TY - JOUR
T1 - Structure-preserving reduced basis methods for Poisson systems
AU - Hesthaven, Jan S.
AU - Pagliantini, Cecilia
PY - 2021/7
Y1 - 2021/7
N2 - We develop structure-preserving reduced basis methods for a large class of nondissipative problems by resorting to their formulation as Hamiltonian dynamical systems. With this perspective, the phase space is naturally endowed with a Poisson manifold structure which encodes the physical properties, symmetries, and conservation laws of the dynamics. The goal is to design reduced basis methods for the general state-dependent degenerate Poisson structure based on a two-step approach. First, via a local approximation of the Poisson tensor, we split the Hamiltonian dynamics into an "almost symplectic" part and the trivial evolution of the Casimir invariants. Second, canonically symplectic reduced basis techniques are applied to the nontrivial component of the dynamics, preserving the local Poisson tensor kernel exactly. The global Poisson structure and the conservation properties of the phase flow are retained by the reduced model in the constant-valued case and up to errors in the Poisson tensor approximation in the state-dependent case. A priori error estimates for the solution of the reduced system are established. A set of numerical simulations is presented to corroborate the theoretical findings.
AB - We develop structure-preserving reduced basis methods for a large class of nondissipative problems by resorting to their formulation as Hamiltonian dynamical systems. With this perspective, the phase space is naturally endowed with a Poisson manifold structure which encodes the physical properties, symmetries, and conservation laws of the dynamics. The goal is to design reduced basis methods for the general state-dependent degenerate Poisson structure based on a two-step approach. First, via a local approximation of the Poisson tensor, we split the Hamiltonian dynamics into an "almost symplectic" part and the trivial evolution of the Casimir invariants. Second, canonically symplectic reduced basis techniques are applied to the nontrivial component of the dynamics, preserving the local Poisson tensor kernel exactly. The global Poisson structure and the conservation properties of the phase flow are retained by the reduced model in the constant-valued case and up to errors in the Poisson tensor approximation in the state-dependent case. A priori error estimates for the solution of the reduced system are established. A set of numerical simulations is presented to corroborate the theoretical findings.
KW - Hamiltonian dynamics
KW - Poisson manifolds
KW - invariants of motion
KW - reduced basis methods (RBM)
KW - structure-preserving schemes
KW - symplectic structure
UR - http://www.scopus.com/inward/record.url?scp=85107573649&partnerID=8YFLogxK
U2 - 10.1090/mcom/3618
DO - 10.1090/mcom/3618
M3 - Article
SN - 0025-5718
VL - 90
SP - 1701
EP - 1740
JO - Mathematics of Computation
JF - Mathematics of Computation
IS - 330
ER -