to Examples 7.1 and 7.2 are unknown, so we use double mesh principle to compute the absolute maximum point wise error. Let be the computed numerical solution on the fine mesh with and mesh intervals in the temporal and spatial directions respectively. We obtain from by adding M and N additional points in the temporal and spatial directions respectively by including the mid-points and into the mesh points. Therefore, for each value of ε, we estimate the absolute maximum point wise error as Since we know the exact solution to Example 7.3, we can use the following to find the maximum pointwise error for each ε. The corresponding order of convergence is estimated by The uniform error is estimated by using and the corresponding uniform order of convergence is estimated by Tables show the maximum point wise error before and after extrapolation for various values of ε and order of convergence. From these tables, it can be observed that decreases as N increases for each value of ε and the errors are stabilized as . Thus, the proposed numerical scheme is ε-uniformly convergent on Bakhvalov–Shishkin mesh. The convergence order is nearly linear.After using Richardson extrapolation, we are getting second-order convergence, which is almost double before extrapolation technique and the accuracy of computed results improved. Tables and display comparison of the proposed scheme with the previous works. It is observed the proposed scheme is better than those considered in [Citation57] and [Citation58].
Tables and display that, as values of ε goes to zero, the proposed scheme does not achieve uniform convergence if we use uniform mesh,as the error (and also ) increase when the number of mesh points increases. This drawback is solved by using one of the Shishkin type mesh called Bakhvalov–Shishkin mesh.
The maximum point wise errors for the solutions also plotted on log–log scale in Figures , , and , before and after extrapolation. The straight line that the plots follow demonstrates to us that one variable changes as a constant power of another; the maximum absolute point-wise error changes as a constant power of mesh parameter N. In addition, according to the negative slope, the maximum absolute error decreases as the number of grid points rises. In these figures the plots are parallel and overlap as . This shows that the proposed scheme converges regardless of the value of the perturbation parameter ε. Further, we note that all computations have been performed using MATLAB® R2022b software.
8. Conclusion
A numerical scheme for solving a class of singularly perturbed delay parabolic partial differential equations with Dirichlet boundary conditions is investigated. The scheme comprises implicit Euler for time discretization on uniform mesh and a hybrid scheme for spatial discretization on Bakhvalov–Shishkin mesh. The hybrid scheme is composed of midpoint upwind for coarse mesh and non-polynomial spline on fine mesh. Richardson extrapolation technique has been used to improve the accuracy of computed solutions and rate of convergence of the scheme. The delay term is treated by making the delay argument as one of nodal points. It has been shown that the present method is uniformly convergent with respect to the perturbation parameter ε and achieves second-order accuracy. Three test examples are presented that numerically validate the theoretical result. Further, it has been shown by the first two examples that the scheme is not convergent on a uniform mesh. The proposed method combined the unconditional stability and flexibility in choosing the basis function advantages from midpoint upwind and non-polynomial spline scheme respectively. These advantages are significant for solving singularly perturbed delay differential equations involving boundary layer phenomenon. Accordingly, as future directions of this research we extend the proposed scheme for solving time fractional singularly perturbed convection–diffusion problems with a delay in time.
Disclosure statement
No potential conflict of interest was reported by the author(s).
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