Goal-oriented time adaptivity for port-Hamiltonian systems
arxiv(2024)
摘要
Port-Hamiltonian systems provide an energy-based modeling paradigm for
dynamical input-state-output systems. At their core, they fulfill an energy
balance relating stored, dissipated and supplied energy. To accurately resolve
this energy balance in time discretizations, we propose an adaptive grid
refinement technique based on a posteriori error estimation. The evaluation of
the error estimator includes the computation of adjoint sensitivities. To
interpret this adjoint equation as a backwards-in-time equation, we show
piecewise weak differentiability of the dual variable. Then, leveraging
dissipativity of the port-Hamiltonian dynamics, we present a parallelizable
approximation of the underlying adjoint system in the spirit of a block-Jacobi
method to efficiently compute error indicators. We illustrate the performance
of the proposed scheme by means of numerical experiments showing that it yields
a smaller violation of the energy balance when compared to uniform refinements
and traditional step-size controlled time stepping.
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