Stability, convergence and order of the extrapolations of the residual smoothing scheme in energy norm

Ribot, Magali; Schatzman, Michelle

Confluentes Mathematici, Tome 3 (2011) no. 3, pp. 495-521.

The Residual Smoothing Scheme is a numerical method which consists in preconditioning at each time step the method of lines. In this paper, RSS is defined and analyzed in an abstract linear parabolic case, i.e. for an abstract ordinary differential equation of the form

$du/dt+Au=0$,

with A a self-adjoint non negative operator, and it can be written

$(U^{n+1}-U^n)/\mathrm{\Delta }t+\tau B(U^{n+1}-U^n)+A{U}^n=0$,

where B is a preconditioner of A.

We show that RSS is stable, convergent and of order one in energy norm. We also prove that its kth Richardson's extrapolation is stable and of order k.

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