3D eikonal equation is a partial differential equation for the calculation of first-arrival traveltimes and has been widely applied in many scopes such as ray tracing,source localization,reflection migration,seismic m...3D eikonal equation is a partial differential equation for the calculation of first-arrival traveltimes and has been widely applied in many scopes such as ray tracing,source localization,reflection migration,seismic monitoring and tomographic imaging.In recent years,many advanced methods have been developed to solve the 3D eikonal equation in heterogeneous media.However,there are still challenges for the stable and accurate calculation of first-arrival traveltimes in 3D strongly inhomogeneous media.In this paper,we propose an adaptive finite-difference(AFD)method to numerically solve the 3D eikonal equation.The novel method makes full use of the advantages of different local operators characterizing different seismic wave types to calculate factors and traveltimes,and then the most accurate factor and traveltime are adaptively selected for the convergent updating based on the Fermat principle.Combined with global fast sweeping describing seismic waves propagating along eight directions in 3D media,our novel method can achieve the robust calculation of first-arrival traveltimes with high precision at grid points either near source point or far away from source point even in a velocity model with large and sharp contrasts.Several numerical examples show the good performance of the AFD method,which will be beneficial to many scientific applications.展开更多
In seismic exploration, it is common practice to separate the P-wavefield from the S-wavefield by the elastic wavefield decomposition technique, for imaging purposes. However, it is sometimes difficult to achieve this...In seismic exploration, it is common practice to separate the P-wavefield from the S-wavefield by the elastic wavefield decomposition technique, for imaging purposes. However, it is sometimes difficult to achieve this, especially when the velocity field is complex. A useful approach in multi-component analysis and modeling is to directly solve the elastic wave equations for the pure P- or S-wavefields, referred as the separate elastic wave equa- tions. In this study, we compare two kinds of such wave equations： the first-order （velocity-stress） and the second- order （displacement-stress） separate elastic wave equa- tions, with the first-order （velocity-stress） and the second- order （displacement-stress） full （or mixed） elastic wave equations using a high-order staggered grid finite-differ- ence method. Comparisons are given of wavefield snap- shots, common-source gather seismic sections, and individual synthetic seismogram. The simulation tests show that equivalent results can be obtained, regardless of whether the first-order or second-order separate elastic wave equations are used for obtaining the pure P- or S-wavefield. The stacked pure P- and S-wavefields are equal to the mixed wave fields calculated using the corre- sponding first-order or second-order full elastic wave equations. These mixed equations are computationallyslightly less expensive than solving the separate equations. The attraction of the separate equations is that they achieve separated P- and S-wavefields which can be used to test the efficacy of wave decomposition procedures in multi-com- ponent processing. The second-order separate elastic wave equations are a good choice because they offer information on the pure P-wave or S-wave displacements.展开更多
Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of ...Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.展开更多
This paper investigates the phenomenon of three-pulse photon echo in thick rare-earth ions doped crystal whose thickness is far larger than 0.002 cm which is adopted in previous works.The influence of thickness on the...This paper investigates the phenomenon of three-pulse photon echo in thick rare-earth ions doped crystal whose thickness is far larger than 0.002 cm which is adopted in previous works.The influence of thickness on the three-pulse photon echo＇s amplitude and efficiency is analyzed with the Maxwell-Bloch equations solved by finite-difference timedomain method.We demonstrate that the amplitude of three-pulse echo will increase with the increasing of thickness and the optimum thickness to generate three-pulse photon echo is 0.3 cm for Tm^（3＋）：YAG when the attenuation of the input pulse is taken into account.Meanwhile,we find the expression 0.09 exp（α＇L）,which is previously employed to describe the relationship between echo＇s efficiency and thickness,should be modified as 1.3 · 0.09 exp（2.4 ·α＇L） with the propagation of echo considered.展开更多
