Nonpolynomial Spline Method for Singularly Perturbed Time-Dependent Parabolic Problem with Two Small Parameters

. Tis study deals with the numerical solution of parabolic convection-difusion problems involving two small positive parameters and arising in modeling of hydrodynamics. To approximate the solution, the backward Euler method for time stepping and ftted trigonometric-spline scheme for spatial discretization are considered on uniform meshes. Te resulting scheme is shown to be uniformly convergent and its rate of convergence is one in the time variable and two in the space variable. Te accuracy and rate of convergence are enhanced by using the Richardson extrapolation. To support the theoretically shown convergence analysis, we have taken some numerical examples and compared the absolute maximum error of the current method with some methods existing in the literature.


Introduction
Consider the following time-dependent singularly perturbed one-dimensional parabolic convection-difusion problem with initial and boundary conditions of Dirichlet type on the domain D � Ω × (0, T], Ω � (0, 1).
constrained to boundary conditions where 0 < ε and μ ≪ 1 are two small positive parameters. We assume that the coefcients a(x, t), b(x, t), and f(x, t) are sufciently regular functions with the conditions a(x, t) ≥ α > 0 and b(x, t) ≥ β > 0 for all (x, t) ∈ D. We also assume that the given data satisfy sufcient smoothness and compatibility conditions on the corner of the domains (0, 0) and (1, 0), i.e., s(0) � u(0, 0) and s(1) � u (1,0). Tese conditions confrm the existence of a unique solution to the problem. Physical occurrences, where energy, heat, or fuid is transformed inside a physical system due to the movement of molecules within the system (convection) and the spread of particles through the random motion from regions of higher concentration to regions of lower concentration (difusion) are described by parabolic partial diferential equations called convection-difusion problems. In the case when the spatial derivatives are multiplied by small positive parameters (such as the partial diferential equations of hydrodynamics), the problems are said to be singularly perturbed problems (SPPs) [1,2]. Te solution to these problems can have a steep gradient where boundary layers occur [3]. Using an analytic method or asymptotic expansions may be impossible to construct a solution or may fail to simplify the given problem. As a result, numerical approximations are the only option.
Classical numerical methods are inefcient to approximate their solution as the perturbation parameter goes to zero. In recent years, more due attention has been paid by diferent scholars on the special purpose numerical methods which are not afected by this parameter. For instance, Clavero et al. [4] defned a higher-order fnite diference scheme on a piecewise-uniform Shishkin mesh, Jha and Kadalbajoo [5] constructed a robust layer adapted diference method with Shishkin-Bakhvalov mesh, and Zahra et al. [6] have developed an exponential-spline scheme on spatial piecewise-uniform mesh of Shishkin type. Das and Natesan [7] and Das and Mehrmann [8] obtained a numerical solution via adaptively generated mesh with the concept of equidistribution, Beckett and Mackenzie [9] solved singularly perturbed reaction-difusion problem using grid equidistribution, Kadalbajoo and Jha [10] have used exponentially ftted cubic spline to solve singularly perturbed convection-difusion problem, and Mittal and Jain [11] defned cubic B-spline collocation method for solving convection-difusion equations. Kadalbajoo et al. [12] have solved the convection-difusion problem with dominating convection term using a ftted collocation method. More recently, ftted operators on uniform meshes have been used to generate parameter-uniform methods like ftting the two small parameters through a fnite diference [13], ftting the parameters through nonstandard method of Micken's [14,15], B-spline technique [16], central diference approximation [17], and piece-wise uniform mesh [18] for singularly perturbed parabolic diferential equation of convectiondifusion type with two small parameters.
Te numerical solution of two-point boundary-value problems which depend only on the spatial variable has been considered by Aziz and Khan [19], Bawa [20], Phaneendra et al. [21], Sirisha et al. [22], and Kumara Swamy et al. [23]. Time dependent parabolic problems subjected to initial and boundary conditions have been treated by Mohanty et al. [24] using nonpolynomial splines involving trigonometric functions. All the problems considered by these authors are with one perturbation parameter and each of the proposed methods is nonftted. With this enthusiasm, in this work, we discuss trigonometric spline with ftted operator fnite diference method on a uniform mesh to use it as a tool to solve time-dependent singularly perturbed onedimensional initial-boundary value problem (IBVP) (1), involving two small positive perturbation parameters. Tis approach has the advantage over fnite diference methods in which it provides continuous approximations not only for the solution u(x, t) but also for its considered derivatives at every point of the range of integration. Also, the continuous diferentiability of the trigonometric part of the splines considered here reduces the loss of smoothness inherited by polynomial splines [21].
Te manuscript is organized as follows: in Section 2, the properties of the continuous problem are given. Error estimates and detailed procedure for the construction of the trigonometric spline scheme are presented in Section 3. In Section 4, the parameter-uniform convergence analysis has been carried out. Numerical examples and results are illustrated by tables and graphs in Section 5. At last, discussion of the results obtained and conclusion are given in Section 6.

