25,233 research outputs found
Singular Lagrangians and precontact Hamiltonian Systems
In this paper we discuss singular Lagrangian systems on the framework of
contact geometry. These systems exhibit a dissipative behavior in contrast with
the symplectic scenario. We develop a constraint algorithm similar to the
presymplectic one studied by Gotay and Nester (the geometrization of the
well-known Dirac-Bergman algorithm). We also construct the Hamiltonian
counterpart and prove the equivalence with the Lagrangian side. A Dirac-Jacobi
bracket is constructed similar to the Dirac bracket
Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods
In this paper, we develop a new tensor-product based preconditioner for
discontinuous Galerkin methods with polynomial degrees higher than those
typically employed. This preconditioner uses an automatic, purely algebraic
method to approximate the exact block Jacobi preconditioner by Kronecker
products of several small, one-dimensional matrices. Traditional matrix-based
preconditioners require storage and
computational work, where is the degree of basis polynomials used, and
is the spatial dimension. Our SVD-based tensor-product preconditioner requires
storage, work in two spatial
dimensions, and work in three spatial dimensions.
Combined with a matrix-free Newton-Krylov solver, these preconditioners allow
for the solution of DG systems in linear time in per degree of freedom in
2D, and reduce the computational complexity from to
in 3D. Numerical results are shown in 2D and 3D for the
advection and Euler equations, using polynomials of degree up to . For
many test cases, the preconditioner results in similar iteration counts when
compared with the exact block Jacobi preconditioner, and performance is
significantly improved for high polynomial degrees .Comment: 40 pages, 15 figure
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