Initialization Issues in General Differential Algebraic Equation Integrators

摘要

The past several years have yielded much research into the development of generalnumerical integrators for nonlinear unstructured higher index differential algebraic equations (DAEs) of the form F(y', y, t) = 0 where Fy' is identically singular. These methods are based on integrating an implicitly defined ODE produced by forming the derivative array equations G(y', w, y, t) = 0. This larger nonlinear system may have components that are not uniquely determined. Prior work has examined the theoretical aspects of these methods. There has also been considerable work done on the efficient implementation of these integrators. This thesis will examine two initialization problems that arise when numerically solving DAEs. One problem is the computation of the consistent initial conditions required to begin the integration of DAEs. Various line search strategies will be compared and examples provided showing where these strategies are needed. The second problem is the initialization of the iterative solvers used during the integration of a DAE. Integration of the implicitly defined ODE depends on solving the nonlinear system G = 0 at each time step. For fully implicit nonlinear systems the possible effects of polynomial prediction on the nonunique components will be examined. A complete analysis is given in the case of higher index linear time varying DAEs. It is shown that the standard ODE theory does not hold and a different prediction strategy must be used.