Finite-difference modeling with adaptive variable-length spatial operators

Geophysics ◽  
2011 ◽  
Vol 76 (4) ◽  
pp. T79-T89 ◽  
Author(s):  
Yang Liu ◽  
Mrinal K. Sen
Keyword(s):  
Finite Difference ◽  
Dispersion Analysis ◽  
Variable Length ◽  
Finite Difference Schemes ◽  
Low Velocity ◽  
Absorbing Boundary ◽  
Local Grid ◽  
Spatial Derivatives ◽  
Spatial Operators ◽  
And Inversion

Most finite-difference simulation algorithms use fixed-length spatial operators to compute spatial derivatives. The choice of length is dictated by computing cost, stability, and dispersion criteria that are satisfied globally. We propose finite-difference schemes with adaptive variable-length spatial operators to decrease computing costs significantly without reducing accuracy. These schemes adopt long operators in regions of low velocity and short operators in regions of high velocity. Two methods automatically determine variable operator lengths. Dispersion analysis, along with 1D and 2D modeling, demonstrates the validity and efficiency of our schemes. In addition, a hybrid absorbing boundary condition helps reduce unwanted reflections from model boundaries. Our scheme is more efficient than those based on variable-grid methods for modeling, migration, and inversion of models with complex velocity structures because the latter require local grid refinement, which usually increases memory requirements and computing costs.

Prefactored optimized compact finite-difference schemes for second spatial derivatives

Geophysics ◽  
2011 ◽  
Vol 76 (5) ◽  
pp. WB87-WB95 ◽  
Author(s):  
Hongbo Zhou ◽  
Guanquan Zhang
Keyword(s):  
Finite Difference ◽  
Difference Schemes ◽  
Compact Scheme ◽  
Finite Difference Schemes ◽  
Difference Operators ◽  
Compact Schemes ◽  
Compact Finite Difference ◽  
Wide Range ◽  
Compact Finite Difference Schemes ◽  
Spatial Derivatives

We have described systematically the processes of developing prefactored optimized compact schemes for second spatial derivatives. First, instead of emphasizing high resolution of a single monochromatic wave, we focus on improving the representation of the compact finite difference schemes over a wide range of wavenumbers. This leads to the development of the optimized compact schemes whose coefficients will be determined by Fourier analysis and the least-squares optimization in the wavenumber domain. The resulted optimized compact schemes provide the maximum resolution in spatial directions for the simulation of wave propagations. However, solving for each spatial derivative using these compact schemes requires the inversion of a band matrix. To resolve this issue, we propose a prefactorization strategy that decomposes the original optimized compact scheme into forward and backward biased schemes, which can be solved explicitly. We achieve this by ensuring a property that the real numerical wavenumbers of both the forward and backward biased stencils are the same as that of the original central compact scheme, and their imaginary numerical wavenumbers have the same values but with opposite signs. This property guarantees that the original optimized compact scheme can be completely recovered after the summation of the forward and backward finite difference operators. These prefactored optimized compact schemes have smaller stencil sizes than even those of the original compact schemes, and hence, they can take full advantage of the computer caches without sacrificing their resolving power. Comparisons were made throughout with other well-known schemes.

scholarly journals Numerical approximation of one model of bacterial self-organization

2012 ◽  
Vol 17 (3) ◽  
pp. 253-270 ◽  
Author(s):  
Raimondas Čiegis ◽  
Andrej Bugajev
Keyword(s):  
Finite Difference ◽  
Method Of Lines ◽  
Self Organization ◽  
Finite Difference Schemes ◽  
Physical Processes ◽  
Finite Difference Approximations ◽  
The Method Of Lines ◽  
Ill Posed ◽  
Spatial Derivatives ◽  
Implicit And Explicit

This paper presents finite difference approximations of one dimensional in space mathematical model of a bacterial self-organization. The dynamics of such nonlinear systems can lead to formation of complicated solution patterns. In this paper we show that this chemotaxisdriven instability can be connected to the ill-posed problem defined by the backward in time diffusion process. The method of lines is used to construct robust numerical approximations. At the first step we approximate spatial derivatives in the PDE by applying approximations targeted for special physical processes described by differential equations. The obtained system of ODE is split into a system describing separately fast and slow physical processes and different implicit and explicit numerical solvers are constructed for each subproblem. Results of numerical experiments are presented and convergence of finite difference schemes is investigated. 

scholarly journals Acoustic wave propagation with new spatial implicit and temporal high-order staggered-grid finite-difference schemes

2021 ◽  
Vol 18 (5) ◽  
pp. 808-823
Author(s):  
Jing Wang ◽  
Yang Liu ◽  
Hongyu Zhou
Keyword(s):  
Finite Difference ◽  
High Order ◽  
Staggered Grid ◽  
Finite Difference Schemes ◽  
Spatial Accuracy ◽  
Time Step ◽  
Dispersion Formula ◽  
Time Space ◽  
L2 Norm ◽  
Spatial Derivatives

Abstract The implicit staggered-grid (SG) finite-difference (FD) method can obtain significant improvement in spatial accuracy for performing numerical simulations of wave equations. Normally, the second-order central grid FD formulas are used to approximate the temporal derivatives, and a relatively fine time step has to be used to reduce the temporal dispersion. To obtain high accuracy both in space and time, we propose a new spatial implicit and temporal high-order SG FD stencil in the time–space domain by incorporating some additional grid points to the conventional implicit FD one. Instead of attaining the implicit FD coefficients by approximating spatial derivatives only, we calculate the coefficients by approximating the temporal and spatial derivatives simultaneously through matching the dispersion formula of the seismic wave equation and compute the FD coefficients of our new stencil by two schemes. The first one is adopting a variable substitution-based Taylor-series expansion (TE) to derive the FD coefficients, which can attain (2M + 2)th-order spatial accuracy and (2N)th-order temporal accuracy. Note that the dispersion formula of our new stencil is non-linear with respect to the axial and off-axial FD coefficients, it is complicated to obtain the optimal spatial and temporal FD coefficients simultaneously. To tackle the issue, we further develop a linear optimisation strategy by minimising the L2-norm errors of the dispersion formula to further improve the accuracy. Dispersion analysis, stability analysis and modelling examples demonstrate the accuracy, stability and efficiency advantages of our two new schemes.

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