Seismic Wave Propagation in Media with Complex Geometries, Simulation of

Glossary Numerical methods Processes in nature are often described by partial differential equations. Finding solutions to those equations is at the heart of many studies aiming at the explanation of observed data. Simulations of realistic physical processes requires generally the use of numerical methods – a special branch of applied mathematics – that approximate the partial differential equations and allows solving them on computers. Examples are the finite-difference, finite-element, or finite-volume methods. Spectral elements The spectral element method is an extension of the finite element method that makes use of specific basis functions describing the solutions inside each element. These basis functions (e. g., Cheby-shev or Legendre polynomials) allow the interpolation of functions exactly at certain collocation points. This is often termed spectral accuracy. Discontinuous Galerkin method The discontinuous Ga-lerkin method is a flavor of the finite-element method that allows discontinuous behavior of the spatial or temporal fields at the element boundaries. The discon-tinuities – that might be small in the case of continuous physical fields such as seismic waves – then define so-called Riemann problems that can be handled using the concepts from finite-volume techniques. Therefore , the approximate solution is updated via numerical fluxes across the element boundaries. Parallel algorithms All modern supercomputers make use of parallel architectures. This means that a large number of processors are performing (different) tasks on different data at the same time. Numerical algorithms need to be adapted to these hardware architec-tures by using specific programming paradigms (e. g., the message passing interface MPI). The computational efficiency of such algorithms strongly depends on the specific parallel nature of the problem to be solved, and the requirement for inter-processor communication. Grid generation Most numerical methods are based on the calculation of the solutions at a large set of points (grids) that are either static or depend on time (adap-tive grids). These grids often need to be adapted to the specific geometrical properties of the objects to be modeled (volcano, reservoir, globe). Grids may be designed to follow domain boundaries and internal surfaces. Before specific numerical solvers are employed the grid points are usually connected to form triangles or rectangles in 2D or hexahedra or tetrahedra in 3D. Seismology is the science that aims at understanding the Earth's interior and its seismic sources from measurements of vibrations of the solid Earth. The resulting images of the physical properties of internal structures and the spatio-temporal behavior …