On matrix-free pseudo-arclength continuation methods applied to a nonlocal partial differential equation in 1+1D with pseudo-spectral time-stepping

Abstract

In this thesis we examine a recent animal aggregation model which describes the evolution of two populations of animals moving on a 1-dimensional spatial domain differing only by the direction they travel. The equations describing the evolution of the populations is a hyperbolic, nonlocal partial di erential equation with periodic boundary conditions [5]. We apply pseudo-spectral methods to numerically integrate initial states of the populations given as small perturbations from a homogeneous steady state from which bifurcations and dynamics have been studied from a linear and weakly nonlinear analysis perspective [10, 11]. The existence of transcendental nonlinearities within the equations makes this application of pseudo-spectral methods interestingly nontrivial and simulations do display dynamics similar to those observed in Eftimie et al. [5]. Finally we apply matrix-free, pseudo-arclength continuation methods with consideration given to symmetries within the model in an attempt to trace curves from known states to more dynamically exotic regions of parameter space. The ow operator is used to condition the Newton systems arising from the continuation and to allow for a matrix-free continuation algorithm [13]. However, unforeseen degeneracies arise within the Newton system which necessitates further research in order to build a robust continuation software.