mixed spectral finite elements for the linear elasticity system in unbounded domains in this paper we present a mixed formulation of a spectral element approximation of the linear elasticity system after studying the main features of this approach we construct perfectly matched layers pmls for modeling unbounded domains then algorithmic issues are discussed and numerical results are given

dynamic earthquake rupture simulations on nonplanar faults embedded in 3d geometrically complex heterogeneous elastic solids dynamic propagation of shear ruptures on a frictional interface in an elastic solid is a useful idealization of natural earthquakes the conditions relating discontinuities in particle velocities across fault zones and tractions acting on the fault are often expressed as nonlinear friction laws the corresponding initial boundary value problems are both numerically and computationally challenging in addition seismic waves generated by earthquake ruptures must be propagated for many wavelengths away from the fault therefore reliable and efficient numerical simulations require both provably stable and high order accurate numerical methods we present a high order accurate finite difference method for a enforcing nonlinear friction laws in a consistent and provably stable manner suitable for efficient explicit time integration b dynamic propagation of earthquake ruptures along nonplanar faults and c accurate propagation of seismic waves in heterogeneous media with free surface topography we solve the first order form of the 3d elastic wave equation on a boundary conforming curvilinear mesh in terms of particle velocities and stresses that are collocated in space and time using summation by parts sbp finite difference operators in space boundary and interface conditions are imposed weakly using penalties by deriving semi discrete energy estimates analogous to the continuous energy estimates we prove numerical stability the finite difference stencils used in this paper are sixth order accurate in the interior and third order accurate close to the boundaries however the method is applicable to any spatial operator with a diagonal norm satisfying the sbp property time stepping is performed with a 4th order accurate explicit low storage runge kutta scheme thus yielding a globally fourth order accurate method in both space and time we show numerical simulations on band limited self similar fractal faults revealing the complexity of rupture dynamics on rough faults

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

multidirectional vector based elastic reverse time migration and angle domain common image gathers with approximate wavefield decomposition of p and s waves abstractelastic reverse time migration e rtm has limitations when the migration velocities contain strong contrasts first the traditional scheme of p s wave mode separation is based on helmholtz s equations which ignore the conversion between p and s waves at the current separation time thus it contains an implicit assumption of the constant shear modulus and requires smoothing the heterogeneous model to approximately satisfy a locally constant condition second the vector based imaging condition needs to use the reflection image normal and it also cannot give the correct polarity of the pp image in all possible conditions third the angle domain common image gathers adcigs calculated using the poynting vectors pvs do not consider the wave interferences that happen at each reflector therefore smooth models are often used for e rtm we relax this condition by proposing an improved data flow that involves three new contributions the first contribution is an improved system of p s wave mode

input optimization i optimal realizations of mathematical models mathematical models are considered as input output systems the input is data technological coefficients available energy prices and the output is the feasible set the set of optimal solutions and the optimal value we study when output is a continuous function of input and identify optimal minimal realizations of mathematical models these are states of the model having the property that every stable perturbation of input results in a locally worse higher value of the optimal value function in input optimization we optimize mathematical model rather than a specific mathematical program

a perfectly matched layer for the time dependent wave equation in heterogeneous and layered media a mathematical analysis of the perfectly matched layer pml for the time dependent wave equation in heterogeneous and layered media is presented we prove the stability of the pml for discontinuous media with piecewise constant coefficients and derive energy estimates for discontinuous media with piecewise smooth coefficients we consider a computational setup consisting of smaller structured subdomains that are discretized using high order accurate finite difference operators for approximating spatial derivatives the subdomains are then patched together into a global domain by a weak enforcement of interface conditions using penalties in order to ensure the stability of the discrete pml it is necessary to transform the interface conditions to include the auxiliary variables in the discrete setting the transformed interface conditions are crucial in deriving discrete energy estimates analogous to the continuous energy estimates thus proving stability and convergence of the numerical method finally we present numerical experiments demonstrating the stability of the pml in a layered medium and high order accuracy of the proposed interface conditions

characterizing an optimal input in perturbed convex programming when every feasible stable perturbation of data results in a non improvement of the optimal value function then we talk about an optimal input or an optimal selection of data in this paper we describe such data for convex programs using perturbed saddle points

