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Peano
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Peano
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ExaSeis is a dynamically adaptive discontinuous Galerkin framework for the simulation of seismic wave propagation and earthquakes. It targets the scattering of linear elastic waves in heterogeneous, isotropic and anisotropic Earth models with complex free-surface topography, while avoiding traditional mesh-generation pipelines.
The key idea of ExaSeis is a curvilinear formulation of the elastic wave equation. Instead of solving directly in Cartesian coordinates \((x,y,z)\), ExaSeis introduces computational coordinates \((q, r, s) \in \tilde{Omega} = [0,1] \times [0,1] \times [0,1] \), and stores a smooth mapping between the computational and the physical domain.
\[ (x (q, r, s) , y (q, r, s) , z (q, r, s)) \leftrightarrow (q (x, y, z) , r (x, y, z) , s (x, y, z)) \]
This allows ExaSeis to represent complicated geometries (e.g. mountain ranges in the Zugspitze setup) on topologically simple meshes: the computational grid remains Cartesian, while the curvilinear mapping conforms to the physical free surface.
On this curvilinear mesh, we consider the elastic equation of motion and split the spatial operator into conservative fluxes and non-conservative products,
\[ \textbf{Flux}(\bar{Q}) := \sum_\mathbf{\xi = q,r,s} H_\xi^{-1}(\textbf{e}_\mathbf{\xi}(0)[\textbf{J}\sqrt{\xi_x^2 + \xi_y^2 + \xi_z^2}\textbf{FL}(\bar{Q}(t))]|_{\mathbf{\xi}=0} + \textbf{e}_\mathbf{\xi}(1)[\textbf{J}\sqrt{\xi_x^2 + \xi_y^2 + \xi_z^2}\textbf{FR}(\bar{Q}(t))]|_{\mathbf{\xi}=1} ) \]
where the antisymmetric split form is useful for designing provably stable discretisations on curvilinear meshes with complex geometries and strongly varying material properties. While the ExaHyPE engine provides a Rusanov flux as a default numerical flux, ExaSeis replaces it with a metric-aware numerical flux that incorporates the Jacobian and metric terms arising from the curvilinear transformation in order to correctly treat free surfaces and material interfaces.
Beyond the curvilinear geometry, ExaSeis provides two major building blocks for computational seismology:
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