Properties of the Continuous Problem
In this section, we consider some properties of the continuous problem through maximum principle and bounds on the solution and on its derivatives that enable us to see a priori estimates for the solution u(x, t) and its derivatives.
Proof. Te proof is given in [16]. □ Lemma 2 (Bounds of the solution). Let u(x, t) be the solution of equation (1) and b(x, t) ≥ β > 0, then we have the estimate where ‖.‖ is the maximum norm.
Proof. Te proof is given in [16].
□ Lemma 3 (Bounds of the derivatives). Te derivatives of the solution u(x, t) satisfy the following bound for all nonnegative integers i, j such that i + 3j ≤ 4.
where the constant C is independent of ε and μ and depends only on the bounded derivatives of the coefcients and the source term.
Proof. Te proof is given in [25].
and the global error satisfes with a constant C.
Proof. From Taylor's series expansion we have Since u(x, t m ) is also expected to satisfy equation (6), we have and U(x, t m ) satisfes equation (6), then we can write it as Simplifying further, we can arrive at from which we can deduce Moreover, the global error is given by Terefore, the time semidiscretization process is uniformly convergent of order one.

t m ) of the boundary-value problem equation (6) satisfes the estimate
Proof. For the proof, one can see [26].
Let us introduce a ftting factor σ which controls the efect of the perturbation parameter ε. Ten, at x � x n , equation (6) can be written as Let U(x n , t m ) be an approximate solution to u(x n , t m ) of equation (17) obtained by the segment Q n (x, t m ) passing through the points (x n , t m , U(x n , t m )) and (x n+1 , t m , U(x n+1 , t m )). Te segment is supposed to satisfy the interpolation condition, and its frst-order derivative is continuous at the interior nodes. Assume each segment has the form where a n , b n , c n and d n are constants and τ is a free parameter used to maintain consistency of the method. Let us denote Substituting equations (18) into (19) and assigning θ � τh, we can have Solving equation (20) for the coefcients we get From the continuity of frst derivative, we have Ten, from which by simple arithmetic manipulation, we get where λ 1 � (θ − sin (θ))/(θ 2 sin (θ)), λ 2 � 2(sin θ − θ cos θ)/ θ 2 sin θ. Applying L'Hospital's rule as τ ⟶ 0, i.e., θ ⟶ 0 yields a scheme matching that of the ordinary cubic spline in Ω, since Te consistency relation for equation (24) leads us to choose λ 1 and λ 2 in such a way that it satisfes the relation 2λ 1 + λ 2 � 1 [19,20]. Taking the Taylor expansion of U m (x) and its frst derivative about x n only up to second-order gives Similarly, From equation (26), we get Substituting equations (28) into (27) gives and from equation (17) using spline's second derivatives, we have Substituting equations (31)- (33) into (24), we arrive at the following diference scheme: where Terefore, the required scheme developed in equation (34) is termed as a ftted operator fnite diference method obtained through trigonometric-spline and used to solve the problem in equation (1). It can be solved by the matrix inversion method after determining the value of the ftting factor in the next section.

