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.