Arbitrary high order nonoscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems
Dumbser, M., and M. Käser (2007),
Arbitrary high order nonoscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems,
Journal of Computational Physics, 221, 693723, doi:10.1016/j.jcp.2006.06.043.
 Abstract
 In this article we present a nonoscillatory ﬁnite volume scheme of arbitrary accuracy
in space and time for solving linear hyperbolic systems on unstructured grids in two and three space dimensions. The key point is a new reconstruction operator that makes use of techniques developed originally in the discontinuous Galerkin ﬁnite element framework. First, we use a hierarchical orthogonal basis to perform reconstruction. Second, reconstruction is not done in physical coordinates, but in a reference coordinate system which eliminates scaling eﬀects and thus avoids illconditioned reconstruction matrices. In order to achieve nonoscillatory properties, we propose a new WENO reconstruction technique that does not reconstruct pointvalues but entire polynomials which can easily be evaluated and diﬀerentiated at any point. We show that due to the special reconstruction the WENO oscillation indicator can be easily computed as a simple quadratic functional. Our WENO scheme does not suﬀer from the problem of negative weights as previously described in the literature, since the linear weights are not used to increase accuracy. Accuracy is obtained by merely putting a large linear weight on the central stencil. The WENO scheme obtained in this way can be implemented very eﬃciently. In order to get
arbitrary high order of accuracy in time, we use the ADER approach of Toro et al. Due to the general formulation of the method, the implementation is almost identical in two and three dimensions and can produce any order of accuracy. We show convergence results obtained with the proposed scheme up to seventh order on distorted triangular meshes in two dimensions and up to sixth order of accuracy in space and time on regular unstructured tetrahedral grids in three space dimensions. Furthermore, we show results of two and three dimensional test problems with discontinuous solutions where the nonoscillatory properties are demonstrated.
 BibTeX

@article{id600,
author = {M. Dumbser and M. K{\"a}ser},
journal = {Journal of Computational Physics},
pages = {693723},
title = {{Arbitrary high order nonoscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems}},
volume = {221},
year = {2007},
doi = {10.1016/j.jcp.2006.06.043},
}
 EndNote

%0 Journal Article
%A Dumbser, M.
%A Käser, M.
%D 2007
%V 221
%J Journal of Computational Physics
%P 693723
%T Arbitrary high order nonoscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems