a high order discontinuous galerkin method for wave propagation through coupled elastic acoustic media we introduce a high order discontinuous galerkin dg scheme for the numerical solution of three dimensional 3d wave propagation problems in coupled elastic acoustic media a velocity strain formulation is used which allows for the solution of the acoustic and elastic wave equations within the same unified framework careful attention is directed at the derivation of a numerical flux that preserves high order accuracy in the presence of material discontinuities including elastic acoustic interfaces explicit expressions for the 3d upwind numerical flux derived as an exact solution for the relevant riemann problem are provided the method supports h non conforming meshes which are particularly effective at allowing local adaptation of the mesh size to resolve strong contrasts in the local wavelength as well as dynamic adaptivity to track solution features the use of high order elements controls numerical dispersion enabling propagation over many wave periods we prove consistency and stability of the proposed dg scheme to study the numerical accuracy and convergence of the proposed method we compare against analytical solutions for wave propagation problems with interfaces including rayleigh lamb scholte and stoneley waves as well as plane waves impinging on an elastic acoustic interface spectral rates of convergence are demonstrated for these problems which include a non conforming mesh case finally we present scalability results for a parallel implementation of the proposed high order dg scheme for large scale seismic wave propagation in a simplified earth model demonstrating high parallel efficiency for strong scaling to the full size of the jaguar cray xt5 supercomputer

numerical dispersion analysis of discontinuous galerkin method with different basis functions for acoustic and elastic wave equations the discontinuous galerkin method dgm has been applied to investigate seismic wave propagation recently however few studies have examined the dispersion property of dgm with different basis functions therefore three common basis functions legendre polynomial lagrange polynomial with equidistant nodes and lagrange polynomial with gauss lobatto legendre gll nodes are employed for numerical approximation the numerical dispersion and anisotropy numerical behavior of acoustic and elastic waves are compared and the numerical errors of different order methods are analyzed the result shows that the dispersion errors for all basis functions reduce generally with increasing interpolation orders but with large differences in different directions specifically the legendre basis function and lagrange basis function with gll nodes have attractive advantages over the lagrange polynomial with equidistant nodes for numerical computation we verify the dispersion properties by theoretical and numerical ana

two dimensional stationary schrodinger equation via the dressing method new exactly solvable potentials wave functions and their physical interpretation the classes of exactly solvable multiline soliton potentials and corresponding wave functions of two dimensional stationary schrodinger equation via dressing method are constructed and their physical interpretation is discussed

high accuracy wave simulation revised derivation numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation the nearly analytic integration discrete naid method for solving the two dimensional acoustic wave equation has been fully mathematically revised analyzed and tested the naid method is an alternative numerical modeling method for generating synthetic seismograms the acoustic wave equation is first transformed into a system of first order ordinary differential equations odes with respect to time variable t and then directly integrated at a small time interval of tn tn 1 to obtain semi discrete ordinary differential equations the integral kernel is expanded into a truncated taylor series to which the integration operator is explicitly applied the high order temporal derivatives involved in the integral kernel are replaced by high order spatial derivatives which then are approximately calculated as a weighted linear combination of the displacement the particle velocity and their spatial gradients in this article we investigate the theoretical properties of the revised naid method including the discrete error and the stability criteria numerical results for constant and layered velocity models show that comparing to the lax wendroff correction lwc scheme and the staggered grid finite difference method the naid method can effectively suppress the numerical dispersion and source noises caused by the discretization of the acoustic wave equation with too coarse spatial grids or when models have strong velocity contrasts between adjacent layers the proposed naid method has been applied in computing the acoustic wavefields for two heterogeneous models the corner edge model and the marmousi model promising numerical results illustrate that the naid method provides an encouraging tool for large scale and complex wave simulation and inversion problems based on the acoustic equation

reverse time migration from topography abstractmigration of seismic data from topography using methods based on finite difference fd approximation to acoustic wave propagation commonly suffers from a number of imaging drawbacks due to the difficulty of applying fd stencils to irregular computational meshes altering the computational geometry from cartesian to a topographic coordinate system conformal to the data acquisition surface can circumvent many of these issues the coordinate transformation approach allows for acoustic wave propagation and the crosscorrelation and inverse scattering imaging conditions to be posed and computed directly in topographic coordinates resulting reverse time migration rtm images may then be interpolated back to the cartesian domain using the known inverse mapping orthogonal 2d topographic coordinates can be developed using known conformal mapping transforms and serve as the computational mesh for performing migration from topography impulse response tests demonstrate the accuracy of the 2d generalized a

