Speaker
Ruibin Qin, Applied Mathematics, University of Waterloo
Abstract:
Methods based on Cartesian grids with embedded geometries are an alternative approach to computations performed on unstructured grids. Generation of Cartesian grids is significantly simpler than generation of unstructured meshes around complex geometries. They also provide significant computational savings on the majority of regular grid cells as the uniform size, shape, and orientation of mesh elements allows us to reduce both computing time and storage requirements. This might be especially important for computationally intensive high-order schemes such as the discontinuous Galerkin methods. However, the major challenge of the discontinuous Galerkin method on Cartesian grids is the so called cut cells that arise from cutting out a geometry from the Computational domain. They are difficult to deal with due to their small size and irregular shapes. Arbitrarily small size of cut cells leads to a restrictive CFL condition imposed on the whole mesh while explicit scheme is used. Irregular shapes are difficult to integrate on due to absence of integration rules, and also need extra storage for special quadrature rules on each cut cell. We present an approach of dealing with cut cells based on merging of cut cells with neighbouring elements. This alleviates the CFL restriction but resulting merged cells might have irregular shapes depending on the shapes of cut cells. We discuss how to deal with integration on such elements with the aim to preserve accuracy and without resolving to refining mesh near the embedded boundary. Numerical results will be shown to validate the efficiency of the proposed approach.