Alternating group explicit parallel algorithms for the solution of one-space dimensional non-linear singular parabolic equations using an O(k² + h⁴) difference method

Abstract

In this paper, we report on the application of the alternating group explicit (AGE) and Newton-AGE iterative methods for the numerical solution of one-space dimensional non-linear singular parabolic equations subject to appropriate initial and Dirichlet boundary conditions. The proposed methods are applicable to problems both in cavtesian and in polar coordinates and are suitable for the use on parallel computers. In all cases, we use a two-level implicit finite difference method of O(k ² + h ⁴) accuracy and three spatial grid points, where h > 0 and k > 0 are step lengths in the space- and time-directions, respectively. The stability analysis of the linear difference scheme and the error analysis of the AGE iterative method are briefly discussed. Numerical results are provided to support the proposed theory.

Keywords:

• Non-linear parabolic equation
• Polar coordinates
• Singular equation
• Convection–diffusion equations
• Burgers' equation
• AGE method
• Newton-AGE method
• Reynolds number

Acknowledgement

This research was supported by ‘The Royal Society’ of London, UK under INSA Bilateral Indo-UK Joint Exchange Programme.