A numerical method for long time solutions of integro-differential systems in multiphase flow

A numerical method to solve a class of intego-differential equations asociated with multiphase flow problems is described. The system has at least two time scales, a normal time and a long time. As an initial value problem, the solution is aperiodic in the normal time and approaches a periodic oscillation over a long time. The numerical method is designed to give accurate calculations over the long time. The method is efficient, saving storage space and computational time. Sample numerical calculations and numerical convergence checks are presented to demonstrate the efficiency and the accuracy of the method. The system considered describes the forced nonlinear oscillations of a gas bubble in a liquid. We present a systematic numerical study of the phenomenon of the slow growth of the mean bubble radius to a steady value over a long time. The correlation between the phenomena of a sudden burst of large amplitude and resonance in the forced oscillations (at subcritical frequencies) is also studied.