We propose and analyze the ReLPM (Real Leja Points Method) for evaluating the propagator f(hB)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 f(z) is the entire function f(z) = (exp(z) − 1)/z. The corresponding stiff differential system y'(t) = By(t) + g; y(0) = y0 , is solved by the exact time marching scheme y_{i+1} = y_i +f(h_i B)(By_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.
Interpolating discrete advection-diffusion propagators at Leja sequences
BERGAMASCHI, LUCA;VIANELLO, MARCO
2004
Abstract
We propose and analyze the ReLPM (Real Leja Points Method) for evaluating the propagator f(hB)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 f(z) is the entire function f(z) = (exp(z) − 1)/z. The corresponding stiff differential system y'(t) = By(t) + g; y(0) = y0 , is solved by the exact time marching scheme y_{i+1} = y_i +f(h_i B)(By_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.Pubblicazioni consigliate
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