Implementation of the Lyapunov Functional in Dif Ference Schemes for the Swift Hohenberg Equation

Abstract Using the operator splitting method Christov Pontes developed a second order in time implicit di erence scheme for solving the Swift Hohenberg equation S H which describes pattern formation in Rayleigh Benard cells For each time step the scheme involves internal iterations which improve the stability and increase the accuracy with which the Lyapunov functional for S H is approximated Di erent cases of pattern formation were treated and it was shown that the new scheme reaches the stationary pattern several times faster than the previously used rst order in time schemes In this work we review the main steps concerning the derivation of the second order in time scheme and prove that the scheme strictly satis es a discrete approximation of the Lyapunov functional Results of numerical simulations conducted in four large boxes with two levels of forcings and two di erent random initial conditions are presented illustrating that this scheme preserves the non increasing time dependent behaviour of the functional The rate of change of the functional is generaly slow save the pecipituous downfalls during the time intervals in which the pattern changes qualitatively