Interpolating discrete advection-diffusion propagators at Leja sequences

  • Authors:

  • M. Caliari;M. Vianello;L. Bergamaschi

  • Affiliations:

  • Dip.to di Informatica, Università di Verona, Italy;Dip.to di Matematica Pura ed Applicata, Università di Padova, Italy;Dip.to di Metodi e Modelli Matematici per le Scienze Applicate, Università di Padova, via Belzoni 7, Padova 35131, Italy

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2004

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Abstract

We propose and analyze the ReLPM (Real Leja Points Method) for evaluating the propagator ϕ(ΔtB)v via matrix interpolation polynomials at spectral Leja sequences. Here B is the large, sparse, nonsymmetric matrix arising from stable 2D or 3D finite-difference discretization of linear advection-diffusion equations, and ϕ(z) is the entire function ϕ(z)= (ez-1)/z. The corresponding stiff differential system y(t)=By(t)+g,y(0)=Y0, is solved by the exact time marching scheme yi+1=yi+Δtiϕ(ΔtiB)(Byi+g), i=0, 1,..., where the time-step is controlled simply via the variation percentage of the solution, and can be large. Numerical tests show substantial speed-ups (up to one order of magnitude) with respect to a classical variable step-size Crank-Nicolson solver.