Seismic anisotropy has been extensively acknowledged as a crucial element that influences the wave propagation characteristic during wavefield simulation,inversion and imaging.Transversely isotropy(TI)and orthorhombic...Seismic anisotropy has been extensively acknowledged as a crucial element that influences the wave propagation characteristic during wavefield simulation,inversion and imaging.Transversely isotropy(TI)and orthorhombic anisotropy(OA)are two typical categories of anisotropic media in exploration geophysics.In comparison of the elastic wave equations in both TI and OA media,pseudo-acoustic wave equations(PWEs)based on the acoustic assumption can markedly reduce computational cost and complexity.However,the presently available PWEs may experience SV-wave contamination and instability when anisotropic parameters cannot satisfy the approximated condition.Exploiting pure-mode wave equations can effectively resolve the above-mentioned issues and generate pure P-wave events without any artifacts.To further improve the computational accuracy and efficiency,we develop two novel pure qP-wave equations(PPEs)and illustrate the corresponding numerical solutions in the timespace domain for 3D tilted TI(TTI)and tilted OA(TOA)media.First,the rational polynomials are adopted to estimate the exact pure qP-wave dispersion relations,which contain complicated pseudo-differential operators with irrational forms.The polynomial coefficients are produced by applying a linear optimization algorithm to minimize the objective function difference between the expansion formula and the exact one.Then,the developed optimized PPEs are efficiently implemented using the finite-difference(FD)method in the time-space domain by introducing a scalar operator,which can help avoid the problem of spectral-based algorithms and other calculation burdens.Structures of the new equations are concise and corresponding implementation processes are straightforward.Phase velocity analyses indicate that our proposed optimized equations can lead to reliable approximation results.3D synthetic examples demonstrate that our proposed FD-based PPEs can produce accurate and stable P-wave responses,and effectively describe the wavefield features in complicated TTI and TOA media.展开更多
PFMM(perspective fast marching method)是一种有效解决透视投影下从明暗恢复形状(SFS)问题的方法,但是适应条件受限,且对初始数据的精度较为敏感。本文通过对Eikonal方程系数的分析,提出了在透视投影下基于自适应Eikonal方程的PFMM,...PFMM(perspective fast marching method)是一种有效解决透视投影下从明暗恢复形状(SFS)问题的方法,但是适应条件受限,且对初始数据的精度较为敏感。本文通过对Eikonal方程系数的分析,提出了在透视投影下基于自适应Eikonal方程的PFMM,解决了PFMM对初始数据过于依赖的问题,是PFMM的推广。对合成图像的实验表明本文算法比PFMM精度更高,对透视投影下SFS问题可以得到比较好的结果。展开更多
The relativistic Hartree-Bogoliubov(RHB)theory is a powerful tool for describing exotic nuclei near drip lines.The key technique is to solve the RHB equation in the coordinate space to obtain the quasi-particle states...The relativistic Hartree-Bogoliubov(RHB)theory is a powerful tool for describing exotic nuclei near drip lines.The key technique is to solve the RHB equation in the coordinate space to obtain the quasi-particle states.In this paper,we solve the RHB equation with the Woods-Saxon-type mean-field and Delta-type pairing-field potentials by using the finite-difference method(FDM).We inevitably obtain spurious states when using the common symmetric central difference formula(CDF)to construct the Hamiltonian matrix,which is similar to the problem resulting from solving the Dirac equation with the same method.This problem is solved by using the asymmetric difference formula(ADF).In addition,we show that a large enough box is necessary to describe the continuum quasi-particle states.The canonical states obtained by diagonalizing the density matrix constructed by the quasi-particle states are not particularly sensitive to the box size.Part of the asymptotic wave functions can be improved by applying the ADF in the FDM compared to the shooting method with the same box boundary condition.展开更多
To the most of velocity fields, the traveltimes of the first break that seismic waves propagate along rays can be computed on a 2-D or 3-D numerical grid by finite-difference extrapolation. Under ensuring accuracy, t...To the most of velocity fields, the traveltimes of the first break that seismic waves propagate along rays can be computed on a 2-D or 3-D numerical grid by finite-difference extrapolation. Under ensuring accuracy, to improve calculating efficiency and adaptability, the calculation method of first-arrival traveltime of finite-difference is de- rived based on any rectangular grid and a local plane wavefront approximation. In addition, head waves and scat- tering waves are properly treated and shadow and caustic zones cannot be encountered, which appear in traditional ray-tracing. The testes of two simple models and the complex Marmousi model show that the method has higher accuracy and adaptability to complex structure with strong vertical and lateral velocity variation, and Kirchhoff prestack depth migration based on this method can basically achieve the position imaging effects of wave equation prestack depth migration in major structures and targets. Because of not taking account of the later arrivals energy, the effect of its amplitude preservation is worse than that by wave equation method, but its computing efficiency is higher than that by total Green′s function method and wave equation method.展开更多
