11-15 July 2011

SciCADE 2011

hosted by the Fields Institute,
held at the University of Toronto

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Advances in Methods for Hyperbolic Equations, II
Organized by
Martin Berzins, University of Utah
Lilia Krivodonova, University of Waterloo

This Minisyposium, the first of two, will address recent advances in the solution of hyperbolic equations. Topics included will be include methods such as Discontinuous Galerkin methods as well as new finite volume schemes and substantial impriovemnets of existing schemes to preserve positivity or to better postprocess the existing high-order solutions.

L. Ivan,
University of Toronto
A Parallel High-Order Solution-Adaptive CENO Scheme on 3D Cubed-Sphere Grids for Space-Physics Flows
high-order central essentially non-oscillatory (CENO) ?nite-volume scheme in com- bination with a block-based adaptive mesh re?nement (AMR) algorithm is developed for three-dimensional (3D) cubed-sphere grids and applied to space-physics ?ows governed by hyperbolic partial di?erential equations. The proposed cubed-sphere simulation framework is based on a genuine multiblock implementation, leading to ?ux calculations, adaptivity, implicit solves and parallelism that are fully transparent to the boundaries between the six cubed-sphere grid sectors. The high-order CENO formulation is naturally uniformly high- order on the whole cubed-sphere grid including at sector boundaries. Numerical results to demonstrate the accuracy and capability of the proposed framework are presented.

Ruibin Qin
University of Waterloo
A Discontinuous Galerkin Method for Hyperbolic Problems on Cartesian Grids with Embedded Geometries
Methods based on Cartesian grids with embedded geometries have advantages over computations on unstructured grids in terms of mesh generation and computing costs. However, cut cells are difficult to deal with due to their small size which leads to a restrictive CFL condition and irregular shapes which are difficult to integrate on. We present an approach of dealing with cut cells based on cell merging in the discontinuous Galerkin method. How to deal with integration on such elements and to
Martin Berzins
University Of Utah
High-Order ENO and DG Positivity Preserving Methods for Time Dependent Problems
The derivation and theoretical properties of a class of high-order data-bounded polynomials on general meshes will be given. Such polynomials make it possible to circumvent the problem of Runge-type oscillations on evenly spaced meshes by adaptively varying the stencil and order used, but at the cost of only enforcing C0 solution continuity at data points. It will be shown that the use of these high-order provably data-bounded polynomials,provides a way to develop positivity preserving polynomial approximations as well as methods of potentially high orders for hyperbolic equations using WENO and DG Methods.
Lilia Krivodonov,
University of Waterloo
Limiters for high-order discontinuous Galerkin methods