Numerical efficiency of some exponential methods for an advection–diffusion equation

In this paper, we investigate several modified exponential finite-difference methods to approximate the solution of the one-dimensional viscous Burgers' equation. Burgers' equation admits solutions that are positive and bounded under appropriate conditions. Motivated by these facts, we propose nonsingular exponential methods that are capable of preserving some structural properties of the solutions of Burgers' equation. The fact that some of the techniques preserve structural properties of the solutions is thoroughly established in this work. Rigorous analyses of consistency, stability and numerical convergence of these schemes are presented for the first time in the literature, together with estimates of the numerical solutions. The methods are computationally improved for efficiency using the Pade approximation technique. As a result, the computational cost is substantially reduced in this way. Comparisons of the numerical approximations against the exact solutions of some initial-boundary-value...