A nonstandard finite difference scheme for a PDE modeling combustion with nonlinear advection and diffusion

We construct a discrete model for a particular PDE having nonlinear advection, diffusion, and reaction. The use of the author’s nonstandard finite difference methods form the basis for this construction. We demonstrate that the solutions to the scheme satisfy positivity and boundedness conditions. Further, an explicit functional relation between the time and space step-sizes is obtained. A discussion is given as to how these results can be generalized to a broad class of PDE’s having these same structural properties.