Differential quadrature solution of nonlinear reaction–diffusion equation with relaxation-type time integration

Abstract

This paper presents the combined application of differential quadrature method (DQM) and finite-difference method (FDM) with a relaxation parameter to nonlinear reaction–diffusion equation in one and two dimensions. The polynomial-based DQM is employed to discretize the spatial partial derivatives by using Gauss–Chebyshev–Lobatto points. The resulting system of ordinary differential equations is solved, discretizating the time derivative by an explicit FDM. A relaxation parameter is used to position the solution from the two time levels, aiming to increase the convergence rate with a moderate time step to the steady state and also to obtain stable solution. Numerical experiments are given to illustrate the scheme for one-dimensional Fisher-type problems and also for two-dimensional reaction–diffusion boundary-value problems. The agreement of the solution with the exact solution is very good in two-dimensional case while some other numerical schemes may result in some unwanted oscillations in the computed solution. Optimal value of the relaxation parameter is obtained numerically to prevent the use of very small time steps and to achieve stable solutions. The DQM with a relaxation-type finite-difference time integration scheme exhibits superior accuracy at large time values for the problems tending towards a steady state.

Keywords:

• DQM
• nonlinear reaction–diffusion equation

2000 AMS Subject Classification :

• 65M06
• 65M99