3,106 research outputs found
Numerical Methods for a Nonlinear BVP Arising in Physical Oceanography
In this paper we report and compare the numerical results for an ocean
circulation model obtained by the classical truncated boundary formulation, the
free boundary approach and a quasi-uniform grid treatment of the problem. We
apply a shooting method to the truncated boundary formulation and finite
difference methods to both the free boundary approach and the quasi-uniform
grid treatment. Using the shooting method, supplemented by the Newton's
iterations, we show that the ocean circulation model cannot be considered as a
simple test case. In fact, for this method we are forced to use as initial
iterate a value close to the correct missing initial condition in order to be
able to get a convergent numerical solution. The reported numerical results
allow us to point out how the finite difference method with a quasi-uniform
grid is the less demanding approach and that the free boundary approach
provides a more reliable formulation than the classical truncated boundary
formulation.Comment: 25 pages, 12 figures, 5 table
Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions
We study the finite-horizon optimal control problem with quadratic
functionals for an established fluid-structure interaction model. The coupled
PDE system under investigation comprises a parabolic (the fluid) and a
hyperbolic (the solid) dynamics; the coupling occurs at the interface between
the regions occupied by the fluid and the solid. We establish several trace
regularity results for the fluid component of the system, which are then
applied to show well-posedness of the Differential Riccati Equations arising in
the optimization problem. This yields the feedback synthesis of the unique
optimal control, under a very weak constraint on the observation operator; in
particular, the present analysis allows general functionals, such as the
integral of the natural energy of the physical system. Furthermore, this work
confirms that the theory developed in Acquistapace et al. [Adv. Differential
Equations, 2005] -- crucially utilized here -- encompasses widely differing PDE
problems, from thermoelastic systems to models of acoustic-structure and, now,
fluid-structure interactions.Comment: 22 pages, submitted; v2: misprints corrected, a remark added in
section
Efficient hyperbolic-parabolic models on multi-dimensional unbounded domains using an extended DG approach
We introduce an extended discontinuous Galerkin discretization of
hyperbolic-parabolic problems on multidimensional semi-infinite domains.
Building on previous work on the one-dimensional case, we split the
strip-shaped computational domain into a bounded region, discretized by means
of discontinuous finite elements using Legendre basis functions, and an
unbounded subdomain, where scaled Laguerre functions are used as a basis.
Numerical fluxes at the interface allow for a seamless coupling of the two
regions. The resulting coupling strategy is shown to produce accurate numerical
solutions in tests on both linear and non-linear scalar and vectorial model
problems. In addition, an efficient absorbing layer can be simulated in the
semi-infinite part of the domain in order to damp outgoing signals with
negligible spurious reflections at the interface. By tuning the scaling
parameter of the Laguerre basis functions, the extended DG scheme simulates
transient dynamics over large spatial scales with a substantial reduction in
computational cost at a given accuracy level compared to standard single-domain
discontinuous finite element techniques.Comment: 28 pages, 13 figure
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