32 research outputs found
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
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
One- and multi-dimensional CWENOZ reconstructions for implementing boundary conditions without ghost cells
We address the issue of point value reconstructions from cell averages in the
context of third order finite volume schemes, focusing in particular on the
cells close to the boundaries of the domain. In fact, most techniques known in
the literature rely on the creation of ghost cells outside the boundary and on
some form of extrapolation from the inside that, taking into account the
boundary conditions, fills the ghost cells with appropriate values, so that a
standard reconstruction can be applied also in boundary cells. In (Naumann,
Kolb, Semplice, 2018), motivated by the difficulty of choosing appropriate
boundary conditions at the internal nodes of a network, a different technique
was explored that avoids the use of ghost cells, but instead employs for the
boundary cells a different stencil, biased towards the interior of the domain.
In this paper, extending that approach, which does not make use of ghost
cells, we propose a more accurate reconstruction for the one-dimensional case
and a two-dimensional one for Cartesian grids. In several numerical tests we
compare the novel reconstruction with the standard approach using ghost cells
One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells
We address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes, focusing in particular on the cells close to the boundaries of the domain. In fact, most techniques in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that, taking into account the boundary conditions, fills the ghost cells with appropriate values, so that a standard reconstruction can be applied also in the boundary cells. In Naumann et al. (Appl. Math. Comput. 325: 252-270. https://doi.org110.1016/j.amc.2017.12.041, 2018), motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network, a different technique was explored that avoids the use of ghost cells, but instead employs for the boundary cells a different stencil, biased towards the interior of the domain. In this paper, extending that approach, which does not make use of ghost cells, we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids. In several numerical tests, we compare the novel reconstruction with the standard approach using ghost cells
RBF Based CWENO Method
Solving hyperbolic conservation laws on general grids can be important to reduce the computational complexity and increase accuracy in many applications. However, the use of non-uniform grids can introduce challenges when using high-order methods. We propose to use a Central WENO (CWENO) scheme based on radial basis function (RBF) interpolation, which is applicable to scattered data. We develop a smoothness indicator, based on RBFs, and CWENO specific weights which depend on the mesh size of the grid to construct an arbitrarily high order RBF-CWENO method. We evaluate the method with multiple examples in one dimension
High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes
We present a new family of very high order accurate direct
Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous
Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on
moving 2D Voronoi meshes that are regenerated at each time step and which
explicitly allow topology changes in time.
The Voronoi tessellations are obtained from a set of generator points that
move with the local fluid velocity. We employ an AREPO-type approach, which
rapidly rebuilds a new high quality mesh rearranging the element shapes and
neighbors in order to guarantee a robust mesh evolution even for vortex flows
and very long simulation times. The old and new Voronoi elements associated to
the same generator are connected to construct closed space--time control
volumes, whose bottom and top faces may be polygons with a different number of
sides. We also incorporate degenerate space--time sliver elements, needed to
fill the space--time holes that arise because of topology changes. The final
ALE FV-DG scheme is obtained by a redesign of the fully discrete direct ALE
schemes of Boscheri and Dumbser, extended here to moving Voronoi meshes and
space--time sliver elements. Our new numerical scheme is based on the
integration over arbitrary shaped closed space--time control volumes combined
with a fully-discrete space--time conservation formulation of the governing PDE
system. In this way the discrete solution is conservative and satisfies the GCL
by construction.
Numerical convergence studies as well as a large set of benchmarks for
hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and
robustness of the proposed method. Our numerical results clearly show that the
new combination of very high order schemes with regenerated meshes with
topology changes lead to substantial improvements compared to direct ALE
methods on conforming meshes