Improving the monodromy matrix computation in pathfollowing schemes for nonsmooth dynamics

An enhanced pathfollowing strategy was recently proposed employing a modified Newton–Raphson solver accelerated by a low-cost routine via a Krylov subspace iterator. One of the key points of the numerical strategy is the computation of the Jacobian of the Poincaré map (i.e., the monodromy matrix), which, by serving as iteration matrix, governs both the accuracy and robustness of the whole strategy. In the context of nonsmooth nonlinear dynamic systems, this matrix can only be computed via numerical differentiation, which leads to subtractive cancellation errors. A new technique is here proposed by performing a central finite difference of the vector field associated with the ODE problems within the time-step integration to assemble the monodromy matrix. This technique allows us to bound the errors induced by the standard numerical differentiation. By means of a wide numerical campaign for meaningful multi-dof dynamical systems, the enhanced pathfollowing scheme is shown to yield a higher level of accuracy than the standard one, as well as higher robustness, with improved convergence properties for the whole numerical strategy. The C++ code implementing the proposed methodology is freely available at https://zenodo.org/record/7245478, and includes code for both the classic method as well as the new approaches for the computation of the Jacobian matrix.