A scheme to treat the singularity in global seismic wavefield simulation using pseudospectral method with staggered grids

The pseudospectral method has been applied to the simulation of seismic wave propagation in 2-D global Earth model. When a whole Earth model is considered, the center of the Earth is included in the model and then singularity arises at the center of the Earth where r=0 since the 1/r term appears in the wave equations. In this paper, we extended the global seismic wavefield simulation algorithm for regular grid mesh to staggered grid configuration and developed a scheme to solve the numerical problems associated with the above singularity for a 2-D global Earth model defined on staggered grid using pseudospectral method. This scheme uses a coordinate transformation at the center of the model, in which the field variables at the center are calculated in Cartesian coordinates from the values on the grids around the center. It allows wave propagation through the center and hence the wavefield at the center can be stably calculated. Validity and accuracy of the scheme was tested by compared with the discrete wavenumber method. This scheme could also be suitable for other numerical methods or models parameterized in cylindrical or spherical coordinates when singularity arises at the center of the model.

Comparison between staggered grid finite difference method and stochastic method in simulating strong ground motions

Strong ground motion of an earthquake is simulated by using both staggered grid finite difference method (FDM) and stochastic method, respectively. The acceleration time histories obtained from the both ways and their response spectra are compared. The result demonstrates that the former is adequate to simulate the low-frequency seismic wave; the latter is adequate to simulate the high-frequency seismic wave. Moreover, the result obtained from FDM can better reflect basin effects.

A Staggered Grid Method for Solving Incompressible Flow on Unstructured Meshes

A finite volume method based unstructured grid is presented to solve the two dimensional viscous and incompressible flow.The method is based on the pressure-correction concept and solved by using a semi-staggered grid technique.The computational procedure can handle cells of arbitrary shapes,although solutions presented in this paper were only involved with triangular and quadrilateral cells.The pressure or pressure-correction value was stored on the vertex of cells.The mass conservation equation was discretized on the dual cells surrounding the vertex of primary cells,while the velocity components and other scale variables were saved on the central of primary cells.Since the semi-staggered arrangement can’t guarantee a strong coupling relationship between pressure and velocity,thus a weak coupling relationship leads to the oscillations for pressure and velocity.In order to eliminate such an oscillation,a special interpolation scheme was used to construct the pressure-correction equation.Computational results of several viscous flow problems show good agreement with the analytical or numerical results in previous literature.This semi-staggered grid method can be applied to arbitrary shape elements,while it has the most efficiency for triangular cells.

SIMULATION SEISMIC WAVE PROPAGATION IN TOPOGRAPHIC STRUCTURES USING ASYMMETRIC STAGGERED GRIDS

A new 3D finite-difference(FD) method of spatially asymmetric staggered grids was presented to simulate elastic wave propagation in topographic structures.The method approximated the first-order elastic wave equations by irregular grids finite difference operator with second-order time precise and fourth-order spatial precise.Additional introduced finite difference formula solved the asymmetric problem arisen in non-uniform staggered grid scheme.The method had no interpolation between the fine and coarse grids.All grids were computed at the same spatial iteration.Complicated geometrical structures like rough submarine interface,fault and nonplanar interfaces were treated with fine irregular grids.Theoretical analysis and numerical simulations show that this method saves considerable memory and computing time,at the same time,has satisfactory stability and accuracy.

Calculation of cell face velocity of non-staggered grid system

In this paper, the cell face velocities in the discretization of the continu- ity equation, the momentum equation, and the scalar equation of a non-staggered grid system are calculated and discussed. Both the momentum interpolation and the linear interpolation are adopted to evaluate the coefficients in the discretized momentum and scalar equations. Their performances are compared. When the linear interpolation is used to calculate the coefficients, the mass residual term in the coefficients must be dropped to maintain the accuracy and convergence rate of the solution.

