Accuracy and Stability of the Hyperbolic Model Time Integration Scheme Revisited

The hyperbolic model (HM) time integration scheme tackles parabolic problems by adding a small artificial second order time derivative term. Described by Samarskii in his 1971 book, the scheme reappeared as the generalized Du Fort–Frankel scheme in a 1976 paper by Gottlieb and Gustafsson. In this note, we revisit accuracy and stability properties of the scheme. In particular, we show that the stability condition, formulated by Samarskii based on operator inequalities, coincides with the requirement that the eigenvalues of the amplification matrix (the stability function operator) are smaller than one in absolute value. However, under this condition, the norm of this matrix may exceed one and this, as recently pointed out by Corem and Ditkowski (2012), may corrupt convergence of the scheme. Hence, we also discuss whether this eventual stability lack can be detected and mitigated in practice.