discontinuous galerkin finite element method for the wave equation the symmetric interior penalty discontinuous galerkin finite element method is presented for the numerical discretization of the second order wave equation the resulting stiffness matrix is symmetric positive definite and the mass matrix is essentially diagonal hence the method is inherently parallel and leads to fully explicit time integration when coupled with an explicit time stepping scheme optimal a priori error bounds are derived in the energy norm and the l 2 norm for the semidiscrete formulation in particular the error in the energy norm is shown to converge with the optimal order cal o h min s ell with respect to the mesh size h the polynomial degree ell and the regularity exponent s of the continuous solution under additional regularity assumptions the l 2 error is shown to converge with the optimal order cal o h min s ell numerical results confirm the expected convergence rates and illustrate the versatility of the method

depth imaging with multiples depth imaging with multiples is a prestack depth migration method that uses multiples as the signal for more accurate boundary mapping and amplitude recovery the idea is partially related to model based multiple suppression techniques and reverse time depth migration conventional reverse time migration uses the two way wave equation for the backward wave propagation of recorded seismic traces and ray tracing or the eikonal equation for the forward traveltime computation the excitation time imaging principle consequently reverse time migration differs little from most other one way wave equation or ray tracing migration methods which expect only primary reflection events because it is almost impossible to attenuate multiples without degrading primaries there has been a compelling need to devise a tool to use multiples constructively in data processing rather than attempting to destroy them furthermore multiples and other nonreflecting wave types can enhance boundary imaging and amplitude recovery

an efficient symplectic reverse time migration method using a local nearly analytic discrete operator in acoustic transversely isotropic media with a vertical symmetry axis abstractreverse time migration rtm has become an extremely important imaging method due to its ability of handling complicated velocity models in recent years there has been an increasing need to perform rtm with anisotropic velocity models such as transversely isotropic media with a vertical symmetry axis vti finite difference fd is one of the most widely used numerical methods in rtm however it often suffers from serious numerical dispersion resulting in significant loss of imaging resolution over the years a nearly analytic discrete operator has been developed to approximate the high order spatial partial differential operators which is effective in suppressing numerical dispersion using this operator we developed a nearly analytic symplectic nas method to numerically solve acoustic vti wave equations which preserves the symplectic structure we have investigated its properties including the stability condition numerical dispersion relation and computational efficiency also we a

grid dispersion and stability criteria of some common finite element methods for acoustic and elastic wave equations purely numerical methods based on finite element approximation of the acoustic or elastic wave equation are becoming increasingly popular for the generation of synthetic seismograms we present formulas for the grid dispersion and stability 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 difficulties caused by irregular node spacing within the element and the use of high order polynomials as is the case for spectral 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 dispersion 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 simulations with long propagation times

a nearly analytic discrete method for solving the acoustic wave equations in the frequency domain abstractwe have developed a nearly analytic discrete nad method to discretize frequency domain acoustic wave equations with an absorbing boundary condition we evaluate in detail the discrete process of wave equations to derive a linear system the sparse structure and eigenproperties of its coefficient matrix also called the impedance matrix were analyzed to reveal the intrinsic difficulty in solving the linear system efficiently to accelerate the forward modeling process in the frequency domain we introduce a class of inexact rotated block triangular preconditioners incorporated with krylov subspace methods to solve this linear system and test their numerical behaviors by comparing with other two commonly used preconditioners to this end we perform wavefield simulation by the nad method and another two conventional numerical schemes in various media numerical dispersion analysis and waveform comparison are also implemented for these numerical schemes our results show the superiority of our pro

inversion of seismic reflection data in the acoustic approximation the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation the method is based on the generalized least squares criterion and it can handle errors in the data set and a priori information on the model multiply reflected energy is naturally taken into account as well as refracted energy or surface waves the inverse problem can be solved using an iterative algorithm which gives at each iteration updated values of bulk modulus density and time source function each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation backward in time of the data residuals the correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models this shows in particular that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commerc

optimal discontinuous galerkin methods for wave propagation we have developed and analyzed a new class of discontinuous galerkin methods dg which can be seen as a compromise between standard dg and the finite element fe method in the way that it is explicit like standard dg and energy conserving like fe in the literature there are many methods that achieve some of the goals of explicit time marching unstructured grid energy conservation and optimal higher order accuracy but as far as we know only our new algorithms satisfy all the conditions we propose a new stability requirement for our dg the stability analysis is based on the careful selection of the two fe spaces which verify the new stability condition the convergence rate is optimal with respect to the order of the polynomials in the fe spaces moreover the convergence is described by a series of numerical experiments

