Abstract Life. Cauchy’s father, Louis-François Cauchy, was born in Rouen in 1760. A brilliant student of classics at Paris University, after graduating he established himself as a barrister at the parlement of Normandy. At the age of twenty-three he became secretary general to Thiroux de Crosnes, the intendant of Haute Normandie. Two years later he followed Thiroux to Paris, where the latter had been appointed to the high office of lieutenant de police . Louis-François gradually advanced to high administrative positions, such as that of first secretary to the Senate. He died in 1848. In 1787 he married Marie-Madeleine Desestre, who bore him four sons and two daughters. She died in 1839. Of their daughters, Thérèse died young and Adèle married her cousin G. de Neuburg. She died in 1863. The youngest son, Amédée, died in 1831, at the age of twenty-five; Alexandre (1792–1857) held high judicial posts; and Eugène (1802–1877) held administrative posts and became known as a scholar in the history of law. Augustin was the eldest child. Cauchy enjoyed an excellent education; his father was his first teacher. During the Terror the family escaped to the village of Arcueil, where they were neighbors of Laplace and Berthollet, the founders of the celebrated Société d’Arcueil. Thus, as a young boy Augustin became acquainted with famous scientists. Lagrange is said to have forecast his scientific genius while warning his father against showing him a mathematical text before the age of seventeen.

Abstract We derive a rigorous analytic description of multiple reflection/transmission phenomena of 3D scalar harmonic wave fields in two inhomogeneous half spaces separated by a smooth curvilinear interface of arbitrary shape. The resulting wave field is expressed in terms of a branching sequence of events connected by recurrence relationships. An individual event is a result of the perturbation of boundary conditions by the previous event of the sequence that, in turn, again perturbs the boundary conditions. Such an event is described by the surface Helmholtz integral of any fundamental solution satisfying conditions at infinity. Functions of the current point of the interface and their normal derivatives (Cauchy data) are described by the surface integrals with the fundamental solutions for an absolutely absorbing interface. The Cauchy data of the latter integrals are obtained from the subsequent transformations. The perturbation of the interface and its normal derivatives are mapped into the spatial frequency domain by means of the Fourier transform where they are multiplied by the corresponding reflection/transmission coefficients and then is mapped back to an individual point on the interface by means of the inverse Fourier transform. Parameters of these transformations depend on the point where the result is computed. Therefore the computation at a new point of the interface requires the repetition of these transformations.

Abstract We present a new numerical method to solve the heterogeneous anelastic, seismic wave equations with arbitrary high order accuracy in space and time on 3-D unstructured tetrahedral meshes. Using the velocity–stress formulation provides a linear hyperbolic system of equations with source terms that is completed by additional equations for the anelastic functions including the strain history of the material. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material accounting for the behaviour of elastic solids and viscous fluids. The proposed method combines the Discontinuous Galerkin (DG) finite element (FE) method with the ADER approach using Arbitrary high order DERivatives for flux calculations. The DG approach, in contrast to classical FE methods, uses a piecewise polynomial approximation of the numerical solution which allows for discontinuities at element interfaces. Therefore, the well-established theory of numerical fluxes across element interfaces obtained by the solution of Riemann problems can be applied as in the finite volume framework. The main idea of the ADER time integration approach is a Taylor expansionin time in which all time derivatives are replaced by space derivatives using the so-called Cauchy–Kovalewski procedure which makes extensive use of the governing PDE. Due to the ADER time integration technique the same approximation order in space and time is achieved automatically and the method is a one-step scheme advancing the solution for one time step without intermediate stages. To this end, we introduce a new unrolled recursive algorithm for efficiently computing the Cauchy–Kovalewski procedure by making use of the sparsity of the system matrices. The numerical convergence analysis demonstrates that the new schemes provide very high order accuracy even on unstructured tetrahedral meshes while computational cost and storage space for a desired accuracy can be reduced when applying higher degree approximation polynomials. In addition, we investigate the increase in computing time, when the number of relaxation mechanisms due to the generalized Maxwell body are increased. An application to a well-acknowledged test case and comparisons with analytic and reference solutions, obtained by different well-established numerical methods, confirm the performance of the proposed method. Therefore, the development of the highly accurate ADER–DG approach for tetrahedral meshes including viscoelastic material provides a novel, flexible and efficient numerical technique to approach 3-D wave propagation problems including realistic attenuation and complex geometry.