## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

The finite-difference method on rectangular meshes is widely used for time-domain modelling of the wave equation. It is relatively easy to implement high-order spatial discretization schemes and parallelization. Also, the method is computationally efficient. However, the use of finite elements on tetrahedral unstructured meshes is more accurate in complex geometries near sharp interfaces. We compared the standard eighth-order finite-difference method to fourth-order continuous mass-lumped finite elements in terms of accuracy and computational cost. The results show that, for simple models like a cube with constant density and velocity, the finite-difference method outperforms the finite-element method by at least an order of magnitude. Outside the application area of rectangular meshes, i.e., for a model with interior complexity and topography well described by tetrahedra, however, finite-element methods are about two orders of magnitude faster than finite-difference methods, for a given accuracy.

To read the full-text of this research,

you can request a copy directly from the authors.

... Standard second-order time stepping is combined with higher-order second-derivative operators in space that have a stencil width of M þ 1 points in each coordinate. Expressions for the weights and stability limits can be found elsewhere (Fornberg, 1988;Zhebel et al., 2014). In the context of RTM, all boundaries are assumed to be absorbing. ...

... The Courant-Friedrichs-Lewy (CFL) condition (Courant et al., 1928) limits the numerical time step Δt. Zhebel et al. (2014), e.g., provide expressions for arbitrary spatial orders with secondorder time stepping. The Nyquist-Shannon criterion dictates a maximum sampling interval of Δt max ¼ 1∕ð2f max Þ, where f max is the maximum frequency in the data. ...

... As an aside, mass-lumped (Zhebel et al., 2014;Mulder and Shamasundar, 2016) or discontinuous Galerkin finite elements (Minisini et al., 2013;Modave et al., 2015) would also only require the equivalent of one point: The solution values on a plane or something close to it on a 3D unstructured mesh. ...

One way to deal with the storage problem for the forward source wavefield in reverse time migration and full-waveform inversion is the reconstruction of that wavefield during reverse time stepping along with the receiver wavefield. Apart from the final states of the source wavefield, this requires a strip of boundary values for the whole time range in the presence of absorbing boundaries. The width of the stored boundary strip, positioned in between the interior domain of interest and the absorbing boundary region, usually equals about half that of the finite-difference stencil. The required storage in 3D with high frequencies can still lead to a decrease in computational efficiency, despite the substantial reduction in data volume compared with storing the source wavefields at all or at appropriately subsampled time steps. We have developed a method that requires a boundary strip with a width of just one point and has a negligible loss of accuracy. Stored boundary values over time enable the computation of the second and higher even spatial derivatives normal to the boundary, which together with extrapolation from the interior provides stability and accuracy. Numerical tests show that the use of only the boundary values provides at most fourth-order accuracy for the reconstruction error in the sourcewavefield. The use of higher even normal derivatives, reconstructed from the stored boundary values, allows for higher orders as numerical examples up to order 26 demonstrate. Subsampling in time is feasible with highorder interpolation and provides even more storage reduction but at a higher computational cost.

... In order to validate our implementation, we consider a benchmark used by Zhebel et al. (2014). In that work, a finite difference (FD) method and a mass-lumped finite element (ML) method were compared using OpenMP implementations. ...

... The pressure then is recorded at a 300m−offset receiver situated 250m above the interface, and the signal is compared to an exact solution. More details about this benchmark are given by Minisini et al. (2012) and Zhebel et al. (2014). Figure 2 shows curves of relative errors as function of the characteristic mesh size and the running time. ...

... Figure 2 shows curves of relative errors as function of the characteristic mesh size and the running time. According to Zhebel et al. (2014), we consider both the 2−norm error and the maximum error on the recorded signal, divided by the maximum absolute value of the exact signal. For the DG runs, we use the same numerical setting than for the ML simulations: meshes with 294 508, 567 071 and 2 320 289 tetrahedra, third-order polynomial basis functions, and a CFL number set to 0.3. ...

Improving both the accuracy and computational performance of simulation tools is a major challenge for seismic imaging, and generally requires specialized algorithms and computational implementations to make full use of modern hardware architectures. We present a computational strategy based on a highorder discontinuous Galerkin time-domain method. Our implementation can be run on several architectures thanks to a unified multi-threading programming framework, and exhibits a good load balancing and minimum data movements. A first benchmark validates this implementation and confirms the interest of accelerators in computational geophysics.

... Moreover, a hybrid discretization can be deployed thanks to the flexibility in mixing several kinds of elements and different discretization orders. In comparative work [Baldassari et al., 2012, Moczo et al., 2011, Zhebel et al., 2014, these finite-element methods generally exhibit similar performance results and they surpass classic finitedifference schemes in cases with geometric or physical discontinuities. However, in three dimensions, spectral elements are currently only available for hexahedral meshes, which are less flexible than tetrahedral meshes. ...

... In order to validate our implementation, numerical convergence as well as run times, we use a reference benchmark proposed by Zhebel et al. [2014] and . ...

... m, 1000 m, 746.2 m). The recorded signal is compared to the exact solution over the period [t p , t f ] using the relative error defined as [Zhebel et al., 2014] ...

Improving both accuracy and computational performance of numerical tools is a
major challenge for seismic imaging and generally requires specialized
implementations to make full use of modern parallel architectures. We present a
computational strategy for reverse-time migration (RTM) with accelerator-aided
clusters. A new imaging condition computed from the pressure and velocity
fields is introduced. The model solver is based on a high-order discontinuous
Galerkin time-domain (DGTD) method for the pressure-velocity system with
unstructured meshes and multi-rate local time-stepping. We adopted the MPI+X
approach for distributed programming where X is a threaded programming model.
In this work we chose OCCA, a unified framework that makes use of major
multi-threading languages (e.g. CUDA and OpenCL) and offers the flexibility to
run on several hardware architectures. DGTD schemes are suitable for efficient
computations with accelerators thanks to localized element-to-element coupling
and the dense algebraic operations required for each element. Moreover,
compared to high-order finite-difference schemes, the thin halo inherent to
DGTD method reduces the amount of data to be exchanged between MPI processes
and storage requirements for RTM procedures. The amount of data to be recorded
during simulation is reduced by storing only boundary values in memory rather
than on disk and recreating the forward wavefields Computational results are
presented that indicate that these methods are strong scalable up to at least
32 GPUs for a large-scale three-dimensional case.

... with weights [17,18] ...

... and zero otherwise. Note that the stability limit for σ depends on the order of the scheme [17]. For the subsequent time levels (n > 1), the discrete partial differential equation can be used for i = 1, . . . ...

... 1. Introduction. Mass-lumped tetrahedral element methods are efficient methods for solving linear wave equations, such as the acoustic wave equation, the elastic wave equations, or Maxwell's equations, on complex three-dimensional (3D) domains with sharp material interfaces [25]. They offer the same convergence rate and geometric flexibility as standard continuous tetrahedral element methods, but also allow for explicit time-stepping due to a diagonal mass matrix. ...

... \bullet u (e) : the wave field on e. \bullet u (e) \in \BbbR n : the wave field at the nodes on e. When using exact integration, the stiffness matrix-vector product v (e) := A (e) u (e) \in \BbbR n is given by (25) [ is a tensor field, with J e := \nabla \phi e the Jacobian of the element mapping and J - t e the transposed of J - 1 e . When c is constant within each element, then \c (e) is also constant and we can compute (26) using the algorithm of [19]: ...

We present new and efficient quadrature rules for computing the stiffness matrices of mass-lumped tetrahedral elements for wave propagation modelling. These quadrature rules allow for a more efficient implementation of the mass-lumped finite element method and can handle materials that are heterogeneous within the element without loss of the convergence rate. The quadrature rules are designed for the specific function spaces of recently developed mass-lumped tetrahedra, which consist of standard polynomial function spaces enriched with higher-degree bubble functions. For the degree-2 mass-lumped tetrahedron, the most efficient quadrature rule seems to be an existing 14-point quadrature rule, but for tetrahedra of degrees 3 and 4, we construct new quadrature rules that require less integration points than those currently available in the literature. Several numerical examples confirm that this approach is more efficient than computing the stiffness matrix exactly and that an optimal order of convergence is maintained, even when material properties vary within the element.

... 1. Introduction. Mass-lumped tetrahedral element methods are efficient methods for solving linear wave equations, such as the acoustic wave equation, the elastic wave equations, or Maxwell's equations, on complex three-dimensional (3D) domains with sharp material interfaces [25]. They offer the same convergence rate and geometric flexibility as standard continuous tetrahedral element methods, but also allow for explicit time-stepping due to a diagonal mass matrix. ...