A CONSERVATIVE UNSTRUCTURED STAGGERED GRID SCHEME FOR INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

A finite-volume formulation is proposed to solve the three-dimensional,non-hydrostatic Navier-Stokes equations on an unstructured,staggered,z-lever grid,with the goal of simulating non-hydrostatic processes in the free-surface flows.The advection and diffusion terms in the momentum equation are discretized explicitly with the Eulerian scheme,which has the attractive property of being conservative.An integral method of the top-layer pressure is applied to account for the full effects of non-hydrostatic pressure at the free-surface layer.It is shown that the results obtained with a small number of vertical layers（e.g.,2-3 layers） are in good agreements with experimental data or analytical solutions,demonstrating the efficiency and accuracy of the model in simulating a range of free-surface flow problems including wave motion and tide-induced motion.

Source wavefield reconstruction based on an implicit staggered-grid finite-difference operator for seismic imaging

Reverse time migration and full waveform inversion involve the crosscorrelation of two wavefields,propagated in the forward-and reverse-time directions,respectively.As a result,the forward-propagated wavefield needs to be stored,and then accessed to compute the correlation with the backward-propagated wavefield.Boundary-value methods reconstruct the source wavefield using saved boundary wavefields and can significantly reduce the storage requirements.However,the existing boundary-value methods are based on the explicit finite-difference(FD)approximations of the spatial derivatives.Implicit FD methods exhibit greater accuracy and thus allow for a smaller operator length.We develop two(an accuracy-preserving and a memory-efficient)wavefield reconstruction schemes based on an implicit staggered-grid FD(SFD)operator.The former uses boundary wavefields at M layers of grid points and the spatial derivatives of wavefields at one layer of grid points to reconstruct the source wavefield for a(2M+2)th-order implicit SFD operator.The latter applies boundary wavefields at N layers of grid points,a linear combination of wavefields at M–N layers of grid points,and the spatial derivatives of wavefields at one layer of grid points to reconstruct the source wavefield(0≤N<M).The required memory of accuracy-preserving and memory-efficient schemes is(M+1)/M and(N+2)/M times,respectively,that of the explicit reconstruction scheme.Numerical results reveal that the accuracy-preserving scheme can achieve accurate reconstruction at the cost of storage.The memory-efficient scheme with N=2 can obtain plausible reconstructed wavefields and images,and the storage amount is 4/(M+1)of the accuracy-preserving scheme.

Computing Depth-averaged Flows Using Boundary-fitted Coordinates and Staggered Grids

A depth averaged nonlinear k ε model for turbulent flows in complex geometries has been developed in a boundary fitted coordinate system. The SIMPLEC procedure is used to develop an economical discrete method for staggered grids to analyze flows in a 90° bend. This paper describes how to change a program in rectangular coordinate into a boundary fitted coordinate. The results compare well with experimental data for flow in a meandering channel showing the efficiency of the model and the discrete method.

A Simple 3D Immersed InterfaceMethod for Stokes Flow with Singular Forces on Staggered Grids

In this paper,a fairly simple 3D immersed interface method based on the CG-Uzawa type method and the level set representation of the interface is employed for solving three-dimensional Stokes flow with singular forces along the interface.The method is to apply the Taylor’s expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes.A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy.The Stokes equations are discretized involving the correction terms on staggered grids and then solved by the conjugate gradient Uzawa type method.The major advantages of the present method are the special simplicity,the ability in handling the Dirichlet boundary conditions,and no need of the pressure boundary condition.The method can also preserve the volume conservation and the discrete divergence free condition very well.The numerical results show that the proposed method is second order accurate and efficient.

