On the stability of the ale space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains

Balázsová, Monika; Feistauer, Miloslav

doi:10.1007/s10492-015-0109-3

Cited by 9 publications

(11 citation statements)

References 41 publications

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“…For this purpose, several high order numerical schemes on unstructured meshes were introduced in the past. A series of explicit high order discontinuous Galerkin (DG) schemes for elastic wave propagation on unstructured meshes was proposed in [34,35,36,37,38,39], while the concept of space-time discontinuous Galerkin schemes, originally introduced and analyzed in [40,41,42,43,44,45,46] for computational fluid dynamics (CFD), was later also extended to linear elasticity in [47,48,49]. The space-time DG method used in [49] is based on the novel concept of staggered discontinuous Galerkin finite element schemes, which was introduced for CFD problems in [50,51,52,53,53,54,55,56].…”

Section: Introductionmentioning

confidence: 99%

A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography

Tavelli

Dumbser

Charrier

et al. 2019

Journal of Computational Physics

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In most classical approaches of computational geophysics for seismic wave propagation problems, complex surface topography is either accounted for by boundary-fitted unstructured meshes, or, where possible, by mapping the complex computational domain from physical space to a topologically simple domain in a reference coordinate system. However, all these conventional approaches face problems if the geometry of the problem becomes sufficiently complex. They either need a mesh generator to create unstructured boundary-fitted grids, which can become quite difficult and may require a lot of manual user interactions in order to obtain a high quality mesh, or they need the explicit computation of an appropriate mapping function from physical to reference coordinates. For sufficiently complex geometries such mappings may either not exist or their Jacobian could become close to singular. Furthermore, in both conventional approaches low quality grids will always lead to very small time steps due to the Courant-Friedrichs-Lewy (CFL) condition for explicit schemes. In this paper, we propose a completely different strategy that follows the ideas of the successful family of high resolution shock-capturing schemes, where discontinuities can actually be resolved anywhere on the grid, without having to fit them exactly. We address the problem of geometrically complex free surface boundary conditions for seismic wave propagation problems with a novel diffuse interface method (DIM) on adaptive Cartesian meshes (AMR) that consists in the introduction of a characteristic function 0 ≤ α ≤ 1 which identifies the location of the solid medium and the surrounding air (or vacuum) and thus implicitly defines the location of the free surface boundary. Physically, α represents the volume fraction of the solid medium present in a control volume. Our new approach completely avoids the problem of mesh generation, since all that is needed for the definition of the complex surface topography is to set a scalar color function to unity inside the regions covered by the solid and to zero outside. The governing equations are derived from ideas typically used in the mathematical description of compressible multiphase flows. An analysis of the eigenvalues of the PDE system shows that the complexity of the geometry has no influence on the admissible time step size due to the CFL condition. The model reduces to the classical linear elasticity equations inside the solid medium where the gradients of α are zero, while in the diffuse interface zone at the free surface boundary the governing PDE system becomes nonlinear. We can prove that the solution of the Riemann problem with arbitrary data and a jump in α from unity to zero yields a Godunov-state at the interface that satisfies the free-surface boundary condition exactly, i.e. the normal stress components vanish. In the general case of an interface that is not aligned with the grid and which is not infinitely thin, a systematic study on the distribution of the volume fraction function inside the interfac...

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Section: Introductionmentioning

confidence: 99%

A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography

Tavelli

Dumbser

Charrier

et al. 2019

Journal of Computational Physics

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“…Proof. An analogical result is proved in [2]. Since the proof is long and technical, it is skipped in this paper.…”

Section: Discrete Characteristic Functionmentioning

confidence: 99%

“…Proof. The first inequality in (14) is just a modification of the similar result from [2], where the quadrature Q m is replaced by integral over I m . The proof is analogical.…”

Section: Stability Analysismentioning

confidence: 99%

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Stability of ALE discontinuous Galerkin method with Radau quadrature

Vlasák¹

2019

Programs and Algorithms of Numerical Mathematics 19

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“…The concept of space-time DG schemes was introduced by van der Vegt et al for computational fluid dynamics in [63,64,65,66,67] and has been subsequently analyzed e.g. in [68,69]. The first application of space-time DG schemes to elastodynamics on collocated grids has been reported in [70,71], but to the best of our knowledge there exists no space-time DG scheme for the linear elastic wave equations on staggered grids so far.…”

mentioning

confidence: 99%

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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On the stability of the ale space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains

Cited by 9 publications

References 41 publications

A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography

A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography

Stability of ALE discontinuous Galerkin method with Radau quadrature

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

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