Removing the stability limit of the time–space domain explicit finite difference schemes for acoustic modeling with stability condition-based spatial operators

Geophysics ◽  
2021 ◽  
pp. 1-82
Author(s):  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Stability Condition ◽  
Finite Difference Schemes ◽  
Grid Spacing ◽  
Time Step ◽  
Space Domain ◽  
Wavenumber Range ◽  
Time Space ◽  
Explicit Finite Difference ◽  
The Stability

The time step and grid spacing in explicit finite-difference (FD) modeling are constrained by the Courant-Friedrichs-Lewy (CFL) condition. Recently, it has been found that spatial FD coefficients may be designed through simultaneously minimizing the spatial dispersion error and maximizing the CFL number. This allows one to stably use a larger time step or a smaller grid spacing than usually possible. However, when using such a method, only second-order temporal accuracy is achieved. To address this issue, I propose a method to determine the spatial FD coefficients, which simultaneously satisfy the stability condition of the whole wavenumber range and the time–space domain dispersion relation of a given wavenumber range. Therefore, stable modeling can be performed with high-order spatial and temporal accuracy. The coefficients can adapt to the variation of velocity in heterogeneous models. Additionally, based on the hybrid absorbing boundary condition, I develop a strategy to stably and effectively suppress artificial reflections from the model boundaries for large CFL numbers. Stability analysis, accuracy analysis and numerical modeling demonstrate the accuracy and effectiveness of the proposed method.

Acoustic and elastic modeling by optimal time-space-domain staggered-grid finite-difference schemes

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. T17-T40 ◽  
Author(s):  
Zhiming Ren ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Stability Condition ◽  
Computational Cost ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Equation Modeling ◽  
Optimal Time ◽  
Finite Difference Schemes ◽  
Space Domain ◽  
Time Space

Staggered-grid finite-difference (SFD) methods are widely used in modeling seismic-wave propagation, and the coefficients of finite-difference (FD) operators can be estimated by minimizing dispersion errors using Taylor-series expansion (TE) or optimization. We developed novel optimal time-space-domain SFD schemes for acoustic- and elastic-wave-equation modeling. In our schemes, a fourth-order multiextreme value objective function with respect to FD coefficients was involved. To yield the globally optimal solution with low computational cost, we first used variable substitution to turn our optimization problem into a quadratic convex one and then used least-squares (LS) to derive the optimal SFD coefficients by minimizing the relative error of time-space-domain dispersion relations over a given frequency range. To ensure the robustness of our schemes, a constraint condition was imposed that the dispersion error at each frequency point did not exceed a given threshold. Moreover, the hybrid absorbing boundary condition was applied to remove artificial boundary reflections. We compared our optimal SFD with the conventional, TE-based time-space-domain, and LS-based SFD schemes. Dispersion analysis and numerical simulation results suggested that the new SFD schemes had a smaller numerical dispersion than the other three schemes when the same operator lengths were adopted. In addition, our LS-based time-space-domain SFD can obtain the same modeling accuracy with shorter spatial operator lengths. We also derived the stability condition of our schemes. The experiment results revealed that our new LS-based SFD schemes needed a slightly stricter stability condition.

Efficient Two-dimensional Acoustic Wave Finite-Difference Numerical Simulation in Strongly Heterogeneous Media Using the Adaptive Mesh Refinement (AMR) Technique

Geophysics ◽  
2021 ◽  
pp. 1-76
Author(s):  
Chunli Zhang ◽  
Wei Zhang
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Adaptive Mesh Refinement ◽  
Heterogeneous Media ◽  
Mesh Refinement ◽  
Adaptive Mesh ◽  
Grid Spacing ◽  
Time Step ◽  
Velocity Models ◽  
The Stability

