Attenuation effects in wave propagation are commonly modeled using visco-acoustic formulations based on the Generalized Standard Linear Solid (GSLS) framework. The accuracy of these models depends on the number of relaxation mechanisms used to approximate the desired attenuation behavior. However, each additional mechanism introduces new state variables and increases the computational cost. Existing approaches typically employ a fixed number of relaxation mechanisms throughout the computational domain, regardless of the local complexity of the wave field.
In this work, we develop a model-adaptive strategy for visco-acoustic wave propagation described by the equations $$ \varrho(x) \partial_{t}v(t,x) = \nabla p(t,{x}) + {f}(t,x) $$ $$\partial_{t} p(t,{x}) = \kappa(t,{x}) \nabla \cdot {v}(t,{x}) + \int_{0}^{t} \dot{\kappa}(t-s,{x})\, \nabla \cdot {v}(s,{x})\,\mathrm{d}s $$ with the variables velocity $v$, pressure $p$ and material parameters $\kappa$, $\varrho$ and retardation kernel $\dot{\kappa}$. In order to approximate the delayed material law of the medium we employ the GSLS model [1].
The central idea is to adjust the number of relaxation mechanisms locally in space and time according to the evolving solution. Regions not yet affected by wave propagation, or exhibiting simple wave behavior, are represented using a reduced model with fewer mechanisms. In contrast, regions containing strong heterogeneities or pronounced attenuation effects are enriched with additional relaxation mechanisms to maintain physical accuracy. This adaptive enrichment and coarsening strategy allows the computational effort to be concentrated where it is most beneficial.
A particular challenge is the presence of two independent error sources: the model error introduced by the attenuation model approximation and the discretization error arising from the numerical scheme. Reducing only one of these contributions does not necessarily improve the overall solution quality. We therefore investigate adaptive procedures that simultaneously control both errors through local refinement of the discretization and local adjustment of the attenuation model complexity. Error indicators are used to guide these decisions and to balance accuracy and efficiency.
The resulting adaptive algorithms are embedded in a high-order space-time discontinuous Galerkin (DG) discretization [2], enabling local refinement in both the numerical approximation and the attenuation model. Numerical experiments on heterogeneous benchmark problems, including the Marmousi model, demonstrate the potential of the proposed approach to substantially reduce computational costs while maintaining the accuracy required for realistic large-scale wave propagation simulations.
[1] J.O. Blanch, J.O.A. Robertsson, W.W. Symes: Modeling of a constant-Q methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique. Geophysics 1995.
[2] D.A. Ziegler: A Parallel and Adaptive Space-Time Discontinuous Galerkin Method for Visco-Elastic and Visco-Acoustic Waves. PhD thesis Karlsruhe 2020.