interior estimates for the n dimensional abreuʼs equation abstract we study the abreu s equation in n dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform k stability

time varying boundary conditions in simulation of seismic wave propagation we propose two new boundary conditions to regulate coherent reflections from the model boundaries in numerical solutions of wave equations both boundary conditions have the common feature that the boundary condition is varied with respect to time the first boundary condition expands or contracts the computational model during a modeling simulation the effect is to cause a doppler shift in the reflected wavefield that can be used to shift energy outside a frequency band of interest in addition when the computational domain is expanding the range of possible incidence angles on the boundary is restricted this can be used to increase the effectiveness of many existing absorbing boundary conditions that are more effective for incidence angles close to normal the second boundary condition is an extension of random boundaries by carefully changing the realization of a random boundary over time a more diffusive wavefield can be simulated we show results with 2d numerical simulations of the scalar wave equation for both these boundary conditions the first boundary condition has application to modeling but both these boundary conditions have potential application within algorithms that rely upon modeling kernels such as reverse time migration and full waveform inversion

high order implicit hybridizable discontinuous galerkin methods for acoustics and elastodynamics we present a class of hybridizable discontinuous galerkin hdg methods for the numerical simulation of wave phenomena in acoustics and elastodynamics the methods are fully implicit and high order accurate in both space and time yet computationally attractive owing to their following distinctive features first they reduce the globally coupled unknowns to the approximate trace of the velocity which is defined on the element faces and single valued thereby leading to a significant saving in the computational cost in addition all the approximate variables including the approximate velocity and gradient converge with the optimal order of k 1 in the l 2 norm when polynomials of degree k 0 are used to represent the numerical solution and when the time stepping method is accurate with order k 1 when the time stepping method is of order k 2 superconvergence properties allows us by means of local postprocessing to obtain better yet inexpensive approximations of the displacement and velocity at any time levels for which an enhanced accuracy is required in particular the new approximations converge with order k 2 in the l 2 norm when k 1 for both acoustics and elastodynamics extensive numerical results are provided to illustrate these distinctive features

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

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

simulation of acoustic wavefields in heterogeneous media a robust method for automatic suppression of numerical dispersion numerical dispersion limits the application of numerical simulation methods for solving the acoustic wave equation in largescalecomputation thenearlyanalyticdiscretemethodnadm and its improved version for suppressing numerical dispersion weredevelopedrecently thisnewmethodisarefinementoftwo previousmethodsandfurtherincreasestheabilityofsuppressing numerical dispersion for modeling acoustic wave propagation in 2d heterogeneous media which uses acoustic wave displacement particle velocity and their gradients to restructure the acoustic wavefield via the truncated taylor expansion and the high order interpolation approximate method for the method proposed weinvestigateitsimplementationandcompareitwith the higher order lax wendroff correction lwc scheme the original nearly analytic discrete method nadm and its improved version with regard to numerical dispersion computational costs and computer storage requirements the numerical dispersionrelationsprovidedbytherefinedalgorithmfor1dand 2d cases are analyzed as well as the numerical results obtained bythismethodagainsttheexactsolutionforthe2dacousticcase numerical results show that the refined method gives no visible numericaldispersionforverylargespatialgridincrements itcan simulatehigh frequencywavepropagationforagivengridinterval and automatically suppress the numerical dispersion when the acoustic wave equation is discretized when too few samples per wavelength are used or when models have large velocity contrasts unlike the high order lwc methods our present methodcansavesubstantialcomputationalcostsandmemoryrequirements because very large grid increments can be used the refined method can be used for the simulation of large scale wavefields

an explicit hybridizable discontinuous galerkin method for the acoustic wave equation abstract we present an explicit hybridizable discontinuous galerkin hdg method for numerically solving the acoustic wave equation the method is fully explicit high order accurate in both space and time and coincides with the classic discontinuous galerkin dg method with upwinding fluxes for a particular choice of its stabilization function this means that it has the same computational complexity as other explicit dg methods however just as its implicit version it provides optimal convergence of order k 1 for all the approximate variables including the gradient of the solution and when the time stepping method is of order k 2 it displays a superconvergence property which allow us by means of local postprocessing to obtain new improved approximations of the scalar field variables at any time levels for which an enhanced accuracy is required in particular the new approximations converge with order k 2 in the l 2 norm for k 1 these properties do not hold for all numerical fluxes indeed our results show that when the hdg numerical flux is replaced by the lax friedrichs flux the above mentioned superconvergence properties are lost although some are recovered when the lax friedrichs flux is used only in the interior of the domain finally we extend the explicit hdg method to treat the wave equation with perfectly matched layers we provide numerical examples to demonstrate the performance of the proposed method

