Novel Cubic Trigonometric B-Spline Approach Based on the Hermite Formula for Solving the Convection-Diffusion Equation

This paper introduces a cubic trigonometric B-spline method (CuTBM) based on the Hermite formula for numerically handling the convection-diﬀusion equation (CDE). The method utilizes a merger of the CuTBM and the Hermite formula for the approximation of a space derivative, while the time derivative is discretized using a ﬁnite diﬀerence scheme. This combination has greatly enhanced the accuracy of the scheme. A stability analysis of the scheme is also presented to conﬁrm that the errors do not magnify. The main advantage of the scheme is that the approximate solution is obtained as a smooth piecewise continuous function empowering us to approximate a solution at any location in the domain of interest with high accuracy. Numerical tests are performed, and the outcomes are compared with the ones presented previously to show the superiority of the presented scheme.


Introduction
e CDE describes physical phenomena in which particles, energy, and other physical quantities are transferred within a physical system due to diffusion and convection. e CDE is given as where α is the coefficient of viscosity and β is the phase velocity, respectively, and both are considered positive. Equation (1) is subject to the following initial condition: v(ξ, 0) � ϕ(ξ), a ≤ ξ ≤ d, (2) and the boundary conditions v(a, t) � g 0 (t), v(b, t) � g 1 (t), Here, ϕ, g 0 , and g 1 are known functions of sufficient smoothness.
In the literature, various numerical techniques have been developed for the one-dimensional CDE with specified initial and boundary conditions such as finite differences, finite elements, spectral methods, method of lines, and many more. Mohebbi and Dehghan [1] presented a high-order compact solution of the one-dimensional heat and advection-diffusion equation. Salkuyeh [2] used finite difference approximation to solve the CDE. Karahan [3,4] worked on unconditional stable explicit and implicit finite difference techniques for the advection-diffusion equation using spreadsheets. Restrictive Taylor approximation was used by Ismail et al. [5] to solve the CDE. Cao et al. [6] developed a fourth-order compact finite difference scheme for solving the CDE. e generalized trapezoidal formula was used by Chawla and Al-Zanaidi [7] to solve the CDE. Dehghan [8] used weighted finite difference techniques for the one-dimensional advection-diffusion equation. Furthermore, Dehghan [9] developed a technique for the numerical solution of the three-dimensional advection-diffusion equation. A second-order space and time nodal method for the CDE was conducted by Rizwan [10]. Kara and Zhang [11] introduced an ADI method for an unsteady CDE. Feng and Tian [12] presented an alternating group explicit method for the CDE. Mittal and Jain [13] redefined the cubic B-spline collocation method for solving the CDE. Kadalbajoo and Arora [14] presented the Taylor-Galerkin B-spline finite element method for the one-dimensional advection-diffusion equation. Sari et al. [15] used a high-order finite difference scheme for solving the advection-diffusion equation. Tsai et al. [16] used a characteristics method with cubic interpolation for the advection-diffusion equation. Daig et al. [17] presented a least-squares finite element method for the advection-diffusion equation. Chawla et al. [18] presented extended one-step time-integration schemes for the CDE. Ding and Zhang [19] presented a highly accurate difference scheme for CDE. Nazir et al. [20] obtained numerical solutions of the CDE CuTBS approach. Aminikhah and Alvi [21] solved the CDE using cubic B-spline quasiinterpolation. A new Rabotnov fractional-exponential function based fractional derivative for the diffusion equation under external force was presented by Kumar et al. [22]. A modified analytical approach with existence and uniqueness was presented by Kumar et al. [23] for fractional Cauchy reaction-diffusion equations. A numerical study of modeling and analysis of fractal and fractal-fractional differential equations was initiated in [24,25].
Motivated by the boom of the spline approach in finding the numerical solutions of the partial differential equations, we have utilized a blend of the Hermite formula and the cubic B-spline for the discretization of the space derivative.
is merger has significantly augmented the accuracy of the scheme. Another favorable advantage is that approximate solutions come up as a smooth piecewise continuous function permitting one to obtain approximation at any desired location in the domain. e approach used by von Neumann is utilized to confirm that the presented scheme is unconditionally stable. e scheme is applied to various test problems, and the outcomes are contrasted with those reported in [19][20][21]. e remaining portion of the paper is organized in the following sequence. Section 2 presents the proposed scheme that is derived out for the numerical treatment of the CDE. e stability analysis of the scheme is discussed in Section 3. e comparison of the numerical results is provided in Section 4. e outcomes of this study are presented in Section 5.

