The effect of quadrature on the dynamics of a discretized nonlinear integro-differential equation

The long-term dynamics of a discretized, nonlinear, integro-differential eq uation with convolution kernel are studied. For a constant time-step algori thm the existence and stability of fixed and periodic points are investigat ed. A systematic treatment is given, which quantifies the effect of varying the quadrature rule and integrating the kernel exactly or approximately. S pecial attention is paid to spurious behaviour that occurs below, or around , the "natural" time-step that corresponds to the linear stability limit fo r the correct fixed point. It is shown that spurious solutions exist, and c an be computed, within this linear stability range. In addition to fixed po ints and period two solutions, analysis is performed for a class of period three orbits that are observed to be relevant to the long-term dynamics. Fi nally, an adaptive algorithm, based on local error control, is studied and a simple model describing its long-term behaviour is developed, (C) 2000 IM ACS. Published by Elsevier Science B.V. All rights reserved.