Fast high order finite volume schemes for linear wave propagation, Numerische Mathematik

We present a new numerical method to solve the heterogeneous anelastic seismic wave equa-tions with arbitrary high order of accuracy in space and time on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. Using the velocity-stress formulation provides a linear hyperbolic system of equations with source terms that is com-pleted by additional equations for the anelastic functions including the strain history of the ma-terial. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material ac-counting for the behaviour of elastic solids and viscous ßuids. The proposed method relies on the Finite Volume (FV) approach where cell-averaged quantities are evolved in time by com-puting numerical ßuxes at the element interfaces. The basic ingredient of the numerical ßux function is the solution of Generalized Riemann Problems at the element interfaces according to the ADER approach of Toro et al., where the initial data is piecewise polynomial instead of piecewise constant as it was in the original Þrst order FV scheme developed by Godunov. The ADER approach automatically produces a scheme of uniform high order of accuracy in space and time. The high order polynomials in space, needed as input for the numerical ßux function