Abstract
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.
Highlights
Introduction e convection-diffusion equation (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 zv zv z2v zt + α zξ β zξ2, a ≤ ξ ≤ b, t > 0, (1)
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
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
Summary
Is paper introduces a cubic trigonometric B-spline method (CuTBM) based on the Hermite formula for numerically handling the convection-diffusion equation (CDE). Dehghan [8] used weighted finite difference techniques for the one-dimensional advection-diffusion equation. Dehghan [9] developed a technique for the numerical solution of the three-dimensional advection-diffusion equation. 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. 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. 2 17694.2 + 9112.59 sin(ξ)), ξ ∈ 17, 9 , 20 10 9 19 ξ ∈ , , 10 20 ξ ∈ 19, 1. 20
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