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50
An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes. IV: anisotropy
 GEOPHYS. J. INT
, 2007
"... 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 velocitystress formulation leading to a linear hyperbolic syst ..."
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Cited by 11 (1 self)
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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 velocitystress formulation leading to a linear hyperbolic system which accounts for the variation of the material properties depending on direction. This approach allows for the modeling of triclinic anisotropy, the most general crystalline symmetry class. The proposed method combines the Discontinuous Galerkin method with the ADER time integration approach using arbitrary high order derivatives of the piecewise polynomial representation of the unknown solution. In contrast to classical Finite Element methods discontinuities of this piecewise polynomial approximation are allowed at element interfaces, which allows for the application of the wellestablished theory of Finite Volumes and numerical fluxes across element interfaces obtained by the solution of derivative Riemann problems. Due to the ADER time integration technique the scheme provides the same approximation order in space and time automatically. Furthermore, through the projection of the tetrahedral elements of the physical space onto a canonical reference tetrahedron an efficient implementation is possible as many threedimensional integral computations can be carried out
Parsimonious finitevolume frequencydomain method for 2D PSVwave modelling
"... A new numerical technique for solving 2D elastodynamic equations based on a finitevolume frequencydomain approach is proposed. This method has been developed as a tool to perform twodimensional (2D) elastic frequencydomain fullwaveform inversion. In this context, the system of linear equations ..."
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Cited by 7 (7 self)
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A new numerical technique for solving 2D elastodynamic equations based on a finitevolume frequencydomain approach is proposed. This method has been developed as a tool to perform twodimensional (2D) elastic frequencydomain fullwaveform inversion. In this context, the system of linear equations that results from the discretisation of the elastodynamic equations is solved with a direct solver, allowing efficient multiplesource simulations at the partial expense of the memory requirement. The discretisation of the finitevolume approach is through triangles. Only fluxes with the required quantities are shared between the cells, relaxing the meshing conditions, as compared to finiteelement methods. The free surface is described along the edges of the triangles, which can have different slopes. By applying a parsimonious strategy, the stress components are eliminated from the discrete equations and only the velocities are left as unknowns in the triangles. Together with the local support of the P0 finitevolume stencil, the parsimonious approach allows the minimising of core memory requirements for the simulation. Efficient perfectly matched layer absorbing conditions have been designed for damping the waves around the grid. The numerical dispersion of this FV formulation is similar to that of O(∆x 2) staggeredgrid finitedifference formulations when considering
Adaptive Galerkin finite element methods for the wave equation
 Comput. Meth. Appl. Math
"... Abstract — This paper gives an overview of adaptive discretization methods for linear secondorder hyperbolic problems such as the acoustic or the elastic wave equation. The emphasis is on Galerkintype methods for spatial as well as temporal discretization, which also include variants of the Crank ..."
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Abstract — This paper gives an overview of adaptive discretization methods for linear secondorder hyperbolic problems such as the acoustic or the elastic wave equation. The emphasis is on Galerkintype methods for spatial as well as temporal discretization, which also include variants of the CrankNicolson and the Newmark finite difference schemes. The adaptive choice of space and time meshes follows the principle of “goaloriented ” adaptivity which is based on a posteriori error estimation employing the solutions of auxiliary dual problems.
Arbitrary high order finite volume schemes for seismic . . .
, 2000
"... We present a new numerical method to solve the heterogeneous anelastic seismic wave equations 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 velocitystress formulation provides a l ..."
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Cited by 6 (1 self)
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We present a new numerical method to solve the heterogeneous anelastic seismic wave equations 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 velocitystress 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 ßuids. The proposed method relies on the Finite Volume (FV) approach where cellaveraged quantities are evolved in time by computing 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,
Local time stepping and discontinuous Galerkin methods for symmetric first order hyperbolic systems
 JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
, 2008
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Charge conserving FEMPIC schemes on general grids
, 2014
"... In this article we aim at proposing a general mathematical formulation for charge conserving finite element Maxwell solvers coupled with particle schemes. In particular, we identify the finite element continuity equations that must be satisfied by the discrete current sources for several classes of ..."
