Exact Riemann solver of the discontinuous Galerkin method for seismic wave propagation in anisotropic media with anisotropic fractures

We formulated and performed an exact Riemann solution of the discontinuous Galerkin method (DGM) for seismic propagation simulation in an anisotropic background medium with a general anisotropic fracture. An orthotropic background medium was considered, a combination of the Riemann flux and DGM was described in detail, and a linear slip (LS) model was incorporated into the DGM. A fracture is considered a geometrical interface across the element on which the LS boundary condition is imposed. Thus, we constructed an exact Riemann solver with general anisotropic fractures and derived an analytical Riemann solution with orthotropic cracks. The analytical Riemann solver avoids the numerical errors caused by time discretization at the fracture interfaces. The analytical Riemann flux of the orthotropic fractures was a function of time and was not constant. The proposed approach allows for anisotropy of both the background model and crack, permits discontinuities in the displacement, and is suitable for arbitrarily irregular fracture structures. Using three numerical examples, the results demonstrated the correctness and validity of our new method and revealed its advantageous performance for complex crack structures.