A multiscale technique for finding slow manifolds of stiff mechanical systems

In the limit of infinite stiffness, the differential equations of motion of stiff mechanical systems become differential algebraic equations whose solutions stay in a constraint submanifold P̂ of the phase space. Even though solutions of the stiff differential equations are typically oscillatory with large frequency, there exists a slow manifold P̃ consisting of nonoscillatory solutions; P̃ has the same dimension as P̂ and converges to it as the stiffness approaches infinity. We introduce an iterative projection algorithm, IPA, that projects points in the phase space of a stiff mechanical system onto the associated slow manifold P̃. The algorithm is based on ideas such as micro-integration and filtering coming from the field of multiscale simulation and is applicable to initializing integration algorithms for both stiff ODEs and DAEs, including the initialization of Lagrange multipliers. We also illustrate in a model situation how the algorithm may be combined with numerical integrators for both the stiff system and the limit constrained system. These combinations may speed up the solution of stiff problems and also be used to integrate DAEs with explicit algorithms.