The finite-difference method (FDM) is one of the most popular numerical methods to simulate seismic wave propagation in complex velocity models. If a uniform grid is applied in the FDM for heterogeneous models, the grid spacing is determined by the global minimum velocity to suppress dispersion and dissipation errors in the numerical scheme, resulting in spatial oversampling in higher-velocity zones. Then, the small grid spacing dictates a small time step due to the stability condition of explicit numerical schemes. The spatial oversampling and reduced time step will cause unnecessarily inefficient use of memory and computational resources in simulations for strongly heterogeneous media. To overcome this problem, we propose to use the adaptive mesh refinement (AMR) technique in the FDM to flexibly adjust the grid spacing following velocity variations. AMR is rarely utilized in acoustic wave simulations with the FDM due to the increased complexity of implementation, including its data management, grid generation and computational load balancing on high-performance computing platforms. We implement AMR for 2D acoustic wave simulation in strongly heterogeneous media based on the patch approach with the FDM. The AMR grid can be automatically generated for given velocity models. To simplify the implementation, we employ a well-developed AMR framework, AMReX, to carry out the complex grid management. Numerical tests demonstrate the stability, accuracy level and efficiency of the AMR scheme. The computation time is approximately proportional to the number of grid points, and the overhead due to the wavefield exchange and data structure is small.

Time-space-domain implicit finite-difference methods for modeling acoustic wave equations

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. T175-T193 ◽  
Author(s):  
Enjiang Wang ◽  
Jing Ba ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Wave Equations ◽  
Finite Difference Methods ◽  
Second Order ◽  
Computational Time ◽  
Time Step ◽  
Space Domain ◽  
Time Space ◽  
Temporal Dispersion

It has been proved that the implicit spatial finite-difference (FD) method can obtain higher accuracy than explicit FD by using an even smaller operator length. However, when only second-order FD in time is used, the combined FD scheme is prone to temporal dispersion and easily becomes unstable when a relatively large time step is used. The time-space domain FD can suppress the temporal dispersion. However, because the spatial derivatives are solved explicitly, the method suffers from spatial dispersion and a large spatial operator length has to be adopted. We have developed two effective time-space-domain implicit FD methods for modeling 2D and 3D acoustic wave equations. First, the high-order FD is incorporated into the discretization for the second-order temporal derivative, and it is combined with the implicit spatial FD. The plane-wave analysis method is used to derive the time-space-domain dispersion relations, and two novel methods are proposed to determine the spatial and temporal FD coefficients in the joint time-space domain. First, we fix the implicit spatial FD coefficients and derive the quadratic convex objective function with respect to temporal FD coefficients. The optimal temporal FD coefficients are obtained by using the linear least-squares method. After obtaining the temporal FD coefficients, the SolvOpt nonlinear algorithm is applied to solve the nonquadratic optimization problem and obtain the optimized temporal and spatial FD coefficients simultaneously. The dispersion analysis, stability analysis, and modeling examples validate that the proposed schemes effectively increase the modeling accuracy and improve the stability conditions of the traditional implicit schemes. The computational efficiency is increased because the schemes can adopt larger time steps with little loss of spatial accuracy. To reduce the memory requirement and computational time for storing and calculating the FD coefficients, we have developed the representative velocity strategy, which only computes and stores the FD coefficients at several selected velocities. The modeling result of the 2D complicated model proves that the representative velocity strategy effectively reduces the memory requirements and computational time without decreasing the accuracy significantly when a proper velocity interval is used.

scholarly journals Computational Analysis of the Stability of 2D Heat Equation on Elliptical Domain Using Finite Difference Method

2020 ◽  
pp. 8-19
Author(s):  
Mehwish Naz Rajput ◽  
Asif Ali Shaikh ◽  
Shakeel Ahmed Kamboh
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Heat Equation ◽  
Stability Condition ◽  
Difference Method ◽  
Stable Solution ◽  
Finite Difference Schemes ◽  
Step Size ◽  
Stability Range ◽  
The Stability

