Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems

In this article we present a non-oscillatory 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 non-oscillatory 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 non-oscillatory properties are demonstrated.

@article{id600,
  author = {Dumbser, M. and K\"aser, M.},
  doi = {10.1016/j.jcp.2006.06.043},
  journal = {Journal of Computational Physics},
  language = {en},
  pages = {693-723},
  title = {Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems},
  volume = {221},
  year = {2007},
}

%O Journal Article
%A Dumbser, M.
%A Käser, M.
%R 10.1016/j.jcp.2006.06.043
%J Journal of Computational Physics
%G en
%P 693-723
%T Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems
%V 221
%D 2007