The precise determination of earthquake location is the fundamental basis in seismological community,and is crucial for analyzing seismic activity and performing seismic tomography.First arrivals are generally used to...The precise determination of earthquake location is the fundamental basis in seismological community,and is crucial for analyzing seismic activity and performing seismic tomography.First arrivals are generally used to practically determine earthquake locations.However,first-arrival traveltimes are not sensitive to focal depths.Moreover,they cannot accurately constrain focal depths.To improve the accuracy,researchers have analyzed the depth phases of earthquake locations.The traveltimes of depth phases are sensitive to focal depths,and the joint inversion of depth phases and direct phases can be implemented to potentially obtain accurate earthquake locations.Generally,researchers can determine earthquake locations in layered models.Because layered models can only represent the first-order feature of subsurface structures,the advantages of joint inversion are not fully explored if layered models are used.To resolve the issue of current joint inversions,we use the traveltimes of three seismic phases to determine earthquake locations in heterogeneous models.The three seismic phases used in this study are the first P-,sPg-and PmP-waves.We calculate the traveltimes of the three seismic phases by solving an eikonal equation with an upwind difference scheme and use the traveltimes to determine earthquake locations.To verify the accuracy of the earthquake location method by the inversion of three seismic phases,we take the 2021 M_(S)6.4 Yangbi,Yunnan earthquake as an example and locate this earthquake using synthetic and real seismic data.Numerical tests demonstrate that the eikonal equation-based earthquake location method,which involves the inversion of multiple phase arrivals,can effectively improve earthquake location accuracy.展开更多
In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic ...In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic regularization of the resulting variational problem, and the time discretization by operator-splitting of an initial value problem associated with the Euler-Lagrange equations of the regularized variational problem. A low-order finite element discretization is advocated since it is well-suited to the low regularity of the solutions. Numerical experiments show that the method sketched above can capture efficiently the extremal solutions of various two-dimensional test problems and that it has also the ability of handling easily domains with curved boundaries.展开更多
We develop a second-order continuousfinite element method for solving the static Eikonal equation.It is based on the vanishing viscosity approach with a homotopy method for solving the discretized nonlinear system.Mor...We develop a second-order continuousfinite element method for solving the static Eikonal equation.It is based on the vanishing viscosity approach with a homotopy method for solving the discretized nonlinear system.More specifically,the homotopy method is utilized to decrease the viscosity coefficient gradually,while Newton’s method is applied to compute the solution for each viscosity coefficient.Newton’s method alone converges for just big enough viscosity coefficients on very coarse grids and for simple 1D examples,but the proposed method is much more robust and guarantees the convergence of the nonlinear solver for all viscosity coefficients and for all examples over all grids.Numerical experiments from 1D to 3D are presented to confirm the second-order convergence and the effectiveness of the proposed method on both structured or unstructured meshes.展开更多
We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind sche...We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities; 2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil. We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities.展开更多
The steady Eikonal equation is a prototypical first-order fully nonlinear equation. A numerical method based on elliptic solvers is presented here to solve two different kinds of steady Eikonal equations and compute s...The steady Eikonal equation is a prototypical first-order fully nonlinear equation. A numerical method based on elliptic solvers is presented here to solve two different kinds of steady Eikonal equations and compute solutions, which are maximal and minimal in the variational sense. The approach in this paper relies on a variational argument involving penalty, a biharmonic regularization, and an operator-splitting-based time-discretization scheme for the solution of an associated initial-value problem. This approach allows the decoupling of the nonlinearities and differential operators.Numerical experiments are performed to validate this approach and investigate its convergence properties from a numerical viewpoint.展开更多
In this paper,we study splitting numerical methods for the three-dimensional Maxwell equations in the time domain.We propose a new kind of splitting finitedifference time-domain schemes on a staggered grid,which consi...In this paper,we study splitting numerical methods for the three-dimensional Maxwell equations in the time domain.We propose a new kind of splitting finitedifference time-domain schemes on a staggered grid,which consists of only two stages for each time step.It is proved by the energy method that the splitting scheme is unconditionally stable and convergent for problems with perfectly conducting boundary conditions.Both numerical dispersion analysis and numerical experiments are also presented to illustrate the efficiency of the proposed schemes.展开更多