sh wave propagation in heterogeneous media velocity stress finite difference method a new finite difference fd method is presented for modeling sh wave propagation in a generally heterogeneous medium this method uses both velocity and stress in a discrete grid density and shear modulus are similarly discretized avoiding any spatial smoothing therefore boundaries will be correctly modeled under an implicit formulation standard problems quarter plane propagation sedimentary basin propagation are studied to compare this method with other methods finally a more complex example a salt dome inside a two layered medium shows the effect of lateral propagation on seismograms recorded at the surface a corner wave always in phase with the incident wave and a head wave will appear which will pose severe problems of interpretation with the usual vertical migration methods

stable and high order accurate boundary treatments for the elastic wave equation on second order form stable and high order accurate boundary treatments for the elastic wave equation on second order form

solutions to range dependent benchmark problems by the finite difference method an explicit second order finite difference scheme has been used to solve the elastic wave equation in the time domain solutions are presented for the perfect wedge the lossless penetrable wedge and the plane parallel waveguide that have been proposed as benchmarks by the acoustical society of america good agreement with reference solutions is obtained if the media is discretized at 20 gridpoints per wavelength there is a major discrepancy up to 20 db in reference source level because the reference solutions are normalized to the source strength at 1 m in the model but the finite difference solutions are normalized to the source strength at 1 m in a homogeneous medium the finite difference method requires computational times between 10 and 20 h on a super minicomputer without an array processor the method has the advantage of providing phase information and when run for a pulse source of providing insight into the evolution of the wave field and energy partitioning more complex models including velocity gradients and strong lateral heterogeneities can be solved with no additional computational effort the method has also been formulated to include shear wave effects

multi bump solutions for a strongly indefinite semilinear schrodinger equation without symmetry or convexity assumptions abstract in this paper we study the following semilinear schrodinger equation with periodic coefficient u v x u f x u u h 1 r n the functional corresponding to this equation possesses strongly indefinite structure the nonlinear term f x t satisfies some superlinear growth conditions and need not be odd or increasing in t using a new variational reduction method and a generalized morse theory we proved that this equation has infinitely many geometrically different solutions furthermore if the solutions of this equation under some energy level are isolated then we can show that this equation has infinitely many m bump solutions for any positive integer m 2

ground state and multiple solutions via generalized nehari manifold abstract by using variational methods and the generalized nehari manifold due to szulkin and weth the existence of the ground states and the multiplicity of solutions for a wide class of superlinear elliptic equations is studied

ground state solutions for some indefinite variational problems abstract we consider the nonlinear stationary schrodinger equation u v x u f x u in r n here f is a superlinear subcritical nonlinearity and we mainly study the case where both v and f are periodic in x and 0 belongs to a spectral gap of v inspired by previous work of li et al 2006 11 and pankov 2005 13 we develop an approach to find ground state solutions i e nontrivial solutions with least possible energy the approach is based on a direct and simple reduction of the indefinite variational problem to a definite one and gives rise to a new minimax characterization of the corresponding critical value our method works for merely continuous nonlinearities f which are allowed to have weaker asymptotic growth than usually assumed for odd f we obtain infinitely many geometrically distinct solutions the approach also yields new existence and multiplicity results for the dirichlet problem for the same type of equations in a bounded domain

energy conservative sbp discretizations of the acoustic wave equation in covariant form on staggered curvilinear grids abstract we develop a numerical method for solving the acoustic wave equation in covariant form on staggered curvilinear grids in an energy conserving manner the use of a covariant basis decomposition leads to a rotationally invariant scheme that outperforms a cartesian basis decomposition on rotated grids the discretization is based on high order summation by parts sbp operators and preserves both symmetry and positive definiteness of the contravariant metric tensor to improve accuracy and decrease computational cost we also derive a modified discretization of the metric tensor that leads to a conditionally stable discretization bounds are derived that yield a point wise condition that can be evaluated to check for stability of the modified discretization this condition shows that the interpolation operators should be constructed such that their norm is close to one

