A novel three-level time-split MacCormack scheme for two-dimensional evolutionary linear convection-diffusion-reaction equation with source term

ABSTRACT

This study presents a three-level explicit time-split MacCormack method to compute approximate solutions of two-dimensional time-dependent linear convection-diffusion-reaction equations with source term. The difference operators split the two-dimensional problem into two pieces so that each subproblem is easily solvable using the original MacCormack approach. Second order accuracy in time and fourth-order convergence in space are achieved by the application of the Taylor series expansion. The proposed algorithm minimizes the computational time, computer memory requirement and is easy to implement. Under a suitable time-step restriction, both stability and error estimates of the numerical scheme are deeply analysed in $L^{\mathrm{\infty }}(0,T;L^2)$-norm. Numerical evidences which confirm the theoretical analysis are considered and discussed.

Keywords:

• 2D unsteady linear convection-diffusion-reaction equation
• source term
• one-dimensional operators (splitting)
• explicit MacCormack scheme
• a three-level explicit time-split MacCormack method
• stability and convergence rate

AMS Subject Classification (MSC):

• 65M10
• 65M05

Acknowledgements

The author appreciates the comments of the anonymous referees which helped to improve the numerical evidences presented in this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).