In this paper,we describe how to construct a finite-difference shockcapturing method for the numerical solution of the Euler equation of gas dynamics on arbitrary two-dimensional domainΩ,possibly with moving boundary...In this paper,we describe how to construct a finite-difference shockcapturing method for the numerical solution of the Euler equation of gas dynamics on arbitrary two-dimensional domainΩ,possibly with moving boundary.The boundaries of the domain are assumed to be changing due to the movement of solid objects/obstacles/walls.Although the motion of the boundary could be coupled with the fluid,all of the numerical tests are performed assuming that such a motion is prescribed and independent of the fluid flow.The method is based on discretizing the equation on a regular Cartesian grid in a rectangular domainΩ_(R)⊃Ω.Ωe identify inner and ghost points.The inner points are the grid points located insideΩ,while the ghost points are the grid points that are outsideΩbut have at least one neighbor insideΩ.The evolution equations for inner points data are obtained from the discretization of the governing equation,while the data at the ghost points are obtained by a suitable extrapolation of the primitive variables(density,velocities and pressure).Particular care is devoted to a proper description of the boundary conditions for both fixed and time dependent domains.Several numerical experiments are conducted to illustrate the validity of themethod.Ωe demonstrate that the second order of accuracy is numerically assessed on genuinely two-dimensional problems.展开更多
In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed...In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils.However,one major novelty and difference from the traditional HWENO framework lies in the fact that,we do not need to introduce and solve any additional equations to update the derivatives of the unknown functionϕ.Instead,we use the currentϕand the old spatial derivative ofϕto update them.The traditional HWENO fast sweeping method is also introduced in this paper for comparison,where additional equations governing the spatial derivatives ofϕare introduced.The novel HWENO fast sweeping methods are shown to yield great savings in computational time,which improves the computational efficiency of the traditional HWENO scheme.In addition,a hybrid strategy is also introduced to further reduce computational costs.Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.展开更多
基金The authors thank the funds supported by the China National Nuclear Corporation under Grants Nos.WUQNYC2101 and WUHTLM2101-04National Natural Science Foundation of China(42074132,42274154).
文摘3D eikonal equation is a partial differential equation for the calculation of first-arrival traveltimes and has been widely applied in many scopes such as ray tracing,source localization,reflection migration,seismic monitoring and tomographic imaging.In recent years,many advanced methods have been developed to solve the 3D eikonal equation in heterogeneous media.However,there are still challenges for the stable and accurate calculation of first-arrival traveltimes in 3D strongly inhomogeneous media.In this paper,we propose an adaptive finite-difference(AFD)method to numerically solve the 3D eikonal equation.The novel method makes full use of the advantages of different local operators characterizing different seismic wave types to calculate factors and traveltimes,and then the most accurate factor and traveltime are adaptively selected for the convergent updating based on the Fermat principle.Combined with global fast sweeping describing seismic waves propagating along eight directions in 3D media,our novel method can achieve the robust calculation of first-arrival traveltimes with high precision at grid points either near source point or far away from source point even in a velocity model with large and sharp contrasts.Several numerical examples show the good performance of the AFD method,which will be beneficial to many scientific applications.
基金partially supported by China National Major Science and Technology Project (Subproject No:2011ZX05024-001-03)
文摘In seismic exploration, it is common practice to separate the P-wavefield from the S-wavefield by the elastic wavefield decomposition technique, for imaging purposes. However, it is sometimes difficult to achieve this, especially when the velocity field is complex. A useful approach in multi-component analysis and modeling is to directly solve the elastic wave equations for the pure P- or S-wavefields, referred as the separate elastic wave equa- tions. In this study, we compare two kinds of such wave equations： the first-order （velocity-stress） and the second- order （displacement-stress） separate elastic wave equa- tions, with the first-order （velocity-stress） and the second- order （displacement-stress） full （or mixed） elastic wave equations using a high-order staggered grid finite-differ- ence method. Comparisons are given of wavefield snap- shots, common-source gather seismic sections, and individual synthetic seismogram. The simulation tests show that equivalent results can be obtained, regardless of whether the first-order or second-order separate elastic wave equations are used for obtaining the pure P- or S-wavefield. The stacked pure P- and S-wavefields are equal to the mixed wave fields calculated using the corre- sponding first-order or second-order full elastic wave equations. These mixed equations are computationallyslightly less expensive than solving the separate equations. The attraction of the separate equations is that they achieve separated P- and S-wavefields which can be used to test the efficacy of wave decomposition procedures in multi-com- ponent processing. The second-order separate elastic wave equations are a good choice because they offer information on the pure P-wave or S-wave displacements.