interface waves in almost incompressible elastic materials we study the problem of two elastic half planes in contact and the stoneley interface wave that may exist at the interface between two different elastic materials emphasis being put on the case when the half planes are almost incompressible we show that numerical simulations involving interface waves require an unexpectedly high number of grid points per wavelength as the materials become more incompressible let and be the lame parameters and densities of the first and second half plane respectively a theoretical study shows that if k is a real constant k k k and 0 then for an accurate solution the required number of grid points per wavelength scales as 1 p where p is the order of accuracy of the numerical method this requirement becomes very restrictive close to the incompressible limit especially for lower order methods i e a small p the theoretical findings are supported by numerical experiments that illustrate the demanding resolution requirement as well as the superiority of higher order methods the scaling is also seen to hold for a more general choice of lame parameters numerical experiments when one of the half planes is a vacuum are also presented where the higher resolution requirement is illustrated in a numerical solution of lamb s problem

fourth order finite difference p sv seismograms i describe the properties of a fourth order accurate space second order accurate time two dimensional p sk finite difference scheme based on the madariagavirieux staggered grid formulation the numerical scheme is developed from the first order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses the madariaga virieux staggered grid scheme has the desirable quality that it can correctly model any variation in material properties including both large and small poisson s ratio materials with minimal numerical dispersion and numerical anisotropy dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints wavelength the scheme can be used to accurately simulate wave propagation in mixed acoustic elastic media making it ideal for modeling marine problems explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free surface or within a layer and to satisfy free surface boundary conditions benchmark comparisons of finite difference and analytical solutions to lamb s problem are almost identical as are comparisons of finite difference and reflectivity solutions for elastic elastic and acoustic elastic layered models

boundary waves and stability of the perfectly matched layer for the two space dimensional elastic wave equation in second order form boundary waves and stability of the perfectly matched layer for the two space dimensional elastic wave equation in second order form

short note on high order finite difference metric discretizations satisfying gcl on moving and deforming grids in this note we generalize our previous treatment of the discretizations of geometric conservation laws on steady grids vinokur and yee 2000 to general time dependent grids the commutative property of mixed difference operators is generalized to apply to time metrics and jacobians our treatment uses half the number of terms as those used in a recent paper by abe et al 2012 we also derive the proper temporal discretizations of both runge kutta and linear multistep methods to satisfy the commutativity property for higher than first order

discretizing singular point sources in hyperbolic wave propagation problems we develop high order accurate source discretizations for hyperbolic wave propagation problems in first order formulation that are discretized by finite difference schemes by studying the fourier series expansions of the source discretization and the finite difference operator we derive sufficient conditions for achieving design accuracy in the numerical solution only half of the conditions in fourier space can be satisfied through moment conditions on the source discretization and we develop smoothness conditions for satisfying the remaining accuracy conditions the resulting source discretization has compact support in physical space and is spread over as many grid points as the number of moment and smoothness conditions in numerical experiments we demonstrate high order of accuracy in the numerical solution of the 1 d advection equation both in the interior and near a boundary the 3 d elastic wave equation and the 3 d linearized euler equations

boundary estimates for the elastic wave equation in almost incompressible materials we study the half plane problem for the elastic wave equation subject to a free surface boundary condition with particular emphasis on almost incompressible materials a normal mode analysis is developed to estimate the solution in terms of the boundary data showing that the problem is boundary stable the dependence on the material properties which is difficult to analyze by the energy method is made transparent by our estimates the normal mode technique is used to analyze the influence of truncation errors in a finite difference approximation our analysis explains why the number of grid points per wave length must be increased when the shear modulus mu becomes small compared to the first lame parameter lambda that is for almost incompressible materials when the surface waves are scaled to have unit wave length our analysis predicts that the grid size must be proportional to mu lambda 1 2 for a second order method for a fourth order method the grid size can be proportional t