... \bullet u (e) : the wave field on e. \bullet u (e) \in \BbbR n : the wave field at the nodes on e. When using exact integration, the stiffness matrix-vector product v (e) := A (e) u (e) \in \BbbR n is given by (25) [ is a tensor field, with J e := \nabla \phi e the Jacobian of the element mapping and J - t e the transposed of J - 1 e . When c is constant within each element, then \c (e) is also constant and we can compute (26) using the algorithm of [19]: ...

We present new and efficient quadrature rules for computing the stiffness matrices of mass-lumped tetrahedral elements for wave propagation modeling. These quadrature rules allow for a more efficient implementation of the mass-lumped finite element method and can handle materials that are heterogeneous within the element without loss of the convergence rate. The quadrature rules are designed for the specific function spaces of recently developed mass-lumped tetrahedra, which consist of standard polynomial function spaces enriched with higher-degree bubble functions. For the degree-2 mass-lumped tetrahedron, the most efficient quadrature rule seems to be an existing 14-point quadrature rule, but for tetrahedra of degrees 3 and 4, we construct new quadrature rules that require fewer integration points than those currently available in the literature. Several numerical examples confirm that this approach is more efficient than computing the stiffness matrix exactly and that an optimal order of convergence is maintained, even when material properties vary within the element.

... The spectral-element method on hexahedra (Komatitsch and Vilotte, 1998, e.g.) may offer relatively good performance and accuracy for application in FWI (Brossier et al., 2014). Tetrahedra offer more gridding flexibility and are a good alternative (Zhebel et al., 2014). ...

... shows a comparison of our 1-D finite-difference scheme with a continuous mass-lumped finite element approach (CML-FEM,Zhebel et al., 2014) for a rough 3-D topography in southeastern Europe. ...

We propose a finite-difference scheme for the simulation of seismic waves interacting with 3-D free-surface topography. The intended application is velocity model building by acoustic full-waveform inversion (FWI). The scheme follows an immersed boundary approach for wave equations in the first-order stress-velocity formulation, discretized on a standard staggered grid. Our scheme employs modified 1-D stencils rather than a full 3-D field wavefield extension at the free surface. Although this decreases the accuracy, it improves the scheme's simplicity and robustness. To avoid stability problems, points close to the zero-pressure boundary must be excluded. The scheme, and its adjoint, have been tested by tilted geometry tests and by comparison to a finite-element method. We present a first test result of full-waveform inversion with the new scheme.

... Spectral Element Methods (SEM) [3] address this issue by diagonalizing this mass matrix system through the use of mass-lumping, which co-locates interpolation nodes for Lagrange basis functions and Gauss-Legendre-Lobatto quadrature points. Since SEM is limited to unstructured hexahedral meshes, which are less geometrically flexible than tetrahedral meshes, triangular and tetrahedral mass-lumped spectral element methods have been investigated as alternatives [4,5,6]. However, due to a mismatch in the number of natural quadrature nodes and the dimension of polynomial approximation spaces on simplices, these methods necessitate adding additional nodes in the interior of the element to construct sufficiently accurate nodal points suitable for mass-lumping. ...

... Defineṽ = M k −1 M k c 2 M k −1 v; assuming that 1 + αv Tṽ = 0, For nonlinear hyperbolic problems with non-smooth solutions such as shocks, as a non-conservative scheme can lead to incorrect shock speeds [29]. The exact enforcement of local conservation is especially important in this context, since Theorem 6 suggests that conservation errors depend otherwise on the regularity of u. 6. Numerical examples. ...

Time-domain discontinuous Galerkin (DG) methods for wave propagation require accounting for the inversion of dense elemental mass matrices, where each mass matrix is computed with respect to a parameter-weighted L2 inner product. In applications where the wavespeed varies spatially at a sub-element scale, these matrices are distinct over each element, necessitating additional storage. In this work, we propose a weight-adjusted DG (WADG) method which reduces storage costs by replacing the weighted L2 inner product with a weight-adjusted inner product. This equivalent inner product results in an energy stable method, but does not increase storage costs for locally varying weights. A-priori error estimates are derived, and numerical examples are given illustrating the application of this method to the acoustic wave equation with heterogeneous wavespeed.

... Such sets of points are known up to k = 9 for triangles and k = 4 for tetrahedra, and due to their diagonal mass matrix, they can be used for fast fully explicit numerical wave simulations (Chin-Joe-Kong et al., 1999;Mulder et al., 2013;Geevers et al., 2018b, a;Cui et al., 2017;Liu et al., 2017). These elements have been compared with finite-difference schemes and have favorable results for the forward wave propagation when interior complexity and topography are present that can be adequately modeled with unstructured tetrahedra (Zhebel et al., 2014). However, to the authors' knowledge these elements have not been used in peer-reviewed literature to perform seismic inversions. ...

In this article, we introduce spyro, a software stack to solve wave propagation in heterogeneous domains and perform full waveform inversion (FWI) employing the finite-element framework from Firedrake, a high-level Python package for the automated solution of partial differential equations using the finite-element method. The capability of the software is demonstrated by using a continuous Galerkin approach to perform FWI for seismic velocity model building, considering realistic geophysics examples. A time domain FWI approach that uses meshes composed of variably sized triangular elements to discretize the domain is detailed. To resolve both the forward and adjoint-state equations and to calculate a mesh-independent gradient associated with the FWI process, a fully explicit, variable higher-order (up to degree k=5 in 2D and k=3 in 3D) mass-lumping method is used. We show that, by adapting the triangular elements to the expected peak source frequency and properties of the wave field (e.g., local P-wave speed) and by leveraging higher-order basis functions, the number of degrees of freedom necessary to discretize the domain can be reduced. Results from wave simulations and FWIs in both 2D and 3D highlight our developments and demonstrate the benefits and challenges with using triangular meshes adapted to the material properties.

... The spatial discretisation is reviewed in Appendix A. Geompack [30] was used to construct unstructured meshes. Assembly is performed on the fly in each time step, as in [6,11,[31][32][33], although on modern hardware, a pre-assembled matrix might be more efficient in 2D, in particular for lower orders. ...

When solving the wave equation with finite elements, mass lumping allows for explicit time stepping, avoiding the cost of a lower-upper decomposition of the large sparse mass matrix. Mass lumping on the reference element amounts to numerical quadrature. The weights should be positive for stable time stepping and preserve numerical accuracy. The standard triangular polynomial elements, except for the linear element, do not have these properties. Accuracy can be preserved by augmenting them with higher-degree polynomials in the interior. This leaves the search for elements with positive weights, which were found up to degree 9 by various authors. The classic accuracy condition, however, is too restrictive. A sharper, less restrictive condition recently led to new mass-lumped tetrahedral elements up to degree 4. Compared to the known ones up to degree 3, they have less nodes and are computationally more efficient. The same criterion is applied here to the construction of triangular elements. For degrees 2 to 4, these turn out to be identical to the known ones. For degree 5, the number of nodes is the same as for the known element, but now there are infinitely many solutions. Some of these have a considerably larger stability limit for time stepping. For degree 6, two elements are found with less nodes than the known ones. For degree 7, one element with less nodes was found but with a negative weight, making it useless for time stepping with the wave equation. If the number of nodes is the same as for the classic element, there are now infinitely many solutions. Numerical tests for a homogeneous wave-propagation problem with a point source confirm the expected accuracy of the new elements. Some of them require less compute time than those obtained with the more restrictive accuracy criterion.

... among many others, where EnrichedLagrange refers to a discretization model based on mixed-order Lagrange polynomials suitable for high-order mass lumping on simplicial elements (Chin-Joe- Kong et al. 1999;Cohen et al. 2001;Zhebel et al. 2014;Mulder & Shamasundar 2016). ...