Aims: The aim and objective of the study to derive and analyze the stability of the finite difference schemes in relation to the irregularity of domain. Study Design: First of all, an elliptical domain has been constructed with the governing two dimensional (2D) heat equation that is discretized using the Finite Difference Method (FDM). Then the stability condition has been defined and the numerical solution by writing MATLAB codes has been obtained with the stable values of time domain. Place and Duration of Study: The work has been jointly conducted at the MUET, Jamshoro and QUEST, Nawabshah Pakistan from January 2019 to December 2019.  Methodology: The stability condition over an elliptical domain with the non-uniform step size depending upon the boundary tracing function is derived by using Von Neumann method. Results: From the results it was revealed that stability region for the small number of mesh points remains larger and gets smaller as the number of mesh nodes is increased. Moreover, the ranges for the time steps are defined for varied spatial step sizes that help to find the stable solution. Conclusion: The corresponding stability range for number of nodes N=10, 20, 30, 40, 50, and 60 was found respectively. Within this range the solution remains smooth as time increases. The results of this study attempt to provide the stable solution of partial differential equations on irregular domains.

scholarly journals Analysis and optimization of higher order explicit finite-difference schemes for the advection stage implementation in the lattice Boltzmann method

2017 ◽  
pp. 227-246
Author(s):  
Г.В. Кривовичев ◽  
Е.С. Марнопольская
Keyword(s):  
Finite Difference ◽  
Lattice Boltzmann ◽  
Kinetic Equations ◽  
Difference Schemes ◽  
Numerical Dispersion ◽  
Finite Difference Schemes ◽  
Maximum Function ◽  
Driven Cavity ◽  
Explicit Finite Difference ◽  
The Stability

Статья посвящена анализу и оптимизации явных разностных схем для решения уравнений переноса, возникающих на этапе адвекции метода расщепления по физическим процессам. Метод может применяться как для решеточных уравнений Больцмана, так и при решении кинетических уравнений общего вида. Рассматриваются схемы второго-четвертого порядков аппроксимации. Для уменьшения эффектов численных диссипации и дисперсии используются схемы с параметром. С использованием метода фон Неймана и полиномиальной аппроксимации границ областей устойчивости получены условия устойчивости схем в виде неравенств на значения параметра Куранта. Оптимальные значения параметра для регулирования диссипативных и дисперсионных эффектов предлагается находить посредством решения задач минимизации функций максимума. Схемы с оптимальными значениями параметра применяются при решении тестовых задач - для одномерного и двумерного уравнений переноса, а также при применении метода расщепления к решению задачи о течении в каверне с подвижной крышкой. This paper is devoted to the analysis and optimization of explicit finite-difference schemes for solving the transport equations arising at the advection stage in the method of splitting into physical processes. The method can be applied to the lattice Boltzmann equations and to the kinetic equations of general type. The second-to-fourth order schemes are considered. In order to minimize the effect of numerical dispersion and dissipation, the parametric schemes are used. The Neumann method and the polynomial approximation of the boundaries of stability domains are employed to obtain the stability conditions in the form of inequalities imposed on the Courant parameter. The optimal values of the parameter used to control the dissipation and dispersion effects are found by minimizing the maximum function. The schemes with optimal parameters are applied for the numerical solution of 1D and 2D advection equations and for the problem of lid-driven cavity flow.

scholarly journals Stability theory for some scalar finite difference schemes: validity of the modified equations approach

2021 ◽  
Vol 70 ◽  
pp. 124-136
Author(s):  
Firas Dhaouadi ◽  
Emilie Duval ◽  
Sergey Tkachenko ◽  
Jean-Paul Vila
Keyword(s):  
Fourier Transform ◽  
Stability Analysis ◽  
Finite Difference ◽  
Heat Equation ◽  
Infinite Series ◽  
Stability Condition ◽  
Finite Difference Schemes ◽  
Modified Equations ◽  
The Stability ◽  
Modified Equation

In this paper, we discuss some limitations of the modified equations approach as a tool for stability analysis for a class of explicit linear schemes to scalar partial differential equations. We show that the infinite series obtained by Fourier transform of the modified equation is not always convergent and that in the case of divergence, it becomes unrelated to the scheme. Based on these results, we explain when the stability analysis of a given truncation of a modified equation may yield a reasonable estimation of a stability condition for the associated scheme. We illustrate our analysis by some examples of schemes namely for the heat equation and the transport equation.

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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