A space-time discontinuous Galerkin method for Boussinesq-type equations

Dumbser, Michael; Facchini, M.

doi:10.1016/j.amc.2015.06.052

Cited by 22 publications

(26 citation statements)

References 38 publications

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“…In this paper, we concentrate our attention on compressible viscous Newtonian fluids, which in the classical continuum theory can be described by the hyperbolic-parabolic Navier-Stokes-Fourier (NSF) theory, as well as on elastic solids. It should also be noted that there are several advantages of a first order hyperbolic formulation of viscous fluids: first, the use of explicit Godunov-type shock-capturing finite volume schemes and, even more, the use of high order discontinuous Galerkin finite element methods is -at least in principlestraightforward for first order systems, while DG schemes need some special care in the presence of parabolic and higher order derivative terms, see the very interesting discussions in the well-known papers of Bassi & Rebay [10], Baumann & Oden [11,12], Cockburn and Shu [30,31], Yan and Shu [140,141,89] and others [1,77,78,83,28,60,43]. Second, the use of a parabolic theory can lead to a severe time step size restriction, if explicit time stepping schemes are used, since the infinite propagation speed of perturbations that is intrinsically inherent in parabolic PDEs is reflected in explicit numerical methods by a stability condition on the time step that scales with the square of the mesh size, while it scales only linearly with the mesh size for first order hyperbolic systems due to the classical CFL condition [33].…”

Section: High Order Ader-weno Finite Volume and Ader Discontinuous Gamentioning

confidence: 99%

High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conducting fluids and elastic solids

Dumbser

Peshkov

Romenski

et al. 2016

Journal of Computational Physics

175

417

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This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics recently proposed by Peshkov & Romenski [106], further denoted as HPR model. In that framework, the viscous stresses are computed from the so-called distortion tensor A, which is one of the primary state variables in the proposed first order system. A very important key feature of the HPR model is its ability to describe at the same time the behavior of inviscid and viscous compressible Newtonian and non-Newtonian fluids with heat conduction, as well as the behavior of elastic and visco-plastic solids. Actually, the model treats viscous and inviscid fluids as generalized visco-plastic solids. This is achieved via a stiff source term that accounts for strain relaxation in the evolution equations of A. Also heat conduction is included via a first order hyperbolic evolution equation of the thermal impulse, from which the heat flux is computed. The governing PDE system is hyperbolic and fully consistent with the first and the second principle of thermodynamics. It is also fundamentally different from first order Maxwell-Cattaneo-type relaxation models based on extended irreversible thermodynamics. The HPR model represents therefore a novel and unified description of continuum mechanics, which applies at the same time to fluid mechanics and solid mechanics. In this paper, the direct connection between the HPR model and the classical hyperbolic-parabolic Navier-Stokes-Fourier theory is established for the first time via a formal asymptotic analysis in the stiff relaxation limit.From a numerical point of view, the governing partial differential equations are very challenging, since they form a large nonlinear hyperbolic PDE system that includes stiff source terms and non-conservative products. We apply the successful family of one-step ADER-WENO finite volume (FV) and ADER discontinuous Galerkin (DG) finite element schemes to the HPR model in the stiff relaxation limit, and compare the numerical results with exact or numerical reference solutions obtained for the Euler and Navier-Stokes equations. Numerical convergence results are also provided. To show the universality of the HPR model, the paper is rounded-off with an application to wave propagation in elastic solids, for which one only needs to switch off the strain relaxation source term in the governing PDE system.We provide various examples showing that for the purpose of flow visualization, the distortion tensor A seems to be particularly useful.Key words: ADER-WENO finite volume schemes, arbitrary high-order Discontinuous Galerkin schemes, path-conservative methods and stiff source terms, unified first order hyperbolic formulation of nonlinear continuum mechanics, fluid mechanics and solid mechanics, viscous compressible fluids and elastic solids * Corresponding author * * Ilya Peshkov is on leave from Sobolev

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Section: High Order Ader-weno Finite Volume and Ader Discontinuous Gamentioning

confidence: 99%

High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conducting fluids and elastic solids