In this paper, we present a series of mathematical abstractions for seismologically relevant wave equations discretized using finite-element methods, and demonstrate how these abstractions can be implemented efficiently in computer code. Our motivation is to mitigate the combinatorial complexity present when considering geophysical waveform modelling and inversion, where a variety of spatial discretizations, material models, and boundary conditions must be considered simultaneously. We accomplish this goal by first considering three distinct classes of abstract mathematical models: (1) those representing the physics of an underlying wave equation, (2) those describing the discretization of the chosen equation onto a finite-dimensional basis and (3) those describing any spatial transforms. A full representation of the discrete wave equation can then be constructed using a hierarchical nesting of models from each class. Additionally, each class is functionally orthogonal to the others, and with certain restrictions models within one class can be interchanged independently from changes in another. We then show how this recasting of the relevant equations can be implemented concisely in computer software using an abstract object-oriented design, and discuss how recent developments in the numerical and computational sciences can be naturally incorporated. This builds to a set of results where we demonstrate how the developments presented can lead to an implementation capable of multiphysics waveform simulations in completely unstructured domains, on both hypercubical and simplical spectral-element meshes, in both two and three dimensions, while remaining concise, efficient and maintainable. © The Author(s) 2018. Published by Oxford University Press on behalf of The Royal Astronomical Society.

... The finite-difference method is widely used in the oil industry because it is relatively easy to code up and optimize. The finite-element method is computationally more demanding but may offer better accuracy at a given cost in the presence of topography and large impedance contrast, but only if the mesh follows the interfaces between different rock types (Kononov et al. 2012;Zhebel et al. 2014). ...

Finite elements can, in some cases, outperform finite-difference methods for modelling wave propagation in complex geological models with topography. In the weak form of the finiteelement method, the delta function is a natural way to represent a point source. If, instead of the usual second-order form, the first-order form of the wave equation is considered, this is no longer true. Fourier analysis for a simple case shows that the spatial operator corresponding to the first-order form has short-wavelength null-vectors. Once excited, these modes are not seen by the spatial operator but only by the time- stepping scheme and show up as noise. A source with a larger spatial extent, for instance a Gaussian or a tapered sinc, can avoid the excitation of problematic short wavelengths. A series of numerical experiments on a 2-D problem with an exact solution provides a suggestion for the best choice of parameters for these source term distributions. The tapered sinc provided the best results and the resulting accuracy can be better than that of the second-order form. The higher operation count of the former, however, does not make it more efficient in terms of accuracy for a given computational effort, at least not for the 2-D examples considered here.

... Dans le cadre de géométries complexes, les méthodes variationnelles telles que les méthodes par volumes finis et éléments finis continus ou discontinus sont, en général plus précises. Bien que plus coûteuses en terme de calcul, elles bénéficient d'avantages non négligeables en traitant les interfaces sous forme de flux et disposent de bien meilleures propriétés de dispersion (Zhebel et al., 2014). Cette caractéristique des méthodes par éléments finis justifie le choix de la technique employée dans le chapitre 4 de cette thèse. ...

Cette thèse se penche sur la propagation d’ondes au sein du système coupléTerre-océan-atmosphère. La compréhension de ces phénomènes a une importance majeurepour l’étude de perturbations sismiques et d’explosions atmosphériques notamment dans lecadre de missions spatiales planétaires. Les formes d’ondes atmosphériques issues du couplagefluide-solide permettent d’obtenir de précieuses informations sur la source du signal ou lespropriétés des milieux de propagation. On développe donc deux outils de modélisation numériqued’ordre élevé pour la propagation d’ondes acoustiques et de gravité. Le premier est endifférences finies sur grille en quinconce et se concentre uniquement sur le milieu atmosphérique,permettant la propagation d’ondes linéaires dans un milieu stratifié visqueux et avecdu vent. Cette méthode linéaire est validée par des solutions quasi-analytiques reposant surles équations de dispersion dans une atmosphère stratifiée. Elle est aussi appliquée à deuxcas d’études : la propagation d’ondes liée à l’impact d’une météorite à la surface de Marsdans le cadre de la mission de la NASA INSIGHT, et la propagation d’ondes atmosphériquesliées au tsunami de Sumatra en 2004. La seconde méthode résout la propagation non-linéaired’ondes acoustiques et de gravité dans une atmosphère complexe couplée, avec topographie,à la propagation d’ondes élastiques dans un solide visco-élastique. Cette méthode repose sursur le couplage d’une formulation en éléments finis discontinus, pour résoudre les équationsde Navier-Stokes dans la partie fluide, avec une méthode par éléments finis continus pourrésoudre les équations de l’élastodynamique dans la partie solide. Elle a été validée grâce àdes solutions analytiques ainsi que par des comparaisons avec les résultats de la méthode pardifférences finies. De nombreuses applications de cette méthode sont alors possibles notammentpour l’étude de séismes de sub-surface, d’explosions atmosphériques liées à la rentréede météorites ou pour la caractérisation des phénomènes non-linéaires lors de la propagationd’infrasons et d’ondes de gravité dans l’atmosphère.

... Although Finite Differences (FD) are generally the method of choice, formulations of Finite Element Methods (FEM) are regularly being published (e.g. [7,[29][30][31]34,38,43,54,57]). The motivation for applying FEM over FD includes treatment of free surface topography, discontinuous coefficients and boundary conditions, higher order accuracy, and adaptive gridding. ...

Finite Element Methods (FEM) are becoming increasingly popular in modeling seismic wave propagation. These methods provide higher order accuracy, geometrical flexibility and adaptive gridding capabilities that are not easy to incorporate in traditional finite difference methods employed for generation of synthetic seismograms. Moreover, several studies have shown that Discontinuous Galerkin FEM (DGM) is a promising approach for modeling wave propagation in fractured media. Here we propose an Enriched Galerkin FEM (EGM) for elastic wave propagation. EGM uses the same bilinear form as DGM and the continuous Galerkin finite element spaces enriched by discontinuous piecewise constants or bilinear functions. EGM satisfies local equilibrium while reducing the degrees of freedom in DGM formulations. In this paper, we consider elastic wave propagation and derive optimal a priori error estimates for DGM and EGM. We present numerical examples in two spatial dimensions that confirm these theoretical results. In addition, we provide numerical comparisons with the Spectral element method. In previous work, DGM has been shown to be effective in modeling elastic wave propagation in fractured media using the linear slip model. We now extend these results to EGM with reduced computational costs over DGM.

... While more difficult to implement and requiring more computations, the finite element method can remain accurate on very complex geometries when using a proper mesh. When applied with mass-lumping, the finite element method can in such cases become more efficient than the finite difference method [20]. ...

We present a new accuracy condition for the construction of continuous mass-lumped elements. This condition is less restrictive than the one currently used and enabled us to construct new mass-lumped tetrahedral elements of degrees 2 to 4. The new degree-2 and degree-3 tetrahedral elements require 15 and 32 nodes per element, respectively, while currently, these elements require 23 and 50 nodes, respectively. The new degree-4 elements require 60, 61 or 65 nodes per element. Tetrahedral elements of this degree had not been found yet. We prove that our accuracy condition results in a mass-lumped finite element method that converges with optimal order in the $L^2$-norm and energy-norm. A dispersion analysis and several numerical tests confirm that our elements maintain the optimal order of accuracy and show that the new mass-lumped tetrahedral elements are more efficient than the current ones.

... Figs 17 and 18 show a comparison of our finite-difference scheme based on 1-D field extensions with a continuous mass-lumped finite-element approach (CML-FEM, Zhebel et al. 2014) for a rough 3-D topography somewhere in south-eastern Europe. Both methods were run with a 5-Hz Ricker wavelet and a constant sound speed of 2500 m s −1 . ...

One approach to incorporate topography in seismic finite-difference codes is a local modification of the difference operators near the free surface. An earlier paper described an approach for modelling irregular boundaries in a constant-density acoustic finite-difference code, based on the second-order formulation of the wave equation that only involves the pressure. Here, a similar method is considered for the first-order formulation in terms of pressure and particle velocity, using a staggered finite-difference discretization both in space and in time. In one space dimension, the boundary conditions consist in imposing antisymmetry for the pressure and symmetry for particle velocity components. For the pressure, this means that the solution values as well as all even derivatives up to a certain order are zero on the boundary. For the particle velocity, all odd derivatives are zero. In 2D, the 1-D assumption is used along each coordinate direction, with antisymmetry for the pressure along the coordinate and symmetry for the particle velocity component parallel to that coordinate direction. Since the symmetry or antisymmetry should hold along the direction normal to the boundary rather than along the coordinate directions, this generates an additional numerical error on top of the time stepping errors and the errors due to the interior spatial discretization. Numerical experiments in 2D and 3D nevertheless produce acceptable results.

... Diagonal-norm SBP operators can also be constructed for triangles and tetrahedra [26,24,6]; however, the number of nodal points for such operators is typically greater than the dimension of the natural polynomial approximation space. Furthermore, to the author's knowledge, appropriate point sets have only been constructed for N ≤ 4 in three dimensions [27], and the construction of high order diagonal norm SBP-DG methods has not yet been performed for uncommon elements such as pyramids [28]. ...

