Implementation of Extended Systems Using Symbolic Algebra

Following the pioneering work of Keller and others in the 1970s and ’80s, numerical techniques for solving non-linear systems of equations that exhibit bifurcations have been developed to the point where they can potentially be applied to a wide range of problems arising in continuum mechanics. The central idea is to augment the discretised governing equations with one or more conditions so that the ‘extended system’ characterises a particular bifurcation point. By computing paths of singular points the behaviour of the system under investigation can be mapped out in a comprehensive fashion. Of considerable practical difficulty when implementing these methods is that they require the evaluation of derivatives of the discretised equations with respect to both the independent variables and the parameters. The higher the codimension of the singularity being sought, the higher the order of the derivatives required. Evaluating these derivatives is both tedious and error prone. An efficient method for computing the necessary derivatives for discretisations based on the Galerkin finiteelement method will be presented that takes advantage of a symbolic algebra package. Our method makes it possible to deal with complicated non-linearities in a very straightforward manner. We demonstrate the complexity of systems that may be addressed by considering Marangoni convection in a two-dimensional domain with a deformable free surface.