Interpolating discrete advection–diffusion propagators at Leja sequences

Abstract

We propose and analyze the ReLPM (Real Leja Points Method) for evaluating the propagator ϕ(Δ tB) 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)=(e z −1)/ z. The corresponding stiff differential system y ̇ (t)=B y(t)+ g, y(0)= y 0 , is solved by the exact time marching scheme y i+1 = y i +Δ t i ϕ(Δ t i B)( B y i + 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.