Mathematical Problems in Engineering
where q(x, t m ) ≥ q * > 0. Its characteristic equation on the mth time level is − εr 2 − μαr + q * � 0, whose solutions are given by the relation Assume there are two real solutions r 1 < 0 and r 2 > 0 that describes the boundary layers at x � 0 and x � 1, respectively, based on the following two cases [27]: Equation (1) has two boundary layers which behaves like the reaction-difusion case (μ ≈ 0) with each of width O( � ε √ ) at x � 1 and x � 0. Ten, the complementary function of equation (6) is where C 1 and C 2 are real constant numbers.
In this case, equation (1) has two boundary layers in the neighborhood of x � 0 and x � 1 with diferent width O(ε/μ) and O(μ), respectively. Ten, the complementary function of equation (6) is where C 3 and C 4 are real constant numbers. Te current problem is the generalization of problems with one perturbation parameter which is studied intensively as convection-difusion or reaction-difusion. From the theory of singular perturbations given in [2] and using Taylor's series expansion at layer regions and restriction to their frst terms, we get the asymptotic solution as where u 0 (x, t m ) is the solution of the reduced problem (when ε � 0) of equation (17) that is given by By considering h small enough, the discretized form of equation (42) is Now, use equations (44) into (34) and restrict in each Taylor expansion to the frst terms. Ten, taking limits both sides as h ⟶ 0 and solving for σ, we get σ � μρa 0, t m 2 2λ 1 + λ 2 coth μρa 0, t m 2 , for the left layer, μρa 0, t m 2 2λ 1 + λ 2 coth μρa 1, t m 2 , for the right layer, To this end, using equations (34) and (45), it is observable that Tis shows the coefcient matrix associated to the difference operator L N,M ε,μ is irreducibly diagonally dominant.

Parameter-Uniform Convergence Analysis
So far, we have shown that the continuous solution and its derivatives are bounded and errors due to the introduction of the discrete approximation in the time variable can be estimated (controlled). To realize the stability and consistency of the developed scheme, we further see the error estimate in the spatial variable and the total discrete scheme.

Lemma 7 (Discrete maximum principle). Assume the discrete function
Proof. To follow the proof by contradiction, let there exists a point (ι, m) where ι ∈ 1, 2, . . . , and suppose that Π m ι < 0, then we have ι ≠ 1, N + 1. But by using equation (25), the assumptions a(x, t) ≥ α and b(x, t) ≥ β, and the series representation x coth x � 1 + Tis contradicts the assumption L N,M ε,μ Π m n ≥ 0 on D N x × D M t , and then it completes the proof.
□ From Lemma 7, it follows that the discrete operator L N,M ε,μ satisfes the discrete maximum principle and then the coefcient matrix associated to it is monotone. Moreover, as it is also irreducibly diagonally dominant, then it is an Mmatrix. Tis guarantees for the existence of unique discrete solution. In the next lemma, we discuss the uniform stability of the discrete solution.

Lemma 8. Te solution U m n of the discrete scheme equation (13) satisfes the following bound
where q(x, t) ≥ q * > 0.
Proof. Let Θ � (‖L N,M ε,μ U m n ‖)/q * + max |U m 1 |, |U m N+1 | and defne barrier functions as Te values of these barrier functions on the boundary points are and on the discretized domain is Ten, by applying the discrete maximum principle given in Lemma 7, the proof is immediate. □ As a result, the method is uniformly stable in the maximum norm. To establish a parameter-uniform convergence of the discrete scheme equation (34), let the truncation error be given by

Mathematical Problems in Engineering
(53) Expanding each term by Taylor series method about x n up to third order and simplifying yields (54) After Taylor expansion of each coefcient up to frst term and arithmetic manipulations, we reach at (55) By Lemma 6, (U ″ ) m n is bounded, then the order of the method in the spatial direction is O(h 2 ). (1) at each grid point (x n , t m ) and U(x n , t m ) be its numerical solution obtained by the proposed scheme given in equation (13). Ten, the error estimate for the fully discrete scheme is given by

