Numerical solutions of two-dimensional unsteady convection–diffusion problems using modified bi-cubic B-spline finite elements

ABSTRACT

This research study presents a numerical scheme to compute approximate solutions of two dimensional unsteady convection–diffusion equation. We used collocation of modified bi-cubic B-spline functions for dependent variable u and for its derivatives w.r.t. space variables x and y. Strong stability preserving Runge–Kutta method (SSP-RK54) has been used for solving system of first-order ordinary differential equations obtained from the collocation form of the partial differential equation. We did not linearize the nonlinear terms by using any transformation or linearization method. The number of computations and the required storage space is very less for the proposed scheme. Four examples have been taken as described in available literature to demonstrate the effect and utility of the proposed scheme. These numerical experiments show that the obtained results are not only quite satisfactory w.r.t. the exact solutions but also competent with the solutions available in earlier research studies. Computational complexity of the proposed scheme has been discussed and shown that it is $O(p\log (p\left)\right)$, where p is total number of nodes. The proposed scheme is easy to implement and the size of required computational work is very small. Moreover, using this scheme, we can compute approximate solutions not only at the mesh points but at any other point of the solution domain as well.

KEYWORDS:

• two dimensional unsteady convection–diffusion equation
• two dimensional transport equation
• Hockney method
• collocation method
• Thomas algorithm
• modified bi-cubic B-splines

2010 AMS Subject Classifications:

• 65D07
• 65N30
• 65N35
• 74S05
• 97N40

Acknowledgments

Both the authors express their sincere regards and gratitude to the referees for their valuable comments, through reviews and suggestions to bring proposed study in present form.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The author Amit Tripathi sincerely expresses his deep sense of gratitude to the Ministry of Human Resources and Development, India, for their financial assistance in the form of JRF and SRF.