two dimensional nonlinear inversion of seismic waveforms numerical results the nonlinear problem of inversion of seismic waveforms can be set up using least squares methods the inverse problem is then reduced to the problem of minimizing a lp nonquadratic function in a space of many 104to106 variables using gradient methods leads to iterative algorithms each iteration implying a forward propagation generated by the actual sources a backward propagation generated by the data residuals acting as if they were sources and a correlation at each point of the space of the two fields thus obtained which gives the updated model the quality of the results of any inverse method depends heavily on the realism of the forward modeling finite difference schemes are a good choice relative to realism because although they are time consuming they give excellent results numerical tests performed with multioffset synthetic data from a two dimensional model prove the feasibility of the approach if only surface recorded reflections are used the high spatial frequency content of the

on the continuity of a lagrangian multiplier function in input optimization we show that a lagrangian multiplier function in input optimization is generally discontinuous on regions of stability and then we find conditions for its continuity

stable difference approximations for the elastic wave equation in second order formulation we consider the three dimensional elastic wave equation for an isotropic heterogeneous material subject to a stress free boundary condition building on our recently developed theory for difference methods for second order hyperbolic systems h o kreiss n a petersson j ystrom siam j numer anal 40 2002 pp 1940 1967 we develop an explicit second order accurate technique which is stable for all ratios of longitudinal over transverse phase velocities the spatial discretization is self adjoint and the stability is obtained through an energy estimate seismic events are often modeled using singular source terms and we devise a technique to place sources independently of the grid while retaining second order accuracy away from the source several numerical examples are given

incorporation of attenuation into time domain computations of seismic wave fields the only numerically tractable way yet found to incorporate attenuation into numerical time domain computations of seismic wave fields is to approximate the viscoelastic modulus by a low order rational function of frequency the coefficients of this function can be determined by the pade approximation our test computations show however that this approximation generally is of poor quality therefore we suggest a new approach which is based on the rheological model of the generalized maxwell body which has a modulus of the desired rational form we choose the relaxation frequencies logarithmically equidistant in the frequency band of interest and determine the weight factors by simple numerical curve fitting to an arbitrary q law this approach is superior to the method above both in accuracy and in computational efficiency for most practical applications approximations of orders 2 or 3 are sufficient the computing time and memory requirements for a finite difference calculation are then approximately twice those of a purely elastic calculation as a first application of the method we compute sh channel waves in discontinuous coal seams with q 50 within the coal the results show that the high frequency airy phase is strongly attenuated this indicates that care has to be taken in comparing the results of purely elastic model calculations of the propagation of seam waves with experimental data

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

dispersion relation preserving stereo modeling method beyond nyquist frequency for acoustic wave equation abstractaccording to nyquist shannon sampling theory two sampling points of the displacements per minimal wavelength are required to reconstruct a wavefield however high frequency components above the nyquist frequency can be theoretically recovered by introducing spatial derivatives of the displacements we have developed a new finite difference scheme for solving the acoustic equation named the dispersion relation preserving stereo modeling method which is practical for seismic forward modeling beyond the nyquist frequency the key idea in this model is to approximate the high order spatial derivatives with wavenumber domain optimization in wavefields represented by the wave displacements and their gradients we investigated the theoretical properties of this method including the absolute error of spatial operators numerical error stability criterion and computational efficiency our results indicated that the method can effectively suppress the numerical dispersion on extremely coarse grids wi

unstructured grid method for stress wave propagation in elastic media a new numerical method named unstructured grid method is presented to calculate two dimensional stress wave propagation problems in elastic media this method is based on the dynamic equilibrium equations of the investigated lumps that are formed among the auxiliary unstructured grids the approach to the solution is obtained by calculating the node displacements and the central point stresses of the spatial grids alternately the stability criterion of the method is given and tested numerically analysis of dispersion for numerical phase velocity is studied the rule is provided to determine the spatial step size of discretization mesh so as to ensure the numerical accuracy of the unstructured grid method numerical results are compared with the available exact solutions and with the solutions of central difference method for demonstrating the good accuracy of the new method the unstructured grid method possesses higher calculating speed than the central difference method finally the dynamic behavior of a square plate with a hole under an impact loading is researched

