Estimation of the linear transient growth of perturbations of cellular flames

Abstract

In this work we estimate rates of the linear transient growth of the perturbations of cellular flames governed by the Sivashinsky equation. The possibility and significance of such a growth was indicated ear- lier in both computational and analytical investigations. Numerical investigation of the norm of the resolvent of the linear operator asso- ciated with the Sivashinsky equation linearized in a neighbourhood of the steady coalescent pole solution was undertaken. The results are presented in the form of the pseudospectra and the lower bound of possible transient amplification. This amplification is strong enough to make the round-off errors visible in the numerical simulations in the form of small cusps appearing on the flame surface randomly in time. Performance of available numerical approaches was compared to each other and the results are checked versus directly calculated norms of the evolution operator.