A difference expansion technique to analyse the advection/dispersion/sorption equation for phosphorous

A finite difference technique for numerical analysis of the partial differential equation for advection and dispersion with sorption is presented in detail. The sorption term is non-linear and it is difficult to simulate the breakthrough curves when the rate of change of concentration is large. A general technique is used which expands the derivatives in terms of increasing orders of finite differences. This has become possible with the advent of symbolic manipulation. It is possible to expand to any order and, importantly, separate out the terms for each order. Details are presented, together with methods for exploring the convergence and stability as well as adjusting the time and spatial differencing to lower or minimise· the error

Isogroups of differential equations using algebraic computing

We describe the Liesymm package, implemented in the symbolic computing system MAPLE, for obtaining the determining equations of the isogroup of a system of partial differential equations. Liesymm fully automates the Harrison-Estabrook procedure which uses Cartan's formulation in terms of differential forms. It also includes a number of routines which assist the user in integrating the determining equations. Liesymm has been implemented in such a way as to provide the user with a number of useful interactive tools for working with differential forms. In addition, it can also generate a “suitable” initial set of differential forms directly from the given partial differential equations and as such can be used at a level that obviates any need for familiarity with forms

On the non-uniqueness of solutions in the modelling of steady wind flows

The two-dimensional, steady, turbulent Navier-Stokes Equations are explored for the case of a logarithmic profile. A general, analytical solution technique is presented, using potential functions; it contains two arbitrary functions. Example solutions are derived with the symbolic manipulator Maple; they show various shear stress profiles derived from a single form for the velocity profile. These stress variations occur within an ‘equilibrium profile’ and are affected by boundary conditions and upstream effects. Shear stress and dynamic roughness values derived from velocity profiles should be used with caution