Adaptive spectral least-squares techniques for reaction-diffusion equations

Abstract

In this thesis we are proposing a new numerical method for solving systems of reactiondiffusion equations. In particular we are focussing on Turing mechanisms, which are a special form of reaction-diffusion equations introduced by Alan Turing in 1952. The speciality of these equations is that diffusion is considered destabilising. Thus starting with a fully stable steady state so-called Turing patterns are established in the solution of the system of partial differential equations. Our approach is based on the triangular spectral element methods. To enhance the performance of the triangular spectral element methods, we are extending them with algorithms for spatial adaptation of the considered domain. We have seen in past studies that this provides excellent improvements to spectral methods. For a good comparison of different refinement criteria we are taking account of one heuristic criterion and two spectral criteria closely related to our method. In difference to recent research we are now considering time-dependent problems with triangular spectral element methods. For the temporal discretisation of the Turing mechanisms we are applying a generic theta-integration scheme with different values for theta. To deal with problems arising from a static time step size, we are additionally introducing an algorithm for an adaptive time step control. Finally we are going to validate our method in two ways. First we are considering three different time-independent model problems for which we already know the exact solution. This will be done to generally verify the correctness of the method and the implementation. Second we are applying our approach to the Turing mechanisms. For these equations we do not know the solution, therefore we compare our results to theoretical assumptions. When validating the method we are also comparing different configurations of our approach. We will see in chapter 7 that our method combined with a spectral refinement criterion and using adaptive time step size with the Crank-Nicolson method for temporal discretisation provides good results for any Turing mechanism.