Exponential time differencing methods and asymptotic behaviour of solutions of problems in ground water flow
Abstract
We start this thesis with a numerical study of the convergence of the exponential time
differencing (ETD) schemes and the semi-implicit Euler method for the Allen-Cahn
equation and a reaction-convection-diffusion equation and also compare the accuracy
and efficiency of these methods.
Next, we solve the nonlinear convection-diffusion (green roof) model numerically using
the ETD method and central difference approximation. This numerical solution is
investigated for three different initial values for the saturation.
Finally, we study travelling wave solutions and self-similar solutions for the green
roof, in particular, for the two limiting cases of being close to a saturated region and
a dry region. Travelling waves, in the form of fronts, are found for most realistic
limiting values of saturation; travelling waves are also investigated for some limiting
versions of the model. Self-similar solutions, valid for high or for low saturations, are
additionally investigated.