Interpolating discrete advection–diffusion propagators at Leja sequences

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 ẏ(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.