Efficient massively parallel implementation of the ReLPM exponential integrator for advection-diffusion models∗†

This work considers the Real Leja Points Method (ReLPM), [J. Comput. Appl. Math., 172 (2004), pp. 79–99], for the exponential integration of large-scale sparse systems of ODEs, generated by Finite Element or Finite Difference discretizations of 3-D advectiondiffusion models. We present an efficient parallel implementation of ReLPM for polynomial interpolation of the matrix exponential propagators exp (∆tA) v and φ(∆tA) v, φ(z) = (exp (z)−1)/z. A scalability analysis of the most important computational kernel inside the code, the parallel sparse matrix-vector product has been performed as well as an experimental study of the communication overhead. As a result of this study an optimized parallel sparse matrix-vector product routine has been implemented. The resulting code shows good scaling behavior even when using more than one thousand processors.