High order methods based on diagonal-norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as nodal discontinuous Galerkin methods with diagonal mass matrices. In this work, we describe how to construct discretely entropy conservative schemes to a more general class of DG methods using flux differencing, quadrature-based projections, and specific DG differentiation operators. This approach also recovers existing methods for Burgers' equation involving dense norm and generalized SBP operators without boundary nodes. Numerical experiments confirm the stability and high order accuracy of the proposed methods for the one-dimensional compressible Euler equations.

... Hence, is not clear if the reduction in the number of elements can compensate for the increased computational Paris, France, 12-15 June 2017 cost per element and the smaller time step (Mulder, 2013). For models with sharp interfaces and topography high-order mass-lumped FEM is known to be more efficient than the standard eighth-order finite-differences (Zhebel et al., 2014). ...

... Standard second-order time stepping is combined with higher-order second-derivative operators in space that have a stencil width of M + 1 points in each coordinate. Expressions for the weights and stability limits can be found elsewhere (Fornberg, 1988;Zhebel et al., 2014). In the context of reversetime migration, all boundaries are assumed to be absorbing. ...

... Alternatively, we may look for less compute-intensive formulations. The second-order formulation of the wave equation is often used for modelling seismic wave propagation with spectral methods, both for box-like elements on quadrilaterals and hexahedra (Komatitsch et al., 1999, e.g.) as well as for simplex-based elements on triangles (Mulder, 1996(Mulder, , 2013 or tetrahedra (Zhebel et al., 2014;Mulder and Shamasundar, 2016). For some applications, a first-order formulation may be desirable. ...

... Boore (1972), Robertsson (1996), Mittet (2002), Bohlen and Saenger (2006), and Zeng et al. (2012) consider variants of the vacuum approach. Unfortunately, the loss of accuracy can become significant: Numerical experiments with a higher order finite-difference scheme for the acoustic wave equation indicate that the numerical error increases to only the first order in space (Zhebel et al., 2014) in case of a smoothed density contrast. The cause is the same as for large subsurface contrasts: The solution is not differentiable across sharp interfaces, leading to a second-order spatial error, and if the subsurface model is just sampled, the position of the interface is uncertain within a grid spacing, leading to a first-order error. ...

The presence of topography poses a challenge for seismic modeling with finite-difference codes. The representation of topography by means of an air layer or vacuum often leads to a substantial loss of numerical accuracy. A suitable modification of the finite-difference weights near the free surface can decrease that error. An existing approach requires extrapolation of interior solution values to the exterior while using the boundary condition at the free surface. However, schemes of this type occasionally become unstable and may be impossible to implement with highly irregular topography. One-dimensional extrapolation along coordinate lines results in a simple and efficient scheme. The stability of the 1D scheme is improved by ignoring the interior point nearest to the boundary during extrapolation in case its distance to the boundary is less than half a grid spacing. The generalization of the 1D scheme to more than one dimension requires a modification if the boundary intersects the finite-difference stencil on both sides of the central evaluation point and if there are not enough interior points to build the finite-difference stencil. Examples for the 2D constant-density acoustic case with a fourth-order finite-difference scheme demonstrate the method's capability. Because the 1D assumption is not valid in two dimensions if the boundary does not follow grid lines, the formal numerical accuracy is not always obtained, but the method can handle highly irregular topography. © 2017 Society of Exploration Geophysicists. All rights reserved.

... In the next section, in order to check the accuracy of the OTSEM, we use the L2-norm together with the maximum-norm to obtain the error of the numerical solutions obtained by other family methods at the same receiver point. The L2-norm-based numerical error is estimated as [77] ...

The mass-lumped method avoids the cost of inverting the mass matrix and simultaneously maintains spatial accuracy by adopting additional interior integration points, known as cubature points. To date, such points are only known analytically in tensor domains, such as quadrilateral or hexahedral elements. Thus, the diagonal-mass-matrix spectral element method (SEM) in non-tensor domains always relies on numerically computed interpolation points or quadrature points. However, only the cubature points for degrees 1 to 6 are known, which is the reason that we have developed a p-norm-based optimization algorithm to obtain higher-order cubature points. In this way, we obtain and tabulate new cubature points with all positive integration weights for degrees 7 to 9. The dispersion analysis illustrates that the dispersion relation determined from the new optimized cubature points is comparable to that of the mass and stiffness matrices obtained by exact integration. Simultaneously, the Lebesgue constant for the new optimized cubature points indicates its surprisingly good interpolation properties. As a result, such points provide both good interpolation properties and integration accuracy. The Courant–Friedrichs–Lewy (CFL) numbers are tabulated for the conventional Fekete-based triangular spectral element (TSEM), the TSEM with exact integration, and the optimized cubature-based TSEM (OTSEM). A complementary study demonstrates the spectral convergence of the OTSEM. A numerical example conducted on a half-space model demonstrates that the OTSEM improves the accuracy by approximately one order of magnitude compared to the conventional Fekete-based TSEM. In particular, the accuracy of the 7th-order OTSEM is even higher than that of the 14th-order Fekete-based TSEM. Furthermore, the OTSEM produces a result that can compete in accuracy with the quadrilateral SEM (QSEM). The high accuracy of the OTSEM is also tested with a non-flat topography model. In terms of computational efficiency, the OTSEM is more efficient than the Fekete-based TSEM, although it is slightly costlier than the QSEM when a comparable numerical accuracy is required.

... Several authors compared the various methods (Fornberg, 1987; Moczo et al., 2011; Chaljub et al., 2007; Wang et al., 2010; Pasquetti and Rapetti, 2004; De Basabe and Sen, 2007, e.g.). Zhebel et al. (2012a) considered the continuous mass-lumped and the discontinuous Galerkin finite elements in terms of accuracy , stability and computational cost. Numerical experiments on 3-D problems showed that both methods have similar stability conditions and require a comparable computational time to obtain a result with a given accuracy, assuming that the stiffness and mass matrices are pre-assembled. ...

In a previous study of seismic modeling with radial basis function-generated finite differences (RBF-FD), we outlined a numerical method for solving 2-D wave equations in domains with material interfaces between different regions. The method was applicable on a mesh-free set of data nodes. It included all information about interfaces within the weights of the stencils (allowing the use of traditional time integrators), and was shown to solve problems of the 2-D elastic wave equation to 3rd-order accuracy. In the present paper, we discuss a refinement of that method that makes it simpler to implement. It can also improve accuracy for the case of smoothly-variable model parameter values near interfaces. We give several test cases that demonstrate the method solving 2-D elastic wave equation problems to 4th-order accuracy, even in the presence of smoothly-curved interfaces with jump discontinuities in the model parameters.

We consider isotropic elastic wave propagation with continuous mass-lumped finite elements on tetrahedra with explicit time stepping. These elements require higher-order polynomials in their interior to preserve accuracy after mass lumping and are only known up to degree 3. Global assembly of the symmetric stiffness matrix is a natural approach but requires large memory. Local assembly on the fly, in the form of matrix-vector products per element at each time step, has a much smaller memory footprint. With dedicated expressions for local assembly, our code ran about 1.3 times faster for degree 2 and 1.9 times for degree 3 on a simple homogeneous test problem, using 24 cores. This is similar to the acoustic case. For a more realistic problem, the gain in efficiency was a factor 2.5 for degree 2 and 3 for degree 3. For the lowest degree, the linear element, the expressions for both the global and local assembly can be further simplified. In that case, global assembly is more efficient than local assembly. Among the three degrees, the element of degree 3 is the most efficient in terms of accuracy at a given cost.

