316 research outputs found
Raviart Thomas Petrov-Galerkin Finite Elements
The general theory of Babu\v{s}ka ensures necessary and sufficient conditions
for a mixed problem in classical or Petrov-Galerkin form to be well posed in
the sense of Hadamard. Moreover, the mixed method of Raviart-Thomas with
low-level elements can be interpreted as a finite volume method with a
non-local gradient. In this contribution, we propose a variant of type
Petrov-Galerkin to ensure a local computation of the gradient at the interfaces
of the elements. The in-depth study of stability leads to a specific choice of
the test functions. With this choice, we show on the one hand that the mixed
Petrov-Galerkin obtained is identical to the finite volumes scheme "volumes
finis \`a 4 points" ("VF4") of Faille, Gallo\"uet and Herbin and to the
condensation of mass approach developed by Baranger, Maitre and Oudin. On the
other hand, we show the stability via an inf-sup condition and finally the
convergence with the usual methods of mixed finite elements.Comment: arXiv admin note: text overlap with arXiv:1710.0439
Dispersive wave runup on non-uniform shores
Historically the finite volume methods have been developed for the numerical
integration of conservation laws. In this study we present some recent results
on the application of such schemes to dispersive PDEs. Namely, we solve
numerically a representative of Boussinesq type equations in view of important
applications to the coastal hydrodynamics. Numerical results of the runup of a
moderate wave onto a non-uniform beach are presented along with great lines of
the employed numerical method (see D. Dutykh et al. (2011) for more details).Comment: 8 pages, 6 figures, 18 references. This preprint is submitted to
FVCA6 conference proceedings. Other author papers can be downloaded at
http://www.lama.univ-savoie.fr/~dutykh
A fully semi-Lagrangian discretization for the 2D Navier--Stokes equations in the vorticity--streamfunction formulation
A numerical method for the two-dimensional, incompressible Navier--Stokes
equations in vorticity--streamfunction form is proposed, which employs
semi-Lagrangian discretizations for both the advection and diffusion terms,
thus achieving unconditional stability without the need to solve linear systems
beyond that required by the Poisson solver for the reconstruction of the
streamfunction. A description of the discretization of Dirichlet boundary
conditions for the semi-Lagrangian approach to diffusion terms is also
presented. Numerical experiments on classical benchmarks for incompressible
flow in simple geometries validate the proposed method
Second-order mixed-moment model with differentiable ansatz function in slab geometry
We study differentiable mixed-moment models (full zeroth and first moment,
half higher moments) for a Fokker-Planck equation in one space dimension.
Mixed-moment minimum-entropy models are known to overcome the zero net-flux
problem of full-moment minimum entropy models. Realizability theory for
these modification of mixed moments is derived for second order. Numerical
tests are performed with a kinetic first-order finite volume scheme and
compared with , classical and a reference scheme.Comment: arXiv admin note: text overlap with arXiv:1611.01314,
arXiv:1511.0271
Flux form Semi-Lagrangian methods for parabolic problems
A semi-Lagrangian method for parabolic problems is proposed, that extends
previous work by the authors to achieve a fully conservative, flux-form
discretization of linear and nonlinear diffusion equations. A basic consistency
and convergence analysis are proposed. Numerical examples validate the proposed
method and display its potential for consistent semi-Lagrangian discretization
of advection--diffusion and nonlinear parabolic problems
Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: adjoint approximations and extensions
This paper continues the convergence analysis in [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882–904] of discrete approximations to the linearized and adjoint equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. We consider a simple modified Lax–Friedrichs discretization on a uniform grid, and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined. It is proved that there is convergence in the discrete approximation of linearized output functionals even for Dirac initial perturbations and pointwise convergence almost everywhere for the solution of the adjoint discrete equations. In particular, the adjoint approximation converges to the correct uniform value in the region in which characteristics propagate into the discontinuity. Moreover, it is shown that the results of [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882–904] and the present paper hold also for quite general nonlinear initial data which contain multiple shocks and for which shocks form at a later time and/or merge
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