In this work the numerical discretization of the partial differential governing equations for compressible and incompressible flows is dealt within the discontinuous Galerkin (DG) framework along space–time adaptive meshes. Two main numerical frameworks can be distinguished: (1) fully explicit ADERDG methods on collocated grids for compressible fluids (2) spectral semiimplicit and spectral space–time DG methods on edgebased staggered grids for the incompressible Navier–Stokes equations. In this work, the highresolution properties of the aforementioned numerical methods are significantly enhanced within a 'cellbycell' Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS). It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the socalled 'Gibbs phenomenon'. Over the years, several attempts have been made to cope with this problem and different kinds of limiters have been proposed. In this work the nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a multidimensional optimal order detection (MOOD) criterion. ADERDG is a novel, communicationavoiding family of algorithms, which achieves high order of accuracy in time not via the standard multistage Runge–Kutta (RK) time discretization like most other DG schemes, but at the aid of an elementlocal predictor stage. In practice the method first produces a socalled candidate solution by using a high order accurate unlimited DG scheme. Then, in those cells where at least one of the chosen admissibility criteria is violated the computed candidate solution is detected as troubled and is locally rejected. Next, the numerical solution of the previous time step is scattered onto cell averages on a suitable subgrid in order to preserve the natural subcell resolution of the DG scheme. Then, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADERWENO finite volume scheme on the subgrid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved subcell averages. In the ADERDG framework several PDE system are investigated, ranging from the Euler equations of compressible gas dynamics, over the viscous and resistive magnetohydrodynamics (MHD), to special and general relativistic MHD. Indeed, the adopted formalism is quite general, leading to a novel family of adaptive ADERDG schemes suitable for hyperbolic systems of partial differential equations in which the numerical fluxes also depend on the gradient of the state vector because of the parabolic nature of diffusive terms. The presented results show clearly that the highresolution and shockcapturing capabilities of the news schemes are significantly enhanced within the cellbycell AMR implementation together with time accurate LTS. A special treatment has been followed for the incompressible Navier–Stokes equations. In fact, the elliptic character of the incompressible Navier–Stokes equations introduces an important difficulty in their numerical solution: whenever the smallest physical or numerical perturbation arises in the fluid flow then it will instantaneously affect the entire computational domain. Thus, a semiimplicit approach has been used. The main advantage of making use of a semiimplicit discretization is that the numerical stability can be obtained for large timesteps without leading to an excessive computational demand. In this context, we derived two new families of spectral semiimplicit and spectral space–time DG methods for the solution of the two and three dimensional Navier–Stokes equations on edgebased adaptive staggered Cartesian grids. The discrete solutions of pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on edgebased dual control volumes, leading to a spatially staggered mesh. In the first family, high order of accuracy is achieved only in space, while a simple semiimplicit time discretization is derived by introducing an implicitness factor θ∈ [0.5 , 1] for the pressure gradient in the momentum equation. The real advantages of the staggering arise after substituting the discrete momentum equation into the weak form of the continuity equation. In fact, the resulting linear system for the pressure is symmetric and positive definite and either block pentadiagonal (in 2D) or block heptadiagonal (in 3D). As a consequence, the pressure system can be solved very efficiently by means of a classical matrixfree conjugate gradient method. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary high order accurate in both space and time. This new numerical method has been thoroughly validated for approximation polynomials of degree up to N= 12 , using a large set of nontrivial test problems in two and three space dimensions, for which either analytical, numerical or experimental reference solutions exist.
Discontinuous Galerkin Methods for Compressible and Incompressible Flows on Space–Time Adaptive Meshes: Toward a Novel Family of Efficient Numerical Methods for Fluid Dynamics / Fambri, F..  In: ARCHIVES OF COMPUTATIONAL METHODS IN ENGINEERING.  ISSN 11343060.  27:1(2020), pp. 199283. [10.1007/s11831018093086]
Discontinuous Galerkin Methods for Compressible and Incompressible Flows on Space–Time Adaptive Meshes: Toward a Novel Family of Efficient Numerical Methods for Fluid Dynamics
Fambri F.
20200101
Abstract
In this work the numerical discretization of the partial differential governing equations for compressible and incompressible flows is dealt within the discontinuous Galerkin (DG) framework along space–time adaptive meshes. Two main numerical frameworks can be distinguished: (1) fully explicit ADERDG methods on collocated grids for compressible fluids (2) spectral semiimplicit and spectral space–time DG methods on edgebased staggered grids for the incompressible Navier–Stokes equations. In this work, the highresolution properties of the aforementioned numerical methods are significantly enhanced within a 'cellbycell' Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS). It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the socalled 'Gibbs phenomenon'. Over the years, several attempts have been made to cope with this problem and different kinds of limiters have been proposed. In this work the nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a multidimensional optimal order detection (MOOD) criterion. ADERDG is a novel, communicationavoiding family of algorithms, which achieves high order of accuracy in time not via the standard multistage Runge–Kutta (RK) time discretization like most other DG schemes, but at the aid of an elementlocal predictor stage. In practice the method first produces a socalled candidate solution by using a high order accurate unlimited DG scheme. Then, in those cells where at least one of the chosen admissibility criteria is violated the computed candidate solution is detected as troubled and is locally rejected. Next, the numerical solution of the previous time step is scattered onto cell averages on a suitable subgrid in order to preserve the natural subcell resolution of the DG scheme. Then, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADERWENO finite volume scheme on the subgrid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved subcell averages. In the ADERDG framework several PDE system are investigated, ranging from the Euler equations of compressible gas dynamics, over the viscous and resistive magnetohydrodynamics (MHD), to special and general relativistic MHD. Indeed, the adopted formalism is quite general, leading to a novel family of adaptive ADERDG schemes suitable for hyperbolic systems of partial differential equations in which the numerical fluxes also depend on the gradient of the state vector because of the parabolic nature of diffusive terms. The presented results show clearly that the highresolution and shockcapturing capabilities of the news schemes are significantly enhanced within the cellbycell AMR implementation together with time accurate LTS. A special treatment has been followed for the incompressible Navier–Stokes equations. In fact, the elliptic character of the incompressible Navier–Stokes equations introduces an important difficulty in their numerical solution: whenever the smallest physical or numerical perturbation arises in the fluid flow then it will instantaneously affect the entire computational domain. Thus, a semiimplicit approach has been used. The main advantage of making use of a semiimplicit discretization is that the numerical stability can be obtained for large timesteps without leading to an excessive computational demand. In this context, we derived two new families of spectral semiimplicit and spectral space–time DG methods for the solution of the two and three dimensional Navier–Stokes equations on edgebased adaptive staggered Cartesian grids. The discrete solutions of pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on edgebased dual control volumes, leading to a spatially staggered mesh. In the first family, high order of accuracy is achieved only in space, while a simple semiimplicit time discretization is derived by introducing an implicitness factor θ∈ [0.5 , 1] for the pressure gradient in the momentum equation. The real advantages of the staggering arise after substituting the discrete momentum equation into the weak form of the continuity equation. In fact, the resulting linear system for the pressure is symmetric and positive definite and either block pentadiagonal (in 2D) or block heptadiagonal (in 3D). As a consequence, the pressure system can be solved very efficiently by means of a classical matrixfree conjugate gradient method. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary high order accurate in both space and time. This new numerical method has been thoroughly validated for approximation polynomials of degree up to N= 12 , using a large set of nontrivial test problems in two and three space dimensions, for which either analytical, numerical or experimental reference solutions exist.File  Dimensione  Formato  

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