Finite-element discretizations of the acoustic wave equation in the time domain often employ mass lumping to avoid the cost of inverting a large sparse mass matrix. For the second-order formulation of the wave equation, mass lumping on Legendre–Gauss–Lobatto points does not harm the accuracy. Here, we consider a first-order formulation of the wave equation. In that case, the numerical dispersion for odd-degree polynomials exhibits super-convergence with a consistent mass matrix but mass lumping destroys that property. We consider defect correction as a means to restore the accuracy, in which the consistent mass matrix is approximately inverted using the lumped one as preconditioner. For the lowest-degree element on a uniform mesh, fourth-order accuracy in 1D can be obtained with just a single iteration of defect correction.
The numerical dispersion curve describes the error in the eigenvalues of the discrete set of equations. However, the error in the eigenvectors also play a role, in two ways. For polynomial degrees above one and when considering a 1-D mesh with constant element size and constant material properties, a number of modes, equal to the maximum polynomial degree, are coupled. One of these is the correct physical mode that should approximate the true eigenfunction of the operator, the other are spurious and should have a small amplitude when the true eigenfunction is projected onto them. We analyze the behaviour of this error as a function of the normalized wavenumber in the form of the leading terms in its series expansion and find that this error exceeds the dispersion error, except for the lowest degree where the eigenvector error is zero. Numerical 1-D tests confirm this behaviour.
We briefly analyze the 2-D case, where the lowest-degree polynomial also appears to provide fourth-order accuracy with defect correction, if the grid of squares or triangles is highly regular and material properties are constant.

Seismic exploration is the primary tool for finding and mapping out hydrocarbon deposits. We have considered the 2D forward modeling problem. The subsurface structures were assumed to be known, and the task was to simulate elastic-wave propagation throughout the medium. Many traditional approaches to solving this problem account for material interfaces by allowing model parameter values (such as wave speed and density) to vary either sharply or smoothly between adjacent data points. Although simple to implement, these strategies typically produce solutions that feature a suboptimal first-order convergence to the true solution. Spatial discretization based on radial-basis-functiongenerated finite differences (RBF-FDs) has previously been shown to offer high accuracy and algebraic simplicity when using scattered layouts of computational nodes. We have now developed this method further, to provide third-order accuracy not only throughout smooth regions, but also for wave reflections and transmissions at arbitrarily curved material interfaces. The key step is supplementing the RBFs that underlie the RBF-FD approximation with a specific space of piecewise polynomials. The nonsmoothness of these polynomials across an interface is designed to enforce continuity of traction and motion at that interface. The highorder accuracy of the method is illustrated on a couple of test problems for the 2D isotropic elastic-wave equation.

Finite-element modelling of seismic wave propagation on tetrahedra requires meshes that accurately follow interfaces between impedance contrasts or surface topography and have element sizes proportional to the local velocity. We explain a mesh generation approach by example. Starting from a finite-difference representation of the velocity model, triangulated surfaces are generated along impedance discontinuities. These define subdomains that are meshed independently and in parallel, honouring the local velocity values. The resulting volumetric meshes are merged into a single mesh. The approach is flexible, efficient, scalable and capable of producing quality meshes.

ated the ability to handle high-resolution simulations of seismic wave propagation in 3D complex domains. However, the exponential accuracy of the method and the reduction of the computational effort rely on the use of conformal hexahedral meshes. Generating an all-hexahedral mesh based upon the available meshing can be difficult. We propose to use a 3D unstructured tetrahedral mesh generator and then split the resulting mesh into hexahedral elements. This approach allows using the same modeling SEM program without any modifications, and it remains faster than dealing with tetrahedral spectral element due to the tensorisation property for spatial derivatives. With this approach, the SEM method doesn’t have restrictions to solving complex problems in sonic or seismic data with respect to methods using tetrahedral meshes. Two examples dealing, respectively, with sonic modeling and the other with surface seismic modeling illustrate the feasibility of the proposed method.

Mass-lumped continuous finite elements allow for explicit time stepping with the second-order wave equation if the resulting integration weights are positive and provide sufficient accuracy. To meet these requirements on triangular and tetrahedral meshes, the construction of continuous finite elements for a given polynomial degree on the edges involves polynomials of higher degree in the interior. The parameters describing the supporting nodes of the Lagrange interpolating polynomials and the integration weights are the unknowns of a polynomial system of equations, which is linear in the integration weights. To find candidate sets for the nodes, it is usually required that the number of equations equals the number of unknowns, although this may be neither necessary nor sufficient. Here, this condition is relaxed by requiring that the number of equations does not exceed the number of unknowns. This resulted in two new types elements of degree 6 for symmetrically placed nodes. Unfortunately, the first type is not unisolvent. There are many different elements of the second type with a large range in their associated time-stepping stability limit. To assess the efficiency of the elements of various degrees, numerical tests on a simple problem with an exact solution were performed. Efficiency was measured by the computational time required to obtain a solution at a given accuracy. For the chosen example, elements of degree 4 with fourth-order time stepping appear to be the most efficient.

We present a review of the application of the spectral-element method to regional and global seismology. This technique is a high-order variational method that allows one to compute accurate synthetic seismograms in three-dimensional heterogeneous Earth models with deformed geometry. We first recall the strong and weak forms of the seismic wave equation with a particular emphasis set on fluid regions. We then discuss in detail how the conditions that hold on the boundaries, including coupling boundaries, are honored. We briefly outline the spectral-element discretization procedure and present the time-marching algorithm that makes use of the diagonal structure of the mass matrix. We show examples that illustrate the capabilities of the method and its interest in the context of the computation of three-dimensional synthetic seismograms.

We analyse the time-stepping stability for the 3-D acoustic wave
equation, discretized on tetrahedral meshes. Two types of methods are
considered: mass-lumped continuous finite elements and the symmetric
interior-penalty discontinuous Galerkin method. Combining the spatial
discretization with the leap-frog time-stepping scheme, which is
second-order accurate and conditionally stable, leads to a fully
explicit scheme. We provide estimates of its stability limit for simple
cases, namely, the reference element with Neumann boundary conditions,
its distorted version of arbitrary shape, the unit cube that can be
partitioned into six tetrahedra with periodic boundary conditions and
its distortions. The Courant-Friedrichs-Lewy stability limit contains an
element diameter for which we considered different options. The one
based on the sum of the eigenvalues of the spatial operator for the
first-degree mass-lumped element gives the best results. It resembles
the diameter of the inscribed sphere but is slightly easier to compute.
The stability estimates show that the mass-lumped continuous and the
discontinuous Galerkin finite elements of degree 2 have comparable
stability conditions, whereas the mass-lumped elements of degree one and
three allow for larger time steps.

Finite-difference (FD) techniques have established themselves as viable tools for the numerical modeling of wave propagation. The accuracy and the computational efficiency of numerical modeling can be enhanced by using high-order spatial differential operators (Dablain,1986).

A spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two- and three-dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, defined with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor-product of a Gauss–Lobatto–Legendre 1-D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor/multicorrector format. Long term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two-dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three-dimensional model. Very accurate modelling of Rayleigh wave propagation and surface diffraction is obtained at a low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three-dimensional surface topographies. Copyright © 1999 John Wiley & Sons, Ltd.

We present the spectral element method to simulate elastic-wave propagation in realistic geological structures involving complicated free-surface topography and material interfaces for two- and three-dimensional geometries. The spectral element method introduced here is a high-order variational method for the spatial approximation of elastic-wave equations. The mass matrix is diagonal by construction in this method, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor/multi-corrector format. Long-term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The associated Courant condition behaves as Δt C < O (n el-1/nd N -2), with n el the number of elements, n d the spatial dimension, and N the polynomial order. In practice, a spatial sampling of approximately 5 points per wavelength is found to be very accurate when working with a polynomial degree of N = 8. The accuracy of the method is shown by comparing the spectral element solution to analytical solutions of the classical two-dimensional (2D) problems of Lamb and Garvin. The flexibility of the method is then illustrated by studying more realistic 2D models involving realistic geometries and complex free-boundary conditions. Very accurate modeling of Rayleigh-wave propagation, surface diffraction, and Rayleigh-to-body-wave mode conversion associated with the free-surface curvature are obtained at low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves by three-dimensional (3D) surface topographies and the associated local effects on strong ground motion. Complex amplification patterns, both in space and time, are shown to occur even for a gentle hill topography. Extension to a heterogeneous hill structure is considered. The efficient implementation on parallel distributed memory architectures will allow to perform real-time visualization and interactive physical investigations of 3D amplification phenomena for seismic risk assessment.

We present a review of the application of the spectral-element method to regional and global seismology. This technique is a high-order variational method that allows one to compute accurate synthetic seismograms in three-dimensional heterogeneous Earth models with deformed geometry. We first recall the strong and weak forms of the seismic wave equation with a particular emphasis set on fluid regions. We then discuss in detail how the conditions that hold on the boundaries, including coupling boundaries, are honored. We briefly outline the spectral-element discretization procedure and present the time-marching algorithm that makes use of the diagonal structure of the mass matrix. We show examples that illustrate the capabilities of the method and its interest in the context of the computation of three-dimensional synthetic seismograms.

