Arbitrary high order discontinuous Galerkin methods for the shallow water and incompressible Navier-Stokes equations on unstructured staggered meshes

Place: Seminar Room -  Department of Mathematics - Via Sommarive 14 - Povo - Trento
at 10.00 a.m.

• Maurizio Tavelli - PhD in mathematics

Abstract:
In this final seminar I present a new class of arbitrary high order accurate semi-implicit discontinuous Galerkin methods for the solution of the shallow water and incompressible Navier-Stokes equations on staggered unstructured curved meshes.
Regarding two-dimensional shallow water equations, the discrete free surface elevation is defined on a primal triangular grid, while the discrete total height and the discrete velocity field are defined on an edge-based staggered dual grid.
Similarly, for the two-dimensional incompressible Navier-Stokes case, the discrete pressure is defined on the main triangular grid and the velocity field is defined on the edge-based staggered grid. While staggered meshes are state of the art in classical finite difference approximations of the incompressible Navier-Stokes equations, their use in the context of high order DG schemes is novel and still quite rare. 
High order in time can be achieved by using a space-time finite element framework, where the basis and test functions are piecewise polynomials in both space and time. 
Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse system for the scalar pressure involving only the direct neighbor elements, so that it becomes a block four-point system in 2D and a block five-point system for 3D tetrahedral meshes. The resulting linear system is conveniently solved with a matrix-free GMRES algorithm. A very simple and efficient Picard iteration is then used in order to derive a space-time pressure correction algorithm that achieves also high order of accuracy in time. 
The special case of high order in space low order in time allows us to recover further regularity about the main linear system for the pressure, such as the symmetry and the positive semi-definiteness in the general case. This allows us to use a very fast linear solver such as the conjugate gradient method. 
The flexibility and accuracy of high order space-time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time. 
We will further extend the previous method to three-dimensional incompressible Navier-Stokes system using a tetrahedral main grid and a corresponding face-based hexaxedral dual grid. The resulting dual mesh consists in non-standard 5-vertex hexahedral elements that cannot be represented using tensor products of one dimensional basis functions. Indeed a modal polynomial basis will be used for the dual mesh.
This new family of numerical schemes is verified by solving a series of typical numerical test problems and by comparing the obtained numerical results with available exact analytical solutions or other numerical reference data. Furthermore, the comparison with available experimental results will be presented for incompressible Navier-Stokes equations

Supervisor: Michael Dumbser; Vincenzo Casulli