elimination of numerical dispersion in finite difference modeling and migration by flux corrected transport finite difference acoustic wave modeling and reverse time depth migration based on the full wave equation are general approaches that can take into account arbitrary variations in velocity and density and can handle turning waves as well however conventional finite difference methods for solving the acoustic or elastic wave equation suffer from numerical dispersion when too few samples per wavelength are used the flux corrected transport fct algorithm adapted from hydrodynamics reduces the numerical dispersion in finite difference wavefield continuation the flux correction procedure endeavors to incorporate diffusion into the wavefield continuation process only where needed to suppress the numerical dispersion incorporating the flux correction procedure in conventional finite difference modeling or reverse time migration can provide finite difference solutions with no numerical dispersion even for impulsive sources the fct correction which can be applied to finite difference approximations of any order in space and time is an efficient alternative to use for finite difference approximations of increasing order through demonstrations of modeling and migration on both synthetic and field data we show the benefits of the fct algorithm as well as its inability to fully recover resolution lost when the spatial sampling becomes too coarse

discontinuous galerkin methods for wave propagation in poroelastic media we have developed a new numerical method to solve the heterogeneous poroelastic wave equations in bounded three dimensional domains this method is a discontinuous galerkin method that achieves arbitrary high order accuracy on unstructured tetrahedral meshes for the low frequency range and the inviscid case by using biot s equations and darcy s dynamic laws we have built a scheme that can successfully model wave propagation in fluid saturated porous media when anisotropy of the pore structure is allowed zero inflow fluxes are used as absorbing boundary conditions a continuous arbitrary high order derivatives time integration is used for the high frequency inviscid case whereas a space time discontinuous scheme is applied for the low frequency case we conducted a numerical convergence test of the proposed methods we used a series of examples to quantify the quality of our numerical results comparing them to analytic solutions as well as numerical solutions obtained by other methodologies in partic

an effective imaging condition for reverse time migration using wavefield decomposition reverse time migration rtm exhibits great superiority over other imaging algorithms in handling steeply dipping structures and complicated velocity models however low frequency high amplitude noises commonly seen in a typical rtm image have been one of the major concerns because they can seriously contaminate the signals in the image if they are not handled properly we propose a new imaging condition to effectively and efficiently eliminate these specific noises from the image the method works by first decomposing the source and receiver wavefields to their one way propagation components followed by applying a correlation based imaging condition to the appropriate combinations of the decomposed wavefields we first give the physical explanation of the principle of such noises in the conventional rtm image then we provide the detailed mathematical theory for the new imaging condition finally we propose an efficient scheme for its numerical implementation it replaces the computationally intensiv

the application of high order differencing to the scalar wave equation 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 approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

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

source wavefield reconstruction using a linear combination of the boundary wavefield in reverse time migration abstractwe have developed a new scheme with high accuracy and parsimonious memory requirements to reconstruct the source wavefield in reverse time migration rtm this scheme used a linear combination of the boundary wavefield in several layers to reconstruct the wavefield in the imaging domain the value of the linear combination was stored in one buffer making the computer memory requirement of our method equal to that of a method that only stored one layer of the boundary wavefield the coefficients for the linear combination were determined by an optimization technique that minimized the finite difference error in the fourier domain the optimal coefficients were presented for finite difference fd stencils of 5 15 grid points the numerical error of our scheme was analyzed and compared with that of standard fd stencils representing the conventional method that used multiple layers of boundary wavefield to back propagate the source wavefield the accuracy of our method is only less accurate than t

