Malaysian Journal of Mathematical Sciences, December 2025, Vol. 19, No. 4
Numerical Solutions of The One-dimensional Burgers' Equation by A Preconditioned Successive Over Relaxation Method
Ghadamyari, S. and Mojarrab, M.
Corresponding Email: ma_mojarrab@math.usb.ac.ir
Received date: 7 January 2025
Accepted date: 27 May 2025
Abstract:
The Successive Over Relaxation (SOR) method is a well-known iterative method to solve the linear system $Ax=b$. One effective strategy for increasing the speed of convergence or even getting out of a possible recession is to use an effective preconditioner. With this perspective, in this paper, using a preconditioned version of the SOR method, we extract the numerical solutions of the one-dimensional Burgers' equation. At first, we discretize Burgers' equation by using some central and forward finite difference formulas. Then, we eliminate the non-linear term produced in the obtained differential equations by using an average formula called non-local arithmetic discretization scheme. Next, we examine the error analysis and conditional stability of the method. Finally, we modify the resulting system of linear equations with its preconditioned version. Theoretical and numerical results show that the present preconditioned method has increased the convergence rate significantly compared to the standard method and the resulting methods.
Keywords: Burgers' equation; finite difference scheme; preconditioner; linear system; SOR; semi approximate approach