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In this article we aim at proposing a general mathematical formulation for charge conserving finite element Maxwell solvers coupled with particle schemes. In particular, we identify the finite element continuity equations that must be satisfied by the discrete current sources for several classes of time domain VlasovMaxwell simulations to preserve the Gauss law at each time step, and propose a generic algorithm for computing such consistent sources. Since our results cover a wide range of schemes (namely curlconforming finite element methods of arbitrary degree, general meshes in 2 or 3 dimensions, several classes of time discretization schemes, particles with arbitrary shape factors and piecewise polynomial trajectories of arbitrary degree), we believe that they provide a useful roadmap in the design of high order charge conserving FEMPIC numerical schemes. 1
ADERWENO Finite Volume Schemes with SpaceTime Adaptive Mesh Refinement
, 2014
"... We present the first high order onestep ADERWENO finite volume scheme with Adaptive Mesh Refinement (AMR) in multiple space dimensions. High order spatial accuracy is obtained through a WENO reconstruction, while a high order onestep time discretization is achieved using a local spacetime discon ..."
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We present the first high order onestep ADERWENO finite volume scheme with Adaptive Mesh Refinement (AMR) in multiple space dimensions. High order spatial accuracy is obtained through a WENO reconstruction, while a high order onestep time discretization is achieved using a local spacetime discontinuous Galerkin predictor method. Due to the onestep nature of the underlying scheme, the resulting algorithm is particularly well suited for an AMR strategy on spacetime adaptive meshes, i.e.with timeaccurate local time stepping. The AMR property has been implemented ’cellbycell’, with a standard treetype algorithm, while the scheme has been parallelized via the Message Passing Interface (MPI) paradigm. The new scheme has been tested over a wide range of examples for nonlinear systems of hyperbolic conservation laws, including the classical Euler equations of compressible gas dynamics and the equations of magnetohydrodynamics (MHD). High order in space and time have been confirmed via a numerical convergence study and a detailed analysis of the computational speedup with respect to highly refined uniform meshes is also presented. We also show test problems where the presented high order AMR scheme behaves clearly better than traditional second order AMR methods. The proposed scheme that combines for the first time high order ADER methods with space–time adaptive grids in two and three space dimensions is likely to become a useful tool in several fields of computational physics, applied mathematics and mechanics.
7 FrequencyDomain Numerical Modelling of ViscoAcoustic Waves with FiniteDifference and FiniteElement Discontinuous Galerkin Methods
"... Seismic exploration is one of the main geophysical methods to extract quantitative inferences about the Earth’s interior at different scales from the recording of seismic waves near the surface. Main applications are civil engineering for cavity detection and landslide characterization, site effect ..."
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Seismic exploration is one of the main geophysical methods to extract quantitative inferences about the Earth’s interior at different scales from the recording of seismic waves near the surface. Main applications are civil engineering for cavity detection and landslide characterization, site effect modelling for seismic hazard, CO2 sequestration and nuclear
Accurate Calculation of FaultRupture Models Using the HighOrder
 Discontinuous Galerkin Method on Tetrahedral Meshes, Bulletin of the Seismological Society of America
"... Abstract We present a new method for nearsource groundmotion calculations due to earthquake rupture on potentially geometrically complex faults. Following the recently introduced Discontinuous Galerkin approach with local time stepping on tetrahedral meshes, we use piecewise polynomial approximati ..."
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Abstract We present a new method for nearsource groundmotion calculations due to earthquake rupture on potentially geometrically complex faults. Following the recently introduced Discontinuous Galerkin approach with local time stepping on tetrahedral meshes, we use piecewise polynomial approximations of the unknown variables inside each element and achieve the same approximation order in time and space due to the new ADER time integration scheme that uses Arbitrary highorder DERivatives. We show how an external source term and its heterogeneous properties in space and time, given by a fine discretization of an extended rupture surface, can be included in much coarser tetrahedral meshes due to the subcell resolution of the highorder polynomial representation. Hereby, the rupture surface is represented very generally as a point cloud of the center locations of individual subfaults at which each polynomial test function is evaluated exactly inside an element and the space– time integration of the source term is accurately computed at each timestep. Besides the incorporation of complex source kinematics we also present the effects of model boundaries that can degrade the accuracy of seismograms due to weak artificial reflections. We propose an extended computational domain of a coarsely meshed buffer region and show that our scheme using the local timestepping completely avoids such boundary problems with only slightly increasing the computational cost. We validate the new approach against different test cases, comparing our results with analytic, quasianalytic, and a series of reference solutions. Our work shows that adding the functionality of accurately treating finite sourcerupture models into the general framework of the ADERDiscontinuous Galerkin approach is an important contribution to modeling realistic earthquake scenarios, allowing the efficient inclusion of heterogeneous source kinematics and complex rupturesurface geometries in nearsource groundmotion simulations.