optimized explicit finite difference schemes for spatial derivatives using maximum norm we propose an optimized scheme using the maximum norm and the simulated annealing the maximum norm offers the largest set of possible solutions for solvers to search the explicit finite difference operator is greatly improved by our optimized scheme we use a tight error limitation to make accuracy improvement to be high and solid our optimized scheme allows greater saving of computational cost and memory demand conventional explicit finite difference methods have difficulties in handling high frequency components due to strong numerical dispersions one can reduce the numerical dispersions by optimizing the constant coefficients of the finite difference operator different from traditional optimized schemes that use the 2 norm and the least squares we propose to construct the objective functions using the maximum norm and solve the objective functions using the simulated annealing algorithm both theoretical analyses and numerical experiments show that our optimized scheme is superior to traditional optimized schemes with regard to the following three aspects first it provides us with much more flexibility when designing the objective functions thus we can use various possible forms and contents to make the objective functions more reasonable second it allows for tighter error limitation which is shown to be necessary to avoid rapid error accumulations for simulations on large scale models with long travel times finally it is powerful to obtain the optimized coefficients that are much closer to the theoretical limits which means greater savings in computational efforts and memory demand

anisotropic wave propagation through finite difference grids an algorithm is presented to solve the elastic wave equation by replacing the partial differentials with finite differences it enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models the space derivatives are calculated using discrete convolution sums while the time derivatives are replaced by a truncated taylor expansion a centered finite difference scheme in cartesian coordinates is used for the space derivatives leading to staggered grids the use of finite difference approximations to the partial derivatives results in a frequency dependent error in the group and phase velocities of waves for anisotropic media the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the hook sum this interpolation induces an additional error in the wave properties the overall error depends on the precision of the derivative and interpolation operators the anisotropic symmetry system its orientation and the degree of anisotropy the dispersion relation for the homogeneous case was derived for the proposed scheme since we use a general description of convolution sums to describe the finite difference operators the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model in particular phase and group velocities of the three wave types can be determined in any direction we demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed

globally optimal finite difference schemes based on least squares abstractspatial finite difference fd coefficients are usually determined by the taylor series expansion te or optimization methods the former can provide high accuracy on a smaller wavenumber or frequency zone and the latter can give moderate accuracy on a larger zone present optimization methods applied to calculate fd coefficients are generally gradient like or global optimization like algorithms and thus iterations are involved they are more computationally expensive and sometimes the global solution may not be found i examined second order spatial derivatives and computed the optimized spatial fd coefficients over the given wavenumber range using the least squares ls method the results indicated that the fd accuracy increased with increasing operator length and decreasing wavenumber range therefore for the given error and operator length globally optimal spatial fd coefficients can be easily obtained some optimal fd coefficients were given i developed schemes to obtain optimized ls

efficiency of high order elements for continuous and discontinuous galerkin methods summary to evaluate the computational performance of high order elements a comparison based on operation count is proposed instead of runtime comparisons more specifically linear versus high order approximations are analyzed for implicit solver under a standard set of hypotheses for the mesh and the solution continuous and discontinuous galerkin methods are considered in two dimensional and three dimensional domains for simplices and parallelotopes moreover both element wise and global operations arising from different galerkin approaches are studied the operation count estimates show that for implicit solvers high order methods are more efficient than linear ones copyright 2013 john wiley sons ltd

an optimal nearly analytic discrete weighted runge kutta discontinuous galerkin hybrid method for acoustic wavefield modeling abstractthe newly developed optimal nearly analytic discrete onad and the weighted runge kutta discontinuous galerkin wrkdg methods can effectively suppress the numerical dispersion caused by discretizing wave equations but it is difficult for onad to implement on flexible meshes whereas the wrkdg has high computational cost for wavefield simulations we have developed a new hybrid algorithm by combining the onad method with the wrkdg method in this hybrid algorithm the computational domain was split into several subdomains in which the subdomain for the onad method used regular cartesian grids whereas the subdomain for the wrkdg method used triangular grids the hybrid method was at least third order spatially accurate we have applied the proposed method to simulate the scalar wavefields for different models including a homogeneous model a rough topography model a fracture model and a cave model the numerical results found that the hybrid method can deal with complicated geometrical struc

reverse time migration in acoustic vti media using a high order stereo operator abstractwith the capability of handling complicated velocity models reverse time migration rtm has become a powerful imaging method improving imaging accuracy and computational efficiency are two significant but challenging tasks in the applications of rtm despite being the most popular numerical technique applied in rtm finite difference fd methods often suffer from undesirable numerical dispersion leading to a noticeable loss of imaging resolution a new and effective fd operator called the high order stereo operator has been developed to approximate the partial differential operators in the wave equation from which a numerical scheme called the three step stereo method tsm has been developed and has shown effectiveness in suppressing numerical dispersion numerical results show that compared with the conventional numerical methods such as the lax wendroff correction lwc scheme and the staggered grid sg fd method this new method significantly reduces numerical dispersion and computati