462 research outputs found
Adaptive Mesh Refinement for Hyperbolic Systems based on Third-Order Compact WENO Reconstruction
In this paper we generalize to non-uniform grids of quad-tree type the
Compact WENO reconstruction of Levy, Puppo and Russo (SIAM J. Sci. Comput.,
2001), thus obtaining a truly two-dimensional non-oscillatory third order
reconstruction with a very compact stencil and that does not involve
mesh-dependent coefficients. This latter characteristic is quite valuable for
its use in h-adaptive numerical schemes, since in such schemes the coefficients
that depend on the disposition and sizes of the neighboring cells (and that are
present in many existing WENO-like reconstructions) would need to be recomputed
after every mesh adaption.
In the second part of the paper we propose a third order h-adaptive scheme
with the above-mentioned reconstruction, an explicit third order TVD
Runge-Kutta scheme and the entropy production error indicator proposed by Puppo
and Semplice (Commun. Comput. Phys., 2011). After devising some heuristics on
the choice of the parameters controlling the mesh adaption, we demonstrate with
many numerical tests that the scheme can compute numerical solution whose error
decays as , where is the average
number of cells used during the computation, even in the presence of shock
waves, by making a very effective use of h-adaptivity and the proposed third
order reconstruction.Comment: many updates to text and figure
On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes
Third order WENO and CWENO reconstruction are widespread high order
reconstruction techniques for numerical schemes for hyperbolic conservation and
balance laws. In their definition, there appears a small positive parameter,
usually called , that was originally introduced in order to avoid a
division by zero on constant states, but whose value was later shown to affect
the convergence properties of the schemes. Recently, two detailed studies of
the role of this parameter, in the case of uniform meshes, were published. In
this paper we extend their results to the case of finite volume schemes on
non-uniform meshes, which is very important for h-adaptive schemes, showing the
benefits of choosing as a function of the local mesh size . In
particular we show that choosing or is
beneficial for the error and convergence order, studying on several non-uniform
grids the effect of this choice on the reconstruction error, on fully discrete
schemes for the linear transport equation, on the stability of the numerical
schemes. Finally we compare the different choices for in the case of
a well-balanced scheme for the Saint-Venant system for shallow water flows and
in the case of an h-adaptive scheme for nonlinear systems of conservation laws
and show numerical tests for a two-dimensional generalisation of the CWENO
reconstruction on locally adapted meshes
An efficient class of increasingly high-order ENO schemes with multi-resolution
We construct an efficient class of increasingly high-order (up to 17th-order)
essentially non-oscillatory schemes with multi-resolution (ENO-MR) for solving
hyperbolic conservation laws. The candidate stencils for constructing ENO-MR
schemes range from first-order one-point stencil increasingly up to the
designed very high-order stencil. The proposed ENO-MR schemes adopt a very
simple and efficient strategy that only requires the computation of the
highest-order derivatives of a part of candidate stencils. Besides simplicity
and high efficiency, ENO-MR schemes are completely parameter-free and
essentially scale-invariant. Theoretical analysis and numerical computations
show that ENO-MR schemes achieve designed high-order convergence in smooth
regions which may contain high-order critical points (local extrema) and retain
ENO property for strong shocks. In addition, ENO-MR schemes could capture
complex flow structures very well
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