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(@DtA)v and @f(@DtA)v, @f(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.

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