Dumbser

Peshkov

Romenski

et al. 2016

Journal of Computational Physics

175

417

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“…We show now similar results for the fully discrete forms. The first result can be seen as a simple extension of the previous theorem using the ideas presented in [80,37].…”

Section: Stability Analysismentioning

confidence: 75%

Arbitrary high order accurate space–time discontinuous Galerkin finite element schemes on staggered unstructured meshes for linear elasticity

Tavelli

Dumbser

2018

Journal of Computational Physics

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In this paper we propose a new high order accurate space-time discontinuous Galerkin (DG) finite element scheme for the solution of the linear elastic wave equations in first order velocity-stress formulation in two and three-space dimensions on staggered unstructured triangular and tetrahedral meshes. The method reaches arbitrary high order of accuracy in both space and time via the use of space-time basis and test functions. Within the staggered mesh formulation, we define the discrete velocity field in the control volumes of a primary mesh, while the discrete stress tensor is defined on a face-based staggered dual mesh. The space-time DG formulation leads to an implicit scheme that requires the solution of a linear system for the unknown degrees of freedom at the new time level. The number of unknowns is reduced at the aid of the Schur complement, so that in the end only a linear system for the degrees of freedom of the velocity field needs to be solved, rather than a system that involves both stress and velocity. Thanks to the use of a spatially staggered mesh, the stencil of the final velocity system involves only the element and its direct neighbors and the linear system can be efficiently solved via matrix-free iterative methods. Despite the necessity to solve a linear system, the numerical scheme is still computationally efficient. The chosen discretization and the linear nature of the governing PDE system lead to an unconditionally stable scheme, which allows large time steps even for low quality meshes that contain so-called sliver elements. The fully discrete staggered space-time DG method is proven to be energy stable for any order of accuracy, for any mesh and for any time step size. For the particular case of a simple Crank-Nicolson time discretization and homogeneous material, the final velocity system can be proven to be symmetric and positive definite and in this case the scheme is also exactly energy preserving. The new scheme is applied to several test problems in two and three space dimensions, providing also a comparison with high order explicit ADER-DG schemes.Keywords: high order schemes, space-time discontinuous Galerkin methods, staggered unstructured meshes, energy stability, large time steps, linear elasticityAnother challenge is the discretization of complex geometries including both, complex surface topography as well as complex sub-surface fault structures. In this case, the use of unstructured simplex meshes composed of triangles or tetrahedra seems to be beneficial concerning the problem of mesh generation in complex geometries. Concerning high order explicit discontinuous Galerkin (DG) finite element schemes for linear elastic wave propagation on unstructured general meshes the reader is referred to [24,25,26,27,28] and to [29,30,31]. However, since the previous methods are explicit, they are only stable under a CFL-type stability condition on the time step that depends on the mesh quality as well as the polynomial approximation degree used. In particular, unstructured simplex mes...

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“…This is general knowledge, mathematically proven in [105], but the importance is accenturated in wave equation where the numerical dispersion error must be kept small compared to the physical dispersion terms. Additional spectral/hp studies of weakly dispersive Boussinesq equations are [106,107]. Engsig-Karup et al [108,109,110] solved a set highly dispersive and nonlinear Boussinesq-type equations.…”

Section: Fully Nonlinear Potential Flowmentioning

confidence: 99%

Spectral/hp element methods: Recent developments, applications, and perspectives

Cantwell

Monteserin

et al. 2018

J Hydrodyn

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Abstract:The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate a 0 -C continuous expansion. Computationally and theoretically, by increasing the polynomial order p , high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.

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A space-time discontinuous Galerkin method for Boussinesq-type equations

Cited by 22 publications

References 38 publications

High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conducting fluids and elastic solids

High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conducting fluids and elastic solids

Arbitrary high order accurate space–time discontinuous Galerkin finite element schemes on staggered unstructured meshes for linear elasticity

Spectral/hp element methods: Recent developments, applications, and perspectives

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