A massively parallel exponential integrator for advection-diffusion models

This work considers the Real Leja Points Method (ReLPM), [M. Caliari, M. Vianello, L. Bergamaschi, Interpolating discrete advection-diffusion propagators at spectral Leja sequences, J. Comput. Appl. Math. 172 (2004) 79–99], for the exponential integration of large-scale sparse systems of ODEs, generated by Finite Element or Finite Difference discretizations of 3-D advection-diffusion models. We present an efficient parallel implementation of ReLPM for polynomial interpolation of the matrix exponential propagators exp(ΔtA)v and φ(ΔtA)v, φ(z)=(exp(z)−1)/zφ(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. The numerical results presented on a number of very large test cases gives experimental evidence that ReLPM is a reliable and efficient tool for the simulation of complex hydrodynamic processes on parallel architectures.