In this work we present a new class of wellbalanced, arbitrary high order accurate semiimplicit discontinuous Galerkin methods for the solution of the shallow water and incompressible NavierStokes equations on staggered unstructured curved meshes. Isoparametric finite elements are used to take into account curved domain boundaries. Regarding twodimensional 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 edgebased staggered dual grid. Similarly, for the twodimensional incompressible NavierStokes case, the discrete pressure is defined on the main triangular grid and the velocity field is defined on the edgebased staggered grid. While staggered meshes are state of the art in classical finite difference approximations of the incompressible NavierStokes equations, their use in the context of high order DG schemes is novel and still quite rare. High order (better than second order) in time can be achieved by using a spacetime 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 fourpoint system in 2D and a block fivepoint system for 3D tetrahedral meshes. The resulting linear system is conveniently solved with a matrixfree GMRES algorithm. Note that the same spacetime DG scheme on a collocated grid would lead to ten nonzero blocks per element in 2D and seventeen nonzero blocks in 3D, since substituting the discrete velocity into the discrete continuity equation on a collocated mesh would involve also neighbors of neighbors. From numerical experiments we find that our linear system is wellbehaved and that the GMRES method converges quickly even without the use of any preconditioner, which is a unique feature in the context of high order implicit DG schemes. A very simple and efficient Picard iteration is then used in order to derive a spacetime pressure correction algorithm that achieves also high order of accuracy in time, which is in general a nontrivial task in the context of high order discretizations for the incompressible NavierStokes equations. 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 semidefiniteness in the general case. This allows us to use a very fast linear solver such as the conjugate gradient (CG) method. The flexibility and accuracy of high order spacetime 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 threedimensional incompressible NavierStokes system using a tetrahedral main grid and a corresponding facebased hexaxedral dual grid. The resulting dual mesh consists in nonstandard 5vertex 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 NavierStokes equations.
Arbitrary high order discontinuous Galerkin methods for the shallow water and incompressible NavierStokes equations on unstructured staggered meshes / Tavelli, Maurizio.  (2016), pp. 1210.
Arbitrary high order discontinuous Galerkin methods for the shallow water and incompressible NavierStokes equations on unstructured staggered meshes
Tavelli, Maurizio
20160101
Abstract
In this work we present a new class of wellbalanced, arbitrary high order accurate semiimplicit discontinuous Galerkin methods for the solution of the shallow water and incompressible NavierStokes equations on staggered unstructured curved meshes. Isoparametric finite elements are used to take into account curved domain boundaries. Regarding twodimensional 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 edgebased staggered dual grid. Similarly, for the twodimensional incompressible NavierStokes case, the discrete pressure is defined on the main triangular grid and the velocity field is defined on the edgebased staggered grid. While staggered meshes are state of the art in classical finite difference approximations of the incompressible NavierStokes equations, their use in the context of high order DG schemes is novel and still quite rare. High order (better than second order) in time can be achieved by using a spacetime 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 fourpoint system in 2D and a block fivepoint system for 3D tetrahedral meshes. The resulting linear system is conveniently solved with a matrixfree GMRES algorithm. Note that the same spacetime DG scheme on a collocated grid would lead to ten nonzero blocks per element in 2D and seventeen nonzero blocks in 3D, since substituting the discrete velocity into the discrete continuity equation on a collocated mesh would involve also neighbors of neighbors. From numerical experiments we find that our linear system is wellbehaved and that the GMRES method converges quickly even without the use of any preconditioner, which is a unique feature in the context of high order implicit DG schemes. A very simple and efficient Picard iteration is then used in order to derive a spacetime pressure correction algorithm that achieves also high order of accuracy in time, which is in general a nontrivial task in the context of high order discretizations for the incompressible NavierStokes equations. 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 semidefiniteness in the general case. This allows us to use a very fast linear solver such as the conjugate gradient (CG) method. The flexibility and accuracy of high order spacetime 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 threedimensional incompressible NavierStokes system using a tetrahedral main grid and a corresponding facebased hexaxedral dual grid. The resulting dual mesh consists in nonstandard 5vertex 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 NavierStokes equations.File  Dimensione  Formato  

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