On the stability of the space–time discontinuous Galerkin method for the numerical solution of nonstationary nonlinear convection–diffusion problems

Balázsová, Monika; Feistauer, Miloslav; Hadrava, Martin; Kosík, Adam

doi:10.1515/jnma-2015-0014

Cited by 11 publications

(6 citation statements)

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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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“…In paper [10], the theory of the STDGM was developed for the case with nonlinear convection as well as diffusion. The paper [4] is a continuation of the works [26] and [10] devoted to proving unconditional stability of the STDGM. In all the above mentioned theoretical papers, the space domain is independent of time.…”

Section: Introductionmentioning

confidence: 99%

“…The presented paper represents the generalization of results from [4] to the STDGM for the numerical solution of a nonstationary nonlinear convection-diffusion problem in a time-dependent domain, formulated with the aid of the ALE method.…”

Section: Introductionmentioning

confidence: 99%

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

Balázsová

Feistauer

2015

Appl Math

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“…This method is widely applied in solving differential equations, providing a powerful numerical solution to engineering problems [9,10,11,12], including studies on concrete structures to predict the onset time of corrosion of reinforcements [13] and for simulating dynamic fracture in concrete [14].…”

Section: Introductionmentioning

confidence: 99%

Parallel approach of a Galerkin-based methodology for predicting the compressive strength of the lightweight aggregate concrete

Migallón

Navarro-González

Penadés

et al. 2019

Construction and Building Materials

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A methodology based on the Galerkin formulation of the finite element method has been analyzed for predicting the compressive strength of the lightweight aggregate concrete using ultrasonic pulse velocity. Due to both the memory requirements and the computational cost of this technique, its parallelization becomes necessary for solving this problem. For this purpose a mixed MPI/OpenMP parallel algorithm has been designed and different approaches and data distributions analyzed. On the other hand, this Galerkin methodology has been compared with multiple linear regression models, regression trees and artificial neural networks. Based on different measures of goodness of fit, the effectiveness of the Galerkin methodology, compared with these statistical techniques for data mining, is shown.

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On the stability of the space–time discontinuous Galerkin method for the numerical solution of nonstationary nonlinear convection–diffusion problems

Cited by 11 publications

References 0 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

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

Parallel approach of a Galerkin-based methodology for predicting the compressive strength of the lightweight aggregate concrete

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