stable and high order accurate difference methods for the elastic wave equation in discontinuous media in this paper we develop a stable and systematic procedure for numerical treatment of elastic waves in discontinuous and layered media we consider both planar and curved interfaces where media parameters are allowed to be discontinuous the key feature is the highly accurate and provably stable treatment of interfaces where media discontinuities arise we discretize in space using high order accurate finite difference schemes that satisfy the summation by parts rule conditions at layer interfaces are imposed weakly using penalties by deriving lower bounds of the penalty strength and constructing discrete energy estimates we prove time stability we present numerical experiments in two space dimensions to illustrate the usefulness of the proposed method for simulations involving typical interface phenomena in elastic materials the numerical experiments verify high order accuracy and time stability

encapsulated high order difference operators on curvilinear non conforming grids constructing stable difference schemes on complex geometries is an arduous task even fairly simple partial differential equations end up very convoluted in their discretized form making them diff

3d weak dispersion reverse time migration using a stereo modeling operator reliable 3d imaging is a required tool for developing models of complex geologic structures reverse time migration rtm as the most powerful depth imaging method has become the preferred imaging tool because of its ability to handle complex velocity models including steeply dipping interfaces and large velocity contrasts finite difference methods are among the most popular numerical approaches used for rtm however these methods often encounter a serious issue of numerical dispersion which is typically suppressed by reducing the grid interval of the propagation model resulting in large computation and memory requirements in addition even with small grid spacing numerical anisotropy may degrade images or worse provide images that appear to be focused but position events incorrectly recently stereo operators have been developed to approximate the partial differential operator in space these operators have been used to develop several weak dispersion and efficient stereo modeling methods that have been found to be superior to conventional algorithms in suppressing numerical dispersion and numerical anisotropy we generalized one stereo modeling method fourth order nearly analytic central difference nacd from 2d to 3d and applied it to 3d rtm the rtm results for the 3d seg eage phase a classic data set 1 and the seg advanced modeling project model demonstrated that even when using a large grid size the nacd method can handle very complex velocity models and produced better images than can be obtained using the fourth order and eighth order lax wendroff correction lwc schemes we also applied 3d nacd and fourth order lwc to a field data set and illustrated significant improvements in terms of structure imaging horizon layer continuity and positioning we also investigated numerical dispersion and found that not only does the nacd method have superior dispersion characteristics but also that the angular variation of dispersion is significantly less than for lwc

stable and accurate wave propagation in discontinuous media a time stable discretization is derived for the second order wave equation with discontinuous coefficients the discontinuity corresponds to inhomogeneity in the underlying medium and is treated by splitting the domain each homogeneous sub domain is discretized using narrow diagonal summation by parts operators and then patched to its neighbors by using a penalty method leading to fully explicit time integration this discretization yields a time stable and efficient scheme the analysis is verified by numerical simulations in one dimension using high order finite difference discretizations and in three dimensions using an unstructured finite volume discretization

stable and accurate schemes for the compressible navier stokes equations minimal stencil width discretizations of combined mixed and non mixed second order derivatives are analyzed with respect to accuracy and stability we show that these discretizations lead to stability for cauchy problems with a careful boundary treatment we also show that the stability holds for initial boundary value problems the analysis is verified by numerical simulations of burgers and navier stokes equations in two and three space dimensions

wave propagation in anisotropic elastic materials and curvilinear coordinates using a summation by parts finite difference method we develop a fourth order accurate finite difference method for solving the three dimensional elastic wave equation in general heterogeneous anisotropic materials on curvilinear grids the proposed method is an extension of the method for isotropic materials previously described in the paper by sjogreen and petersson 2012 11 the proposed method discretizes the anisotropic elastic wave equation in second order formulation using a node centered finite difference method that satisfies the principle of summation by parts the summation by parts technique results in a provably stable numerical method that is energy conserving we also generalize and evaluate the super grid far field technique for truncating unbounded domains unlike the commonly used perfectly matched layers pml the super grid technique is stable for general anisotropic material because it is based on a coordinate stretching combined with an artificial dissipation as a result the discretization satisfies an energy estimate proving that the numerical approximation is stable we demonstrate by numerical experiments that sufficiently wide super grid layers result in very small artificial reflections applications of the proposed method are demonstrated by three dimensional simulations of anisotropic wave propagation in crystals