A method of solving the stiffness problem in Biot＇s poroelastic equations using a staggered high-order finite-difference

In modelling elastic wave propagation in a porous medium, when the ratio between the fluid viscosity and the medium permeability is comparatively large, the stiffness problem of Blot＇s poroelastic equations will be encountered. In the paper, a partition method is developed to solve the stiffness problem with a staggered high-order finite-difference. The method splits the Biot equations into two systems. One is stiff, and solved analytically, the other is nonstiff, and solved numerically by using a high-order staggered-grid finite-difference scheme. The time step is determined by the staggered finite-difference algorithm in solving the nonstiff equations, thus a coarse time step may be employed. Therefore, the computation efficiency and computational stability are improved greatly. Also a perfect by matched layer technology is used in the split method as absorbing boundary conditions. The numerical results are compared with the analytical results and those obtained from the conventional staggered-grid finite-difference method in a homogeneous model, respectively. They are in good agreement with each other. Finally, a slightly more complex model is investigated and compared with related equivalent model to illustrate the good performance of the staggered-grid finite-difference scheme in the partition method.

Numerical simulation of standing wave with 3D predictor-corrector finite difference method for potential flow equations

A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.

Numerical simulation of seismic wave in elastic and viscoelastic TTI media

Based on the two-dimensional(2D)three-component first-order velocity-stress equation,the high order staggered mesh finite difference numerical simulation method was used to simulate the elastic and viscoelastic tilted transversely isotropic(TTI)media.The perfect matched layer(PML)absorption boundary condition was selected to eliminate the boundary effect.The results show that:(①)Under the condition of fixed elastic parameters of elastic TTI medium,when the polarization angle and azimuth are 60°and 45°respectively,the degree of shear wave splitting is significantly greater than the angle of 0°;②The influence of viscoelasticity on TTI medium is mainly reflected in the amplitude.If the quality factor decreases,the attenuation of the seismic wave amplitude increases,causing the waveform to become wider and distorted.If the quality factor increases,the viscoelastic medium becomes closer to elastic medium;③For TTI medium with different polarization angle and azimuth angle in the upper and lower layers,the shear wave can multiple splits at the interface of medium.The symmetry of seismograms is affected by the polarization angle and azimuth angle of TTI medium;④Viscoelasticity has a great influence on reflected wave,transmitted wave and converted wave in the low-velocity model.When the viscoelasticity is strong,the weaker waves may not be shown.

3DFD Simulation of Asymmetric Wave Field by Side-wall Acoustic Tool in Open and Cased Wells

We developed a parallelized scheme of 3D finite difference （3DFD）with non-oniform staggered grid to simulate the eccentric borehole acoustic field with side-wall acoustic logging tools in open and cased wells. Higher accuracy and lower computation cost were achieved with this scheme for modeling such an asymmetric wave field generated by a high frequency source near or on the borehole wall. We also modeled the cases with and without considering the effects of the tool body. The simulation results demonstrated that the logging tool body would attenuate the direct waves but have only little influence on the interface waves in such a borehole condition. The effects of the tool body on the wave field were significant only when the contrast of the elastic properties between tool body and borehole fluid was large.

Numerical simulation of elastic wave propagation based on the transversely isotropic BISQ equation

The Biot and Squirt-flow are the two most important mechanisms of fluid flow in the porous medium with fluids. Based on the BISQ (Biot-Squirt) model where the two mechanisms are treated simultaneously, dispersion and attenuation of elastic waves in the porous medium are widely investigated in recent years. However, we have not read any reports on numerical simulation based on the BISQ equation. In this paper, following the BISQ equation, elastic wave propagation in the transversely isotropic porous medium filled with fluids is simulated by the stag-gered grid method for different frequency and phase boundary cases and the two-layer medium. And propagating characteristics of seismic and acoustic waves and various phenomena occured in the propagating process are in-vestigated when the two mechanisms are considered simultaneously.

A NUMERICAL METHOD OF MODELING ELASTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA

The velocity-stress finite-difference method is adopted to simulate the elastic wave propa- gation in azimuthal anisotropic media.The difference grids are completely staggered in the numerical im- plementation.To reduce the computational work,the absorbin8 boundary conditions for anisotropic media are introduced first and the corner points are specially treated.Examples show that more accurate results can be obtained from the modeling algorithm,which cost much less computational time than the conven- tional methods.Therefore,the algorithm has broad application prospects in engineering.

