On the Efficient Generation of Taylor Expansions for DAE Solutions by Automatic Differentiation
(2005)
Under certain conditions the signature method suggested by
Pantiledes and Pryce facilitates the local expansion of DAE solutions
by Taylor polynomials of arbitrary order. The successive calculation of
Taylor coefficients involves the solution of nonlinear algebraic equations
by some variant of the Gauss-Newton method. Hence, one needs to evaluate
certain Jacobians and several right hand sides. Without advocating
a particular solver we discuss how this information can be efficiently
obtained using ADOL-C or similar automatic differentiation packages.3
Systems of stiff ordinary differential equations (ODEs) can be integrated
properly only by implicit methods. For that purpose, one usually has to
solve a system of nonlinear equations at each time step. This system of equations
may be solved by variants of Newton's method. Here, the main computing effort lies
in forming and factoring the Jacobian or a suitable approximation to it.
In this paper, we examine a new approach of constructing an appropriate quasi-
Newton approximation for solving stiff ODEs. The method makes for the first time
explicit use of tangent and adjoint information that can be obtained using the forward
and the reverse mode of algorithmic differentiation (AD). We elaborate the
conditions for invariance with respect to linear transformations of the state space
and thus similarity transformations of the Jacobian. One new updating variant that
yields such an invariant method is presented. Numerical results for Runge-Kutta
methods and linear multi-step methods are discussed.