We study a new curvilinear scheme for wave propagation modelling in presence of topography. The discrete scheme takes advantage of recent developments in areoacoustics. Our new scheme relies on the conventional grid coupled with optimized filters to remove numerical noise in case of strong material heterogeneity. We used non-centred stencils for free surface implementation and optimized the explicit Runge-Kutta scheme for the time differencing. We performed a complete theoretical stability and dispersion analysis of the discrete scheme. Finally, we illustrate the numerical accuracy of the new scheme by intensive experiments.

The higher-order finite-element scheme with mass lumping for triangles and tetrahedra is an efficient method for solving the wave equation. A number of lower-order elements have already been found. Here the search for elements of higher order is continued. Elements are constructed in a systematic manner. The nodes are chosen in a symmetric way. Integration rules must be exact up to a certain degree to maintain an overall accuracy that is the same as without mass lumping. First, for given integration degrees, consistent rule structures are derived for which integration formulas are likely to exist. Then, as each rule structure corresponds to a potential element of certain order, the position of element nodes and the integration weights can be found by solving the related system of nonlinear equations. With this systematic approach, a number of new sixth-order triangular elements and a new fourth-order tetrahedral element have been found.

High-order finite elements with mass lumping al-low for explicit time stepping when integrating the wave equa-tion. An earlier study suggests that this approach can be used for two-dimensional triangulations, but cannot be extended to tetrahedra. Here, however, a new element for tetrahedra is presented. Finite elements for triangles and tetrahedra are better suit-ed to model irregular surfaces and sharp contrasts in velocity models than standard finite differences on regular cartesian grids. The question is whether or not the superior accuracy of the finite element method allows for a reduction of the number of degrees of freedom that is large enough to bal-ance its higher cost. Here it is shown by a comparison on a simple two-dimensional reflection problem that the higher-order finite-element method is actually more efficient than the standard finite-difference method. In addition, a comparison between finite-element schemes of various order suggests that the higher-order approximations are more efficient than the lower-order ones.

Purely numerical methods based on finite-element approx-imation of the acoustic or elastic wave equation are becoming increasingly popular for the generation of synthetic seismo-grams. We present formulas for the grid dispersion and stabil-ity criteria for some popular finite-element methods FEM for wave propagation, namely, classical and spectral FEM. We develop an approach based on a generalized eigenvalue formulation to analyze the dispersive behavior of these FEMs for acoustic or elastic wave propagation that overcomes diffi-culties caused by irregular node spacing within the element and the use of high-order polynomials, as is the case for spec-tral FEM.Analysis reveals that for spectral FEM of order four or greater, dispersion is less than 0.2% at four to five nodes per wavelength, and dispersion is not angle dependent. New results can be compared with grid-dispersion results of some classical finite-difference methods FDM used for acoustic or elastic wave propagation. Analysis reveals that FDM and classical FEM require a larger sampling ratio than a spectral FEM to obtain results with the same degree of accuracy. The staggered-grid FDM is an efficient scheme, but the disper-sion is angle dependent with larger values along the grid axes. On the other hand, spectral FEM of order four or greater is isotropic with small dispersion, making it attractive for simu-lations with long propagation times.

One of the nagging problems which arises in application of discrete solution methods for wave‐propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh. These undesired events eventually override the actual seismic signals which propagate in the modeled region. The solution to avoiding boundary effects used to be to enlarge the numerical mesh, thus delaying the side reflections and wraparound longer than the range of times involved in the modeling. Obviously this solution considerably increases the expense of computation. More recently, nonreflecting boundary conditions were introduced for the finite‐difference method (Clayton and Enquist, 1977; Reynolds, 1978). These boundary conditions are based on replacing the wave equation in the boundary region by one‐way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh. This approach has been relatively successful, except that its effectiveness degrades for events which impinge on the boundaries at shallow angles. It is also not clear how to apply this type of boundary condition to global discrete methods such as the Fourier method for which all grid points are coupled.

Prediction of elastic anisotropic full wavefields is required in reverse-time migration, full-waveform inversion, borehole seismology, seismic modelling, and other processes. We propose a novel finite-difference algorithm based on solution of the Navier wave equation using a multi-block methodology. In the current implementation the blocks are sub-horizontal layers. A curvilinear adaptive hexahedral grid in blocks is generated by mapping the original 3D physical domain onto a parametric cube with horizontal layers and interfaces. These interfaces correspond to the main curvilinear physical contrast interfaces of a sub-horizontally layered formation. The top boundary of the parametric cube handles the land surface with a smooth topography. Free surface and solid-solid transmission boundary conditions at interfaces are approximated with the second-order accuracy. Smooth medium in the layers is approximated by second to sixth spatial order schemes. All expected properties of the developed algorithm are demonstrated in numerical tests using correspondent parallel MPI code.

SUMMARY Present-day computers allow for realistic 3D simulations of seismic wave propagation, as well as migration and inversion of seismic data with numerical solutions of the full wave equation. The finitedifference method is popular because of its simplicity but suffers from accuracy degradation for complex models with sharp interfaces between large impedance contrasts and for models with rough topography. A tetrahedral mesh offers more flexibility and maintains its accuracy if element boundaries are aligned with sharp interfaces. Higher-order finite elements with mass lumping provide a fully explicit time-stepping scheme. We have implemented elements of degree one, two, and three for the 3D acoustic wave equation. Numerical tests confirm the accuracy of the mass-lumped elements. There are two different third-degree elements that have almost the same accuracy, but one has a more favourable stability limit than the other. Convergence analysis shows that the higher the order of the element, the better the computational performance is. A low-storage implementation with OpenMP shows good scaling on 4-and 8-node platforms.

In this work, we propose a Spectral Finite Element method (SEM) for 3D acoustic wave equation modelling combined to an unstructured mesh adaptation method, the h-refinement approach. This method is well adapted to complex geometries and unstructured meshes, which is very important dealing with complex velocity models. Impulse response for modeling has been shown to validate the proposed SEM implementation. Moreover, it proved good parallelization speed up, which is critical for high performance computing purposes as required by Reverse Time Migration.

We solve the three-dimensional acoustic wave equation, discretized on tetrahedral meshes. Two methods are considered: mass-lumped continuous finite elements and the symmetric interior-penalty discontinuous Galerkin method (SIP-DG). Combining the spatial discretization with the leap-frog time-stepping scheme, which is second-order accurate and conditionally stable, leads to a fully explicit scheme. We provide estimates of its stability limit for simple cases, namely, the reference element with Neumann boundary conditions, its distorted version of arbitrary shape, the unit cube that can be partitioned into 6 tetrahedra with periodic boundary conditions, and its distortions. The CFL stability limit contains a length scale for which we considered different options. The one based on the sum of the eigenvalues of the spatial operator for the first degree mass-lumped element gives the best results. It resembles the diameter of the inscribed sphere but is slightly easier to compute. The stability estimates show that mass-lumped continuous and SIP-DG finite elements have comparable stability conditions, with the exception of the elements of the first degree. The stability limit for the mass-lumped elements is less restrictive and allows for larger time steps.

The air/earth interface is accurately represented in a 3D finite-difference elastic wave propagation algorithm merely by assigning material properties of air to the spatial grid nodes above the earth's surface. Computational stability is maintained by making the boundary gradational. Synthetic seismic traces calculated by this approach compare favorably with those computed by imposing an explicit stress-free condition on the surface.

The numerical solution of the initial value problem for the wave equation is considered for the case when the equation coefficients are piecewise smooth. This problem models linear wave propagation in a medium in which the properties of the medium change discontinuously at interfaces. Convergent difference approximations can be found that do not require the explicit specification of the boundary conditions at interfaces in the medium and hence are simple to program. Although such difference approximations typically can only be expected to be first-order accurate, the numerical phase velocity has the same accuracy as the difference approximation would if the coefficients in the differential equation were smoooth. This is proved for the one-dimensional case and demonstrated numerically for an example in two space dimensions in which the interface is not aligned with the computational mesh.

