Interpolation and error control schemes for algebraic differential equations using continuous implicit Runge-Kutta methods.

摘要

It is well known in the literature that standard implicit Runge-Kutta methods applied to algebraic differential equations exhibit order reduction in both differential and algebraic components. As a result, interpolation and error control schemes are also affected, especially for higher index problems. This thesis presents alternative interpolation and error controls schemes based on the defects and the algebraic residuals of a corresponding perturbed equation for particular classes of semi explicit index 1, index 2 and index 3 problems.;Due to its explicit nature, the proposed scheme can be implemented efficiently and at the same time yields results comparable with those from the literature. At each level of this "boostrapping" procedure, the differential interpolants and algebraic interpolants can be constructed separately using explicit evaluations of the interpolants constructed from the previous level.;As in the case with any continuous approximation, the perturbed equation corresponding to these interpolation schemes can be converted to an equivalent underlying initial value problem which is independent of the algebraic component. This allows one to apply established results in the literature to a defect-based error control scheme for the differential component. The same idea can be applied to derive an error control scheme for the algebraic component.;Based on the above ideas, we develop the three schemes--namely, Scheme 1, Scheme 2 and Scheme 3--corresponding to semi explicit algebraic differential equations of index 1, index 2 and index 3 respectively. For each of these schemes, we also derive a corresponding error control scheme based on the defect and algebraic residual of an underlying perturbed initial value problem.;The test problems we use to illustrate our approach are the classical pendulum problem, the seven body problem and the driven cavity problem. While the driven cavity problem is of index 2, the other problems are of index 3 or index 3 reformulated as index 2 problems. These problems arise from two important classes of application: constrained mechanical systems and fluid dynamics.;The three implicit continuous Runge-Kutta formulas used as examples in our investigation are SDIRK(2), Gauss(2) and Gauss(3).