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This paper deals with the two classes of non-linear advection–diffusion–reaction (ADR) equations subject to certain initial and boundary conditions. We derive exact finite difference (FD) schemes with the help of solitary wave solutions of ADR equations. Furthermore, we construct non-standard FD schemes for both the equations. The positivity and boundedness properties of the equations are satisfied by the proposed non-standard FD schemes. We also prove that these schemes are stable under certain conditions. Moreover, they are shown to be first order consistent in both time and space. We compute solutions of the ADR equations by applying proposed non-standard FD schemes under given initial and boundary conditions. The obtained results demonstrate that the proposed methods are simple and powerful tool in finding the numerical solutions of ADR equations. The schemes give good accuracy even for few spatial divisions.

On the convergence of NSFD schemes for a new class of advection–diffusion–reaction equations