Efficient approximation of the exponential operator for discrete 2D advection-diffusion problems

In this paper we compare Krylov subspace methods with Faber series expansion for approximating the matrix exponential operator on large, sparse, non-symmetric matrices. We consider in particular the case of Chebyshev series, corresponding to an initial estimate of the spectrum of the matrix by a suitable ellipse. Experimental results upon matrices with large size, arising from space discretization of 2D advection-diffusion problems, demonstrate that the Chebyshev method can be an effective alternative to Krylov techniques.