Theorem . Let u(x n , t m ) be the solution of the problem in equation
Proof. Applying the triangular inequality, the proof is immediate from Lemmas 5 and 6. Terefore, the proposed method is convergent independent of the perturbation parameters and its rate of convergence is O(k + h 2 ), one in the time variable and two in the space variable.
It is well known that the more refned the step sizes, the better accurate results obtained. But this is computationally tough. So, using the Richardson extrapolation which depends on two already calculated approximations with diferent embedded meshes, we can minimize it and enhance the accuracy of our result with less refned mesh.
x , respectively. Te rate of convergence of the proposed method is shown to be O(k + h 2 ) by Teorem 9. On diferent meshes, the approximate U can be given as

Mathematical Problems in Engineering
Te results obtained through the Richardson extrapolation denoted and defned by better approximate the exact solution u than U. From equations (60) and (61), we get the error estimate Furthermore, we have the following error estimate.

Theorem 10.
Let u(x n , t m ) be the solution of equation (1) and U 1Rich (x n , t m ) be the numerical solution obtained through the Richardson extrapolation as defned in equation (18). Ten, the error after extrapolation satisfes Proof. Te proof is given in [28].

Numerical Examples and Results
Under this section, we apply the current method to the following numerical examples with λ 1 � 1/284, λ 2 � 282/284 and verify the theoretical results experimentally.
and after the Richardson extrapolation as where M and N are the numbers of mesh points in t and x directions with k and h step sizes, respectively. U M N is the approximate solution obtained using M and N number of meshes and U 2M 2N is the approximate solution obtained using a double number of meshes 2M and 2N with half step sizes. U 1Rich and U 2Rich are given in equations (60) and (61), respectively. But for Example 2, since we have an exact solution, the absolute maximum errors before and after the Richardson extrapolation are defned, respectively, as where (Uext) M N is the exact solution at the grid points. Te rate of convergence for both examples before and after extrapolation is, respectively, obtained by .

Discussion and Conclusion
We have described a trigonometric-spline method for solving time-dependent singularly perturbed convectiondifusion problems using the ftted operator technique. Applying the backward Euler method, we have frst transformed the continuous time-dependent problem into an ordinary diferential equation (ODE) of space variable x. And then, in turn, using a trigonometric-spline discrete scheme in the space variable, we have obtained a fully discrete scheme. In our numerical experimentation, the calculated maximum point-wise errors before and after extrapolation are presented in Table 1, and the corresponding rates of convergence are given in Table 2. Te overall current results for Examples 1 and 2 are shown in Tables 3 and 4, respectively.  Te outcomes displayed in Tables 1, 3, and 4 clearly show that, as the perturbation parameter goes to zero, the absolute maximum point-wise error remains constant. Tis indicates  the developed scheme given in. Equation (34) and the extrapolated technique are insensitive to the perturbation parameters and gives better results than the results in the literature. Te rate of convergence of the current method is increased from one to two as a consequence of the Richardson extrapolation supporting the theoretically asserted hypothesis. Tis is illustrated in Tables 2 and 5. As the number of mesh points increases, the absolute maximum point-wise errors decrease, and the rate of convergence increases. Tese show us that there is no oscillation or unexpected change in the solution, that is, the efect of the perturbation parameter is controlled by the ftted numerical scheme obtained. From the graphs presented in Figures 1  and 2, one can observe that the problem given in Example 1 has a solution which exhibits a parabolic boundary layer. Besides this, Figure 2(a) indicates that the two boundary layers have equal width, as discussed in Case 1, whereas Figure 2(b) shows the boundary layers have diferent width ratifying in Case 2. From Figure 3, we can observe the profle of the numerical and exact solution of Example 2. Furthermore, since μ � 1 in Example 2, the boundary layer appears at the right end of the domain depending on the sign of the convection coefcient a(x, t) which is illustrated in Figure 4. We have plotted the maximum point-wise errors in the log-log scale for the sake of revealing the numerical order of convergence before and after the Richardson extrapolation technique in Figure 5, and again it endorses the theoretical order of convergence and the error being constant. Generally, the numerical results obtained by our method confrm the theoretical results. In our future works, we consider problems of the type treated in [32,33].

Data Availability
No data were used to support the fndings of this study.

Conflicts of Interest
Te authors declare that they have no conficts of interest.