Abstract
This research work presents some comparisons and analyses of the time discontinuous space–time Galerkin method and the space discontinuous Galerkin method applied to elastic wave propagation in anisotropic and heterogeneous media. Mechanism of both methods to ensure their stability using time or space discontinuities of unknown fields is analyzed and compared. The most general case of anisotropic and heterogeneous media with physical interfaces of discontinuous material properties is considered, especially for the space discontinuous Galerkin method. A new stability result is proved. Numerical applications to different elastic media, more particularly polycrystalline materials containing a large number of physical interfaces, are also presented to confirm theoretical analyses.
Highlights
Nowadays, as a classical problem, the numerical modeling of elastic wave propagation can be done reliably and efficiently in a large number of cases
Contrary to the stability of the time discontinuous Galerkin (dG) method that is unconditional whatever the space and time discretizations used, the stability of the space dG method is proved for its variational formulation discretized in space but not yet discretized in time
When a time discretization scheme is applied, its own stability condition should be taken into account
Summary
As a classical problem, the numerical modeling of elastic wave propagation can be done reliably and efficiently in a large number of cases. Wilcox et al has proved the stability of the velocity–strain dG method in 2D isotropic elastic media including physical interfaces Their result shows that the energies dissipated in the jumps in space of the velocity and strains fields guarantee the stability for the velocity–strain dG formulation that is not yet discretized in time [15]. The exact upwind numerical fluxes developed in [17,18] are given before the presentation of the main result of the present work: the stability of the space dG method in the general case of anisotropic heterogeneous media with physical interfaces of discontinuous material properties. Even when the stability condition is satisfied by the time discretization method, if the stability is not proved at the continuous level for the space dG variational formulation, typically in the case of the use of approximated numerical fluxes, the stability of discretized solutions cannot be guaranteed. Quantitative comparison of the damping behavior between both space and dG methods would be performed
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