Derivation of the Scheme
For positive integers M and N, let k � T/N and h � b − a/M be the time and the space step sizes, respectively. e time domain is discretized as t n � nk, e procedure for finding the approximate solution of (1) involves determination of the approximate solution V(ξ, t) to the exact solution v(ξ, t) as follows [26]: where σ j (t) are time-dependent quantities to be determined and TB 4 j (ξ) are cubic trigonometric basis functions given in [26] as follows:

Mathematical Problems in Engineering
By applying the local support property of TB 4 j (ξ), it is observed that only TB 4 j− 3 (ξ), TB 4 j− 2 (ξ), and TB 4 j− 1 (ξ) are survived. Consequently, the approximation v n j at (ξ j , Now, v n j and its necessary derivatives are approximated by applying the collocation conditions on B 3 j (ξ). e obtained approximations are given by where Consider the Hermite formula at the knot (ξ j , t n ) [27] given by v ξξ Substituting (8) in (10), we obtain v ξξ where Note that (11) provides new approximation of the second derivative. Now, applying the θ-weighted scheme to (1), we obtain v t n j � θh n+1 For the Crank-Nicolson approach, we choose θ � 0.5 so that (14) Inserting (8) in (15) and replacing j with j − 1, we obtain Inserting (8) and (11) in (15), we obtain Mathematical Problems in Engineering (16) and (17) together produce an inconsistent system of (M + 1) equations in (M + 3) unknowns. To obtain a consistent system, we need two additional equations which can be obtained using the given boundary conditions. Consequently, a consistent system of dimension (M + 3) × (M + 3) is obtained which can be solved using any Gaussian elimination-based numerical algorithm.

Initial State.
e initial condition and the derivatives of initial condition are used to find initial vector σ 0 as follows: System (18) produces an (M + 3) × (M + 3) matrix system of the following form: where

Stability Analysis
e von Neumann stability technique is applied in this section to explore the stability of the given scheme. Consider the growth of error in a single Fourier mode, Ω n j � δ n e iηhj , where η is the mode number, h is the step size, and i � �� � − 1 √ . Inserting the Fourier mode into equation (15) yields where Dividing equation (21) by δ n e iη(j− 2)h and rearranging the equation, we obtain Using cos(ηh) � (e iηh + e − iηh )/2 and sin(ηh) � (e iηh − e − iηh )/ 2i in equation (23) and simplifying, we obtain where Note that η ∈ [− π, π]. Without loss of generality, we can assume that η � 0 so that equation (24) takes the following form: which proves that the present computational scheme is unconditionally stable.

Numerical Experiments and Discussion
In this section, some numerical calculations are performed to test the accuracy of the offered scheme.
e numerical results are obtained by utilizing the presented scheme. In Table 1, the absolute errors are compared with those obtained in [19] at various time stages. Figure 1 illustrates the comparison between the exact and numerical solutions at various time stages. Figure 2 shows the 2D and 3D error profiles at T � 1. A 3D comparison between the exact and numerical solutions is presented to exhibit the exactness of the scheme in Figure 3. e approximate solution when t � 1, k � 0.01, and h � 0.05 is given by with the initial condition v(ξ, 0) � exp(0.22ξ)sin(πξ) and the boundary conditions e analytic solution of the given problem is v(ξ, t) � exp(0.22ξ − (0.0242 + 0.5π 2 )t)sin(πξ). By utilizing the proposed scheme, the numerical results are acquired. An excellent comparison between absolute errors computed by our scheme and the scheme in [19] is presented in Table 2. A close comparison between the exact and numerical solutions at different time stages is depicted in Figure 4. Figure 5 plots 2D and 3D absolute errors at T � 1. Figure 6 deals with the 3D comparison that occurs between the exact and numerical solutions. e approximate solution when t � 1, k � 0.01, and h � 0.05 is given as Mathematical Problems in Engineering with the initial condition v(ξ, 0) � exp(0.25ξ)sin(πξ) and the boundary conditions e analytic solution of the given problem is v(ξ, t) � exp(0.25ξ − (0.012ξ + 0.2π 2 )t)sin(πξ). e numerical outcomes are obtained utilizing the proposed scheme. An excellent comparison between absolute errors computed by our scheme and the scheme in [19] is discussed in Table 3. Figure 7 deals with the behavior of exact and approximate solutions at various time stages. Figure 8 plots 2D and 3D absolute errors at T � 1. In Figure 9, a tremendous 3D contrast between the exact and numerical solutions is shown. e approximate solution when t � 1, k � 0.01, and h � 0.05 is given as e analytic solution is v(ξ, t) � ������� � 20/20 + t √ exp (− (ξ − 2 − 0.8t) 2 /0.4(20 + t)). In Table 4, the comparative analysis of absolute errors with those in [20] is provided. Figure 10 illustrates the behavior of numerical solutions at various stages of time. Figure 11 depicts the 2D and 3D graphs of absolute errors. Figure 12 shows the rattling accuracy that exists between the exact and numerical solutions.

Concluding Remarks
In this study, a cubic trigonometric B-spline collocation method based on the Hermite formula is developed for the convection-diffusion equation. e smooth piecewise cubic B-spline has been used to approximate derivatives in space, whereas a standard finite difference has been used to discretize the time derivative. A combination of the Hermite formula and the cubic trigonometric B-splines for approximating the space derivative has considerably augmented the accuracy of the scheme. e solution comes up as a smooth piecewise continuous function so that one can find an approximate solution at any wanted location in the domain of interest. A special attention is devoted to the stability analysis of the scheme to confirm that the errors do not amplify. e numerical results are contrasted with some current numerical techniques. It is inferred that the presented scheme is more precise and provides better accuracy. It is also worthwhile to mention that the offered scheme is applicable to a variety of problems of applied nature in science and engineering.

Data Availability
e experimental data used to support the findings of this study are available within this paper.