A wide class of difference equations is described for approximating discontinuous time dependent solutions, with prescribed initial data, of hyperbolic systems of nonlinear conservation laws. Among these schemes we determine the best ones, i.e., these which have the smallest truncation error and in which the discontinuities are confined to a narrow band of 2 to 3 meshpoints. These schemes are tested for stability and are found to be stable under a mild strengthening of the CourantFriedrichs-Lewy criterion. Test calculations of onedimensional flows of compressible fluids with shocks, rarefaction waves and contact discontinuities show excellent agreement with exact solutions. In particular, when Lagrange coordinates are used, there is no smearing of interfaces. The additional terms introduced into the difference scheme for the purpose of keeping the shock transition narrow are similar to, although not identical with, the artificial viscosity terms, and the like of them introduced by Richtmyer and von Neumann and elaborated by other workers in this field. (auth)

In this paper we use a modified equation analysis to develop formally fourth order accurate finite difference and pseudospectral methods for the one-dimensional wave equation. The difference scheme is constructed by performing a modified equation analysis of a centered, second-order conservative scheme to determine its dominant error term. Subtracting a centered discretization of this term from the scheme cancels the second order truncation errors. This technique yields a formally fourth order accurate explicit difference scheme that employs only three time levels. Similarly, the modified equation technique can be used to achieve fourth order time accuracy for the pseudospectral method with no increase in storage. The difference and pseudospectral schemes are fourth order convergent for constant coefficients even when a spatially singular forcing term is used for a source. Numerical results are given comparing the accuracy and efficiency of these methods for some model problems. Finally, we present a gene...

In this paper three models of parallel speedup are studied. They are fixed-size speedup, fixed-time speedup, and memory-bounded speedup. The latter two consider the relationship between speedup and problem scalability. Two sets of speedup formulations are derived for these three models. One set considers uneven workload allocation and communication overhead and gives more accurate estimation. Another set considers a simplified case and provides a clear picture on the impact of the sequential portion of an application on the possible performance gain from parallel processing. The simplified fixed-size speedup is Amdahl′s law. The simplified fixed-time speedup is Gustafson′s scaled speedup. The simplified memory-bounded speedup contains both Amdahl′s law and Gustafson′s scaled speedup as special cases. This study leads to a better understanding of parallel processing.

Using numerical integration in the formation of the finite element mass matrix and placing the movable nodes at integration points causes it to become lumped or diagonal (block diagonal) with the optimal rate of energy convergence retained.

We analyse 13 3-D numerical time-domain explicit schemes for modelling seismic wave propagation and earthquake motion for their behaviour with a varying P-wave to S-wave speed ratio (VP/VS). The second-order schemes include three finite-difference, three finite-element and one discontinuous-Galerkin schemes. The fourth-order schemes include three finite-difference and two spectral-element schemes. All schemes are second-order in time. We assume a uniform cubic grid/mesh and present all schemes in a unified form. We assume plane S-wave propagation in an unbounded homogeneous isotropic elastic medium. We define relative local errors of the schemes in amplitude and the vector difference in one time step and normalize them for a unit time. We also define the equivalent spatial sampling ratio as a ratio at which the maximum relative error is equal to the reference maximum error. We present results of the extensive numerical analysis.
We theoretically (i) show how a numerical scheme sees the P and S waves if the VP/VS ratio increases, (ii) show the structure of the errors in amplitude and the vector difference and (iii) compare the schemes in terms of the truncation errors of the discrete approximations to the second mixed and non-mixed spatial derivatives.
We find that four of the tested schemes have errors in amplitude almost independent on the VP/VS ratio.
The homogeneity of the approximations to the second mixed and non-mixed spatial derivatives in terms of the coefficients of the leading terms of their truncation errors as well as the absolute values of the coefficients are key factors for the behaviour of the schemes with increasing VP/VS ratio.
The dependence of the errors in the vector difference on the VP/VS ratio should be accounted for by a proper (sufficiently dense) spatial sampling.

In finite‐difference methods a seismic source can be implemented using either initial wavefield values or body forces. However, body forces can only be specified at finite‐difference nodes, and, if using initial values, a source cannot be located close to a reflecting boundary or interface in the model. Hence, difficulties can exist with these schemes when the region surrounding a source is heterogeneous or when a source either is positioned between nodes or is arbitrarily close to a free surface.
A completely general solution to these problems can be obtained by using Kaiser windowed sinc functions to define a small region around the true source location that contains several nodal body forces. Both monopole and dipole point sources can be defined, enabling many source types to be implemented in either acoustic or elastic media. Such a function can also be used to arbitrarily locate receivers. If the number of finite‐difference nodes per wavelength is four or more (and with a source region half‐width of only four nodes) this scheme results in insignificant phase errors and in amplitude errors of no more than 0.1%. Numerical examples for sources located less than one node from either a free surface or an image source demonstrate that the scheme can be used successfully for any surface‐source or multisource configuration.

Stable and accurate numerical modeling of seismic wave propagation in the vicinity of high-contrast interfaces is achieved with straightforward modifications to the conventional, rectangular-staggered-grid, finite-difference (FD) method. Improvements in material parameter averaging and spatial differencing of wavefield variables yield high-quality synthetic seismic data.

We have pursued two-dimensional (2D) finite-difference (FD) modelling of seismic scattering from free-surface topography. Exact free-surface boundary conditions for the particle velocities have been derived for arbitrary 2D topographies. The boundary conditions are combined with a velocity–stress formulation of the full viscoelastic wave equations. A curved grid represents the physical medium and its upper boundary represents the free-surface topography. The wave equations are numerically discretized by an eighth-order FD method on a staggered grid in space, and a leap-frog technique and the Crank–Nicholson method in time.
In order to demonstrate the capabilities of the surface topography modelling technique, we simulate incident point sources with a sinusoidal topography in seismic media of increasing complexities. We present results using parameters typical of exploration surveys with topography and heterogeneous media. Topography on homogeneous media is shown to generate significant scattering. We show additional effects of layering in the medium, with and without randomization, using a von Kármán realization of apparent anisotropy. Synthetic snapshots and seismograms indicate that prominent surface topography can cause back-scattering, wave conversions and complex wave patterns which are usually discussed in terms of inter-crust heterogeneities.

The pseudospectral (or Fourier) method has been used recently by several investigators for forward seis- mic modeling. The method is introduced here in two different ways: as a limit of finite differences of increas- ing orders, and by trigonometric interpolation. An argu- ment based on spectral analysis of a model equation shows that the pseudospectral method (for the accu- racies and integration times typical of forward elastic seismic modeling) may require, in each space dimension, as little as a quarter the number of grid points com- pared to a fourth-order finite-difference scheme and one-sixteenth the number of points as a second-order finite-difference scheme. For the total number of points in two dimensions, these factors become l/16 and l/256, respectively; in three dimensions, they become l/64 and 114 096, repectively In a series of test calculations on the two-dimensional elastic wave equation, only minor degradations are found in cases with variable coefficients and discontinu- ous interfaces.

The second-order central difference is often used to approximate the derivatives of the wave equation. It is demonstrated that gains in computational efficiency can be made by using high-order approximation for these derivatives. A 2-D model is used to illustrate the relative accuracy of O(DELTA t2, DELTA x2), O(DELTA t2, DELTA x4), O(DELTA t4, DELTA x4, and O(DELTA t4m DELTA x10) central- difference schemes. For practical illustrations, a 2-D form of the O(DELTA t4, DELTA x10) algorithm is used to compute the exploding reflector response of a salt-dome model and compared with a fine-grid O(DELTA t2, DELTA x4) result. Transmissive sponge-like boundary conditions are also examined and shown to be effective.-from Author

If higher-order finite elements are used to discretize the wave equation, spurious modes may occur. These modes are classified as unphysical and supposedly make elements of high order useless for accurate computations. This is in conflict with numerical experiments which appear to provide good results. Here Fourier analysis is used to investigate the behaviour of the numerical error for a number of higher-order one-dimensional finite elements. It is shown that the spurious modes have a contribution to the numerical error that behaves in a reasonable manner, and that higher-order elements can be more accurate than lower-order elements. Lumped elements with Gauss–Lobatto nodes appear to be the best choice.

Simple recursions are derived for calculating the weights in compact finite difference formulas for any order of derivative and to any order of accuracy on one-dimensional grids with arbitrary spacing. Tables are included for some special cases (of equispaced grids).

In this article, we construct new higher order finite element spaces for the approximation of the two-dimensional (2D) wave equation. These elements lead to explicit methods after time discretization through the use of appropriate quadrature formulas which permit mass lumping. These formulas are constructed explicitly. Error estimates are provided for the corresponding semidiscrete problem. Finally, higher order finite difference time discretizations are proposed and various numerical results are shown.