文摘Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.
基金Project supported by Tianjin Research Program Application Foundation and Advanced Technology,China(Grant No.15JCQNJC01100)
文摘This paper investigates the phenomenon of three-pulse photon echo in thick rare-earth ions doped crystal whose thickness is far larger than 0.002 cm which is adopted in previous works.The influence of thickness on the three-pulse photon echo＇s amplitude and efficiency is analyzed with the Maxwell-Bloch equations solved by finite-difference timedomain method.We demonstrate that the amplitude of three-pulse echo will increase with the increasing of thickness and the optimum thickness to generate three-pulse photon echo is 0.3 cm for Tm^（3＋）：YAG when the attenuation of the input pulse is taken into account.Meanwhile,we find the expression 0.09 exp（α＇L）,which is previously employed to describe the relationship between echo＇s efficiency and thickness,should be modified as 1.3 · 0.09 exp（2.4 ·α＇L） with the propagation of echo considered.
基金supported by the National Key R&D Program of China(2021YFA0716902)National Natural Science Foundation of China(NSFC)under contract number 42374149 and 42004119National Science and Technology Major Project(2024ZD1002907)。
文摘Seismic anisotropy has been extensively acknowledged as a crucial element that influences the wave propagation characteristic during wavefield simulation,inversion and imaging.Transversely isotropy(TI)and orthorhombic anisotropy(OA)are two typical categories of anisotropic media in exploration geophysics.In comparison of the elastic wave equations in both TI and OA media,pseudo-acoustic wave equations(PWEs)based on the acoustic assumption can markedly reduce computational cost and complexity.However,the presently available PWEs may experience SV-wave contamination and instability when anisotropic parameters cannot satisfy the approximated condition.Exploiting pure-mode wave equations can effectively resolve the above-mentioned issues and generate pure P-wave events without any artifacts.To further improve the computational accuracy and efficiency,we develop two novel pure qP-wave equations(PPEs)and illustrate the corresponding numerical solutions in the timespace domain for 3D tilted TI(TTI)and tilted OA(TOA)media.First,the rational polynomials are adopted to estimate the exact pure qP-wave dispersion relations,which contain complicated pseudo-differential operators with irrational forms.The polynomial coefficients are produced by applying a linear optimization algorithm to minimize the objective function difference between the expansion formula and the exact one.Then,the developed optimized PPEs are efficiently implemented using the finite-difference(FD)method in the time-space domain by introducing a scalar operator,which can help avoid the problem of spectral-based algorithms and other calculation burdens.Structures of the new equations are concise and corresponding implementation processes are straightforward.Phase velocity analyses indicate that our proposed optimized equations can lead to reliable approximation results.3D synthetic examples demonstrate that our proposed FD-based PPEs can produce accurate and stable P-wave responses,and effectively describe the wavefield features in complicated TTI and TOA media.
文摘PFMM(perspective fast marching method)是一种有效解决透视投影下从明暗恢复形状(SFS)问题的方法,但是适应条件受限,且对初始数据的精度较为敏感。本文通过对Eikonal方程系数的分析,提出了在透视投影下基于自适应Eikonal方程的PFMM,解决了PFMM对初始数据过于依赖的问题,是PFMM的推广。对合成图像的实验表明本文算法比PFMM精度更高,对透视投影下SFS问题可以得到比较好的结果。
基金Supported by the National Natural Science Foundation of China(11775119,2175109)the Natural Science Foundation of Tianjin,China(19JCYBJC30800)。
文摘The relativistic Hartree-Bogoliubov(RHB)theory is a powerful tool for describing exotic nuclei near drip lines.The key technique is to solve the RHB equation in the coordinate space to obtain the quasi-particle states.In this paper,we solve the RHB equation with the Woods-Saxon-type mean-field and Delta-type pairing-field potentials by using the finite-difference method(FDM).We inevitably obtain spurious states when using the common symmetric central difference formula(CDF)to construct the Hamiltonian matrix,which is similar to the problem resulting from solving the Dirac equation with the same method.This problem is solved by using the asymmetric difference formula(ADF).In addition,we show that a large enough box is necessary to describe the continuum quasi-particle states.The canonical states obtained by diagonalizing the density matrix constructed by the quasi-particle states are not particularly sensitive to the box size.Part of the asymptotic wave functions can be improved by applying the ADF in the FDM compared to the shooting method with the same box boundary condition.
