TY - JOUR
T1 - Combining finite element space-discretizations with symplectic time-marching schemes for linear Hamiltonian systems
AU - Cockburn, Bernardo
AU - Du, Shukai
AU - Sánchez, Manuel A.
N1 - Publisher Copyright:
Copyright © 2023 Cockburn, Du and Sánchez.
PY - 2023
Y1 - 2023
N2 - We provide a short introduction to the devising of a special type of methods for numerically approximating the solution of Hamiltonian partial differential equations. These methods use Galerkin space-discretizations which result in a system of ODEs displaying a discrete version of the Hamiltonian structure of the original system. The resulting system of ODEs is then discretized by a symplectic time-marching method. This combination results in high-order accurate, fully discrete methods which can preserve the invariants of the Hamiltonian defining the ODE system. We restrict our attention to linear Hamiltonian systems, as the main results can be obtained easily and directly, and are applicable to many Hamiltonian systems of practical interest including acoustics, elastodynamics, and electromagnetism. After a brief description of the Hamiltonian systems of our interest, we provide a brief introduction to symplectic time-marching methods for linear systems of ODEs which does not require any background on the subject. We describe then the case in which finite-difference space-discretizations are used and focus on the popular Yee scheme (1966) for electromagnetism. Finally, we consider the case of finite-element space discretizations. The emphasis is placed on the conservation properties of the fully discrete schemes. We end by describing ongoing work.
AB - We provide a short introduction to the devising of a special type of methods for numerically approximating the solution of Hamiltonian partial differential equations. These methods use Galerkin space-discretizations which result in a system of ODEs displaying a discrete version of the Hamiltonian structure of the original system. The resulting system of ODEs is then discretized by a symplectic time-marching method. This combination results in high-order accurate, fully discrete methods which can preserve the invariants of the Hamiltonian defining the ODE system. We restrict our attention to linear Hamiltonian systems, as the main results can be obtained easily and directly, and are applicable to many Hamiltonian systems of practical interest including acoustics, elastodynamics, and electromagnetism. After a brief description of the Hamiltonian systems of our interest, we provide a brief introduction to symplectic time-marching methods for linear systems of ODEs which does not require any background on the subject. We describe then the case in which finite-difference space-discretizations are used and focus on the popular Yee scheme (1966) for electromagnetism. Finally, we consider the case of finite-element space discretizations. The emphasis is placed on the conservation properties of the fully discrete schemes. We end by describing ongoing work.
KW - Hamiltonian systems
KW - Poisson systems
KW - finite difference methods
KW - finite element methods
KW - symplectic time-marching methods
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U2 - 10.3389/fams.2023.1165371
DO - 10.3389/fams.2023.1165371
M3 - Review article
AN - SCOPUS:85153057768
SN - 2297-4687
VL - 9
JO - Frontiers in Applied Mathematics and Statistics
JF - Frontiers in Applied Mathematics and Statistics
M1 - 1165371
ER -