We present a new numerical method to solve the heterogeneous elastic wave equations formulated as a linear hyperbolic system using first-order derivatives with arbitrary high-order accuracy in space and time on 3-D unstructured tetrahedral meshes. The method combines the Discontinuous Galerkin (DG) Finite Element (FE) method with the ADER approach using Arbitrary high-order DERivatives for flux calculation. In the DG framework, in contrast to classical FE methods, the numerical solution is approximated by piecewise polynomials which allow 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. To define a suitable flux over the element surfaces, we solve so-called Generalized Riemann-Problems (GRP) at the element interfaces. The GRP solution provides simultaneously a numerical flux function as well as a time-integration method. The main idea is a Taylor expansion in time in which all time-derivatives are replaced by space derivatives using the so-called Cauchy-Kovalewski or Lax-Wendroff procedure which makes extensive use of the governing PDE. The numerical solution can thus be advanced for one time step without intermediate stages as typical, for example, for classical Runge-Kutta time stepping schemes. Due to the ADER time-integration technique, the same approximation order in space and time is achieved automatically. Furthermore, the projection of the tetrahedral elements in physical space on to a canonical reference tetrahedron allows for an efficient implementation, as many computations of 3-D integrals can be carried out analytically beforehand. Based on a numerical convergence analysis, we demonstrate that the new schemes provide very high order accuracy even on unstructured tetrahedral meshes and computational cost and storage space for a desired accuracy can be reduced by higher-order schemes. Moreover, due to the choice of the basis functions for the piecewise polynomial approximation, the new ADER-DG method shows spectral convergence on tetrahedral meshes. An application of the new method to a well-acknowledged test case and comparisons with analytical and reference solutions, obtained by different well-established methods, confirm the performance of the proposed method. Therefore, the development of the highly accurate ADER-DG approach for tetrahedral meshes provides a numerical technique to approach 3-D wave propagation problems in complex geometry with unforeseen accuracy.

Conventional finite-difference operators for numerical differentiation become progressively inaccurate at higher frequencies and therefore require very fine computational grids. This problem is avoided when the derivatives are computed by multiplication in the Fourier domain. However, because matrix transpositions are involved, efficient application of this method is restricted to computational environments where the complete data volume required by each computational step can be kept in random access memory.
To circumvent these problems a generalized numerical dispersion analysis for wave equation computations is developed. Operators for spatial differentiation can then be designed by minimizing the corresponding peak relative error in group velocity within a spatial frequency band. For specified levels of maximum relative error in group velocity ranging from 0.03% to 3%, differentiators have been designed that have the largest possible bandwidth for a given operator length.
The relation between operator length and the required number of grid points per shortest wavelength, for a required accuracy, provides a useful starting point for the design of cost-effective numerical schemes. To illustrate this, different alternatives for numerical simulation of the time evolution of acoustic waves in three-dimensional inhomogeneous media are investigated. It is demonstrated that algorithms can be implemented that require fewer arithmetic and I/O operations by orders of magnitude compared to conventional second-order finite-difference schemes to yield results with a specified minimum accuracy.

A 2D numerical finite-difference algorithm accounting for surface topography is presented. Higher-order, dispersion-bounded, cost-optimized finite-difference operators are used in the interior of the numerical grid, while non-reflecting absorbing boundary conditions are used along the edges. Transformation from a curved to a rectangular grid achieves the modelling of the surface topography. We use free-surface boundary conditions along the surface. In order to obtain complete modelling of the effects of wave propagation, it is important to account for the surface topography, otherwise near-surface effects, such as scattering, are not modelled adequately. Even if other properties of the medium, for instance randomization, can improve numerical simulations, inclusion of the surface topography makes them more realistic.

In this paper we present a Legendre spectral element method for solution of multi-dimensional unsteady change-of-phase Stefan problems. The spectral element method is a high-order (p-type) finite element technique, in which the computational domain is broken up into general (curved) quadrilateral macroelements, and the solution, data and geometry are expanded within each element in terms of tensor-product Lagrangian interpolants. The discrete equations are generated by a Galerkin formulation followed by Gauss–Lobatto Legendre quadrature, for which it is shown that exponential convergence to smooth solutions is obtained as the polynomial order of fixed elements is increased. The spectral element equations are inverted by conjugate gradient iteration, in which the matrix-vector products are calculated efficiently using tensor-product sum-factorization.To solve the Stefan problem numerically, the heat equations in the liquid and solid phases are transformed to fixed domains applying an interface-local time-dependent immobilization transformation technique. The modified heat equations are discretized using finite differences in time, resulting at each time step in a Helmholtz equation in space that is solved using Legendre spectral element elliptic discretizations. The new interface position is then computed using a variationally consistent flux treatment along the phase boundary, and the solution is projected upon the corresponding updated mesh. The rapid convergence rate and stability of the method are discussed, and numerical results are presented for a one-dimensional Stefan problem using both a semi-implicit and a fully implicit time-stepping scheme. Finally, a two-dimensional Stefan problem with a complex phase boundary is solved using the semi-implicit scheme.

We present an introduction to the spectral element method, which provides an innovative numerical approach to the calculation of synthetic seismograms in 3-D earth models. The method combines the flexibility of a finite element method with the accuracy of a spectral method. One uses a weak formulation of the equations of motion, which are solved on a mesh of hexahedral elements that is adapted to the free surface and to the main internal discontinuities of the model. The wavefield on the elements is discretized using high-degree Lagrange interpolants, and integration over an element is accomplished based upon the Gauss–Lobatto–Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix, which greatly simplifies the algorithm. We illustrate the great potential of the method by comparing it to a discrete wavenumber/reflectivity method for layer-cake models. Both body and surface waves are accurately represented, and the method can handle point force as well as moment tensor sources. For a model with very steep surface topography we successfully benchmark the method against an approximate boundary technique. For a homogeneous medium with strong attenuation we obtain excellent agreement with the analytical solution for a point force.

Lax-Wendroff and Nyström methods are numerical algorithms of temporal approximations for solving differential equations. These methods provide efficient algorithms for high-accuracy seismic modeling. In the context of spatial pseudospectral discretizations, I explore these two kinds of methods in a comparative way. Their stability and dispersion relation are discussed in detail. Comparison between the fourth-order Lax-Wendroff method and a fourth-order Nyström method shows that the Nyström method has smaller stability limit but has a better dispersion relation, which is closer to the sixth-order Lax-Wendroff method. The structure-preserving property of these methods is also revealed. The Lax-Wendroff methods are a second-order symplectic algorithm, which is independent of the order of the methods. This result is useful for understanding the error growth of Lax-Wendroff methods. Numerical experiments based on the scalar wave equation are performed to test the presented schemes and demonstrate the advantages of the symplectic methods over the nonsymplectic ones.

We investigate the stability of some high-order finite element methods, namely the spectral element method and the interior-penalty discontinuous Galerkin method (IP-DGM), for acoustic or elastic wave propagation that have become increasingly popular in the recent past. We consider the Lax-Wendroff method (LWM) for time stepping and show that it allows for a larger time step than the classical leap-frog finite difference method, with higher-order accuracy. In particular the fourth-order LWM allows for a time step 73 per cent larger than that of the leap-frog method; the computational cost is approximately double per time step, but the larger time step partially compensates for this additional cost. Necessary, but not sufficient, stability conditions are given for the mentioned methods for orders up to 10 in space and time. The stability conditions for IP-DGM are approximately 20 and 60 per cent more restrictive than those for SEM in the acoustic and elastic cases, respectively.

SUMMARYA method is proposed for accurately describing arbitrary-shaped free boundaries in finite-difference schemes for elastodynamics, in a time-domain velocity–stress framework. The basic idea is as follows: fictitious values of the solution are built in vacuum, and injected into the numerical integration scheme near boundaries. The most original feature of this method is the way in which these fictitious values are calculated. They are based on boundary conditions and compatibility conditions satisfied by the successive spatial derivatives of the solution, up to a given order that depends on the spatial accuracy of the integration scheme adopted. Since the work is mostly done during the pre-processing step, the extra computational cost is negligible. Stress-free conditions can be designed at any arbitrary order without any numerical instability, as numerically checked. Using 10 grid nodes per minimal S-wavelength with a propagation distance of 50 wavelengths yields highly accurate results. With 5 grid nodes per minimal S-wavelength, the solution is less accurate but still acceptable. A subcell resolution of the boundary inside the Cartesian meshing is obtained, and the spurious diffractions induced by staircase descriptions of boundaries are avoided. Contrary to what occurs with the vacuum method, the quality of the numerical solution obtained with this method is almost independent of the angle between the free boundary and the Cartesian meshing.