基金National Natural Science Foundation of China (49894190-024) and Geophysical Prospecting Key Laboratory Foundation of China National Petroleum Corporation.
文摘To the most of velocity fields, the traveltimes of the first break that seismic waves propagate along rays can be computed on a 2-D or 3-D numerical grid by finite-difference extrapolation. Under ensuring accuracy, to improve calculating efficiency and adaptability, the calculation method of first-arrival traveltime of finite-difference is de- rived based on any rectangular grid and a local plane wavefront approximation. In addition, head waves and scat- tering waves are properly treated and shadow and caustic zones cannot be encountered, which appear in traditional ray-tracing. The testes of two simple models and the complex Marmousi model show that the method has higher accuracy and adaptability to complex structure with strong vertical and lateral velocity variation, and Kirchhoff prestack depth migration based on this method can basically achieve the position imaging effects of wave equation prestack depth migration in major structures and targets. Because of not taking account of the later arrivals energy, the effect of its amplitude preservation is worse than that by wave equation method, but its computing efficiency is higher than that by total Green′s function method and wave equation method.
基金supported by the National Natural Science Foundation of China(Grant Nos.42174111 and42064004)the Beijing Natural Science Foundation(Grant No.8222033)the Ningxia Science and Technology Leading Talent Training Program(Grant No.2022GKLRLX04)。
文摘The precise determination of earthquake location is the fundamental basis in seismological community,and is crucial for analyzing seismic activity and performing seismic tomography.First arrivals are generally used to practically determine earthquake locations.However,first-arrival traveltimes are not sensitive to focal depths.Moreover,they cannot accurately constrain focal depths.To improve the accuracy,researchers have analyzed the depth phases of earthquake locations.The traveltimes of depth phases are sensitive to focal depths,and the joint inversion of depth phases and direct phases can be implemented to potentially obtain accurate earthquake locations.Generally,researchers can determine earthquake locations in layered models.Because layered models can only represent the first-order feature of subsurface structures,the advantages of joint inversion are not fully explored if layered models are used.To resolve the issue of current joint inversions,we use the traveltimes of three seismic phases to determine earthquake locations in heterogeneous models.The three seismic phases used in this study are the first P-,sPg-and PmP-waves.We calculate the traveltimes of the three seismic phases by solving an eikonal equation with an upwind difference scheme and use the traveltimes to determine earthquake locations.To verify the accuracy of the earthquake location method by the inversion of three seismic phases,we take the 2021 M_(S)6.4 Yangbi,Yunnan earthquake as an example and locate this earthquake using synthetic and real seismic data.Numerical tests demonstrate that the eikonal equation-based earthquake location method,which involves the inversion of multiple phase arrivals,can effectively improve earthquake location accuracy.
基金supported by the National Science Foundation(No.DMS-0913982)
文摘In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic regularization of the resulting variational problem, and the time discretization by operator-splitting of an initial value problem associated with the Euler-Lagrange equations of the regularized variational problem. A low-order finite element discretization is advocated since it is well-suited to the low regularity of the solutions. Numerical experiments show that the method sketched above can capture efficiently the extremal solutions of various two-dimensional test problems and that it has also the ability of handling easily domains with curved boundaries.
基金supported by Natural Science Foundation of Jiangsu Province(Nos.KFR21026,PAF20042)National Natural Science Foundation of China(Nos.GBA20029,GCA20004)+2 种基金Science Challenge Project(No.TZ2018002)National Science and Technology Major Project(No.J2019-II-0007-0027)WH is supported by NSF DMS-1818769.
