A new approach for DAEs

grantor: University of TorontoThis thesis introduces a new 'least squares' implementation of a continuous extension of an underlying discrete Runge-Kutta (RK) formula that is particularly effective for solving DAEs. The resulting method provides a continuous approximation to the solution of the DAE and does not assume that the problem has a special form or that the index of the problem is known. The methods we develop and implement include some that are based on underlying discrete formulas that were previously not considered suitable for DAEs because of severe order reduction or stability restrictions. Specifically we propose a particular implementation of a modified extension of these formulas and develop stability and convergence results which indicate that these methods can be effective when solving DAEs. Numerical results are included to verify this. We also investigate and justify an error control and stepsize choosing strategy that is based on directly monitoring and bounding the continuous residual (or defect) of the numerical solution. We investigate the relationship between the size of the residual and the local error for index one and index two DAEs. We also derive the extra order conditions that arise for special classes of index one and index two problems. Our fixed stepsize and variable stepsize numerical results confirm the improved order and stability that is achievable with our approach. These results also seem to indicate that our approach is less likely to run into convergence difficulties (with nonlinear equations that must be solved on each timestep) than is observed with traditional implementations.Ph.D