A massively parallel exponential integrator for advection-diffusion models

This work considers the Real Leja Points Method (ReLPM) 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. A scalability analysis of the most important computational kernel inside the code, the parallel sparse matrix\u2013vector 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\u2013vector 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

The ReLPM Exponential Integrator for FE Discretizations of Advection-Diffusion Equations

We implement an exponential integrator for large and sparse systems of ODEs, generated by FE (Finite Element) discretization with mass-lumping of advection-diffusion equations. The relevant exponential-like matrix function is approximated by polynomial interpolation, at a sequence of real Leja points related to the spectrum of the FE matrix (ReLPM, Real Leja Points Method). Application to 2D and 3D advection-dispersion models shows speed-ups of one order of magnitude with respect to a classical variable step-size Crank-Nicolson solver

Accurate evaluation of divided differences for polynomial interpolation of exponential propagators

In this paper, we propose an approach to the computation of more accurate divided differences for the interpolation in the Newton form of the matrix exponential propagator phi(hA) v, phi(z) = (e(z)-1)/z. In this way, it is possible to approximate.( hA) v with larger time step size h than with traditionally computed divided differences, as confirmed by numerical examples. The technique can be also extended to "higher" order phi(k) functions, k >= 0