文摘We develop a second-order continuousfinite element method for solving the static Eikonal equation.It is based on the vanishing viscosity approach with a homotopy method for solving the discretized nonlinear system.More specifically,the homotopy method is utilized to decrease the viscosity coefficient gradually,while Newton’s method is applied to compute the solution for each viscosity coefficient.Newton’s method alone converges for just big enough viscosity coefficients on very coarse grids and for simple 1D examples,but the proposed method is much more robust and guarantees the convergence of the nonlinear solver for all viscosity coefficients and for all examples over all grids.Numerical experiments from 1D to 3D are presented to confirm the second-order convergence and the effectiveness of the proposed method on both structured or unstructured meshes.
基金partially supported by ONR Grant N00014-02-1-0090ARO MURI Grant W911NF-07-1-0185NSF Grant DMS0811254
文摘We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities; 2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil. We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities.
基金supported by the National Science Foundation(No.DMS-0913982)
文摘The steady Eikonal equation is a prototypical first-order fully nonlinear equation. A numerical method based on elliptic solvers is presented here to solve two different kinds of steady Eikonal equations and compute solutions, which are maximal and minimal in the variational sense. The approach in this paper relies on a variational argument involving penalty, a biharmonic regularization, and an operator-splitting-based time-discretization scheme for the solution of an associated initial-value problem. This approach allows the decoupling of the nonlinearities and differential operators.Numerical experiments are performed to validate this approach and investigate its convergence properties from a numerical viewpoint.
文摘In this paper,we study splitting numerical methods for the three-dimensional Maxwell equations in the time domain.We propose a new kind of splitting finitedifference time-domain schemes on a staggered grid,which consists of only two stages for each time step.It is proved by the energy method that the splitting scheme is unconditionally stable and convergent for problems with perfectly conducting boundary conditions.Both numerical dispersion analysis and numerical experiments are also presented to illustrate the efficiency of the proposed schemes.
基金The work of A.Chertock was supported in part by the NSF Grants DMS-1216974 and DMS-1521051The work of A.Kurganov was supported in part by the NSF Grants DMS-1216957 and DMS-1521009The work of G.Russo was supported partially by the University of Catania,Project F.I.R.Charge Transport in Graphene and Low Dimensional Systems,and partially by ITN-ETN Horizon 2020 Project Mod Comp Shock,Modeling and Computation on Shocks and Interfaces,Project Reference 642768.
文摘In this paper,we describe how to construct a finite-difference shockcapturing method for the numerical solution of the Euler equation of gas dynamics on arbitrary two-dimensional domainΩ,possibly with moving boundary.The boundaries of the domain are assumed to be changing due to the movement of solid objects/obstacles/walls.Although the motion of the boundary could be coupled with the fluid,all of the numerical tests are performed assuming that such a motion is prescribed and independent of the fluid flow.The method is based on discretizing the equation on a regular Cartesian grid in a rectangular domainΩ_(R)⊃Ω.Ωe identify inner and ghost points.The inner points are the grid points located insideΩ,while the ghost points are the grid points that are outsideΩbut have at least one neighbor insideΩ.The evolution equations for inner points data are obtained from the discretization of the governing equation,while the data at the ghost points are obtained by a suitable extrapolation of the primitive variables(density,velocities and pressure).Particular care is devoted to a proper description of the boundary conditions for both fixed and time dependent domains.Several numerical experiments are conducted to illustrate the validity of themethod.Ωe demonstrate that the second order of accuracy is numerically assessed on genuinely two-dimensional problems.
基金supported by the NSF (Grant No.DMS-1753581)supported by NSFC (Grant No.12071392).
文摘In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils.However,one major novelty and difference from the traditional HWENO framework lies in the fact that,we do not need to introduce and solve any additional equations to update the derivatives of the unknown functionϕ.Instead,we use the currentϕand the old spatial derivative ofϕto update them.The traditional HWENO fast sweeping method is also introduced in this paper for comparison,where additional equations governing the spatial derivatives ofϕare introduced.The novel HWENO fast sweeping methods are shown to yield great savings in computational time,which improves the computational efficiency of the traditional HWENO scheme.In addition,a hybrid strategy is also introduced to further reduce computational costs.Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.