New Sixth-Order Compact Schemes for Poisson/Helmholtz Equations

Some new sixth-order compact finite difference schemes for Poisson/Helmholtz equations on rectangular domains in both two-and three-dimensions are developed and analyzed.Different from a few sixth-order compact finite difference schemes in the literature,the finite difference and weight coefficients of the new methods have analytic simple expressions.One of the new ideas is to use a weighted combination of the source term at staggered grid points which is important for grid points near the boundary and avoids partial derivatives of the source term.Furthermore,the new compact schemes are exact for 2D and 3D Poisson equations if the solution is a polynomial less than or equal to 6.The coefficient matrices of the new schemes are M-matrices for Helmholtz equations with wave number K≤0,which guarantee the discrete maximum principle and lead to the convergence of the new sixth-order compact schemes.Numerical examples in both 2D and 3D are presented to verify the effectiveness of the proposed schemes.

NUMERICAL SIMULATION OF HYDRODYNAMIC CHARACTERISTICS ON AN ARC CROWN WALL USING VOLUME OF FLUID METHOD BASED ON BFC

In the present study, a new algorithm based on the Volume Of Fluid （VOF） method is developed to simulate the hydrodynamic characteristics on an arc crown wall. Structured grids are generated by the coordinate transform method in an arbitrary complex region. The Navier-Stokes equations for two-dimensional incompressible viscous flows are discretized in the Body Fitted Coordinate （BFC） system. The transformed SIMPLE algorithm is proposed to modify the pressure-velocity field and a transformed VOF method is used to trace the free surface. Hydrodynamic characteristics on an arc crown wall are obtained by the improved numerical model based on the BFC system （BFC model）. The velocity field, the pressure field and the time profiles of the water surface near the arc crown wall obtained by using the BFC model and the Cartesian model are compared. The BFC model is verified by experimental results.

TWO-DIMENSIONAL PLANE WATER FLOW AND WATER QUALITY DISTRIBUTION IN BOSTEN LAKE

The two-dimensional plane water flow and water quality was developed by usingthe techniques of coordinate transformation, alternating directions, staggered grid, linearrecurrence, and implicit scheme in the study of large water body in lakes. The model was proved tobe suitable for treating the irregular boundary and predicting quickly water flow and water quality.The application of the model to the Bosten Lake in Xinjiang Uygur Autonomous Region of China showsthat it is reasonable and practicable.

Numerical solution of potential flow equations with a predictor-corrector finite difference method

We develop a numerical solution algorithm of the nonlinear potential flow equations with the nonlinear free surface boundary condition.A finite difference method with a predictor-corrector method is applied to solve the nonlinear potential flow equations in a two-dimensional (2D) tank.The irregular tank is mapped onto a fixed square domain with rectangular cells through a proper mapping function.A staggered mesh system is adopted in a 2D tank to capture the wave elevation of the transient fluid.The finite difference method with a predictor-corrector scheme is applied to discretize the nonlinear dynamic boundary condition and nonlinear kinematic boundary condition.We present the numerical results of wave elevations from small to large amplitude waves with free oscillation motion,and the numerical solutions of wave elevation with horizontal excited motion.The beating period and the nonlinear phenomenon are very clear.The numerical solutions agree well with the analytical solutions and previously published results.

A Multilevel Method for the Solution of Time Dependent Optimal Transport

In this paper we present a new computationally efficientnumerical scheme for the minimizing flow for the computation of the optimal L 2 mass transport map-ping using the fluid approach.We review the method and discuss its numerical properties.We then derive a new scaleable,efficient discretization and a solution technique for the problem and show that the problem is equivalent to a mixed form formulation of a nonlinear fluid flow in porous media.We demonstrate the effec-tiveness of our approach using a number of numerical experiments.