Efficient High-Order Methods for ODEs and DAEs
AbstractWe present methods for the high-order differentiation through ordinary
differential equations (ODEs), and more importantly, differential
algebraic equations (DAEs). First, methods are developed that assert
that the requested derivatives are really those of the solution of the
ODE, and not those of the algorithm used to solve the ODE. Next,
high-order solvers for DAEs are developed that in a fully automatic
way turn an n-th order solution step of the DAEs into a corresponding
step for an ODE initial value problem. In particular, this requires
the automatic high-order solution of implicit relations, which is
achieved using an iterative algorithm that converges to the exact
result in at most (n+1) steps. We give examples of the performance of
J. Hoefkens, M. Berz, K. Makino, in: "Automatic Differentiation: From Simulation to Optimization", G. Corliss, C. Faure, A. Griewank, L. Hascoet, U. Naumann (Eds.) (2001) Springer
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