An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - IV. Anisotropy

• Published in: Geophysical journal international Vol. 169; no. 3; pp. 1210 - 1228
• Main Authors: de la Puente, Josep, Käser, Martin, Dumbser, Michael, Igel, Heiner
• Format: Journal Article
• Language: English
• Published: Oxford, UK Blackwell Publishing Ltd 01.06.2007
• Subjects: ADER approach; anisotropy; discontinuous Galerkin method; high-order accuracy in space and time; unstructured tetrahedral meshes
• ISSN: 0956-540X; 1365-246X
• DOI: 10.1111/j.1365-246X.2007.03381.x

We present a new numerical method to solve the heterogeneous elastic anisotropic wave equation with arbitrary high-order accuracy in space and time on unstructured tetrahedral meshes. Using the most general Hooke's tensor we derive the velocity-stress formulation leading to a linear hyperbolic system which accounts for the variation of the material properties depending on direction. This approach allows for the accurate modelling even of the most general crystalline symmetry class, the triclinic anisotropy, as no interpolation of material properties to particular mesh vertices is necessary. The proposed method combines the Discontinuous Galerkin method with the arbitrary high-order derivatives (ADER) time integration approach using arbitrary high-order derivatives of the piecewise polynomial representation of the unknown solution. The discontinuities of this piecewise polynomial approximation at element interfaces permit the application of the well-established theory of finite volumes and numerical fluxes across element interfaces obtained by the solution of derivative Riemann problems. Due to the novel ADER time integration technique the scheme provides the same approximation order in space and time automatically. A numerical convergence study confirms that the new scheme achieves the desired arbitrary high-order accuracy even for anisotropic material on unstructured tetrahedral meshes. Furthermore, it shows that higher accuracy can be reached with higher-order schemes while reducing computational cost and storage space. To this end, we also present a new Godunov-type numerical flux for anisotropic material and compare its accuracy with a computationally simpler Rusanov flux. As a further extension, we include the coupling of anisotropy and viscoelastic attenuation based on the Generalized Maxwell Body rheology and the mean and deviatoric stress concepts. Finally, we validate the new scheme by comparing the results of our simulations to an analytic solution as well as to spectral element computations.