An explicit method to calculate implicit spatial finite differences

Geophysics ◽  
2021 ◽  
pp. 1-71
Author(s):  
Hongwei Liu ◽  
Yi Luo
Keyword(s):  
Finite Difference ◽  
High Performance ◽  
Computational Cost ◽  
New Method ◽  
Seismic Exploration ◽  
Explicit Method ◽  
Lu Decomposition ◽  
Implicit Methods ◽  
Band Matrices ◽  
Explicit Finite Difference

The finite-difference solution of the second-order acoustic wave equation is a fundamental algorithm in seismic exploration for seismic forward modeling, imaging, and inversion. Unlike the standard explicit finite difference (EFD) methods that usually suffer from the so-called "saturation effect", the implicit FD methods can obtain much higher accuracy with relatively short operator length. Unfortunately, these implicit methods are not widely used because band matrices need to be solved implicitly, which is not suitable for most high-performance computer architectures. We introduce an explicit method to overcome this limitation by applying explicit causal and anti-causal integrations. We can prove that the explicit solution is equivalent to the traditional implicit LU decomposition method in analytical and numerical ways. In addition, we also compare the accuracy of the new methods with the traditional EFD methods up to 32nd order, and numerical results indicate that the new method is more accurate. In terms of the computational cost, the newly proposed method is standard 8th order EFD plus two causal and anti-causal integrations, which can be applied recursively, and no extra memory is needed. In summary, compared to the standard EFD methods, the new method has a spectral-like accuracy; compared to the traditional LU-decomposition implicit methods, the new method is explicit. It is more suitable for high-performance computing without losing any accuracy.

Determining the optimal coefficients of the explicit finite-difference scheme using the Remez exchange algorithm

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. S137-S147 ◽  
Author(s):  
Zheng He ◽  
Jinhai Zhang ◽  
Zhenxing Yao
Keyword(s):  
Finite Difference ◽  
Simulated Annealing Algorithm ◽  
Least Squares Method ◽  
Computational Cost ◽  
Seismic Exploration ◽  
Traditional Methods ◽  
Exchange Algorithm ◽  
Optimal Coefficients ◽  
Constant Coefficients ◽  
Explicit Finite Difference

Explicit finite-difference (FD) schemes are widely used in the seismic exploration field due to their simplicity in implementation and low computational cost. However, they suffer from strong artifacts caused by using coarse grids for high-frequency applications. The optimization of constant coefficients is popular in reducing spatial dispersions, but current methods could not guarantee that the bandwidth of the tolerable dispersion error is the widest. We have applied the Remez exchange algorithm to optimize the constant coefficients of the explicit FD schemes, for conventional and staggered grids. The resulting dispersion errors are distributed alternately between the maxima and minima in the passband of the filter, which is consistent with the most important equal-ripple property of the error magnitude for the optimal solution according to the Chebyshev criterion. The Remez exchange algorithm can determine the optimal coefficients of the FD method with only a few iterations, and the resulting operator has a wider bandwidth compared with previous solutions. It can handle arbitrary orders without the influence of local minima. Its computational cost for solving the objective function is comparable to that of the least-squares method, but its bandwidth is wider. Its accuracy is also higher than that of the maximum norm solved by the simulated annealing algorithm, but its computational cost is much lower. Theoretically, the equal-ripple error can offer the widest bandwidth for suppressing numerical dispersions among all solutions obtained by the constant-coefficient optimization. In other words, we can obtain a smaller error limitation than traditional methods under the same bandwidth. This superiority over traditional methods is essential for reducing the total error accumulation, which is helpful to avoid rapid error accumulations especially for large-scale models and long-term problems.

On the Solution of the Diffusion Equations by Numerical Methods

1966 ◽  
Vol 88 (4) ◽  
pp. 421-427 ◽  
Author(s):  
H. Z. Barakat ◽  
J. A. Clark
Keyword(s):  
Exact Solution ◽  
Finite Difference ◽  
Present Method ◽  
Finite Difference Methods ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Procedure ◽  
Implicit Methods ◽  
Difference Methods ◽  
Explicit Finite Difference

An explicit-finite difference approximation procedure which is unconditionally stable for the solution of the general multidimensional, nonhomogeneous diffusion equation is presented. This method possesses the advantages of the implicit methods, i.e., no severe limitation on the size of the time increment. Also it has the simplicity of the explicit methods and employs the same “marching” type technique of solution. Results obtained by this method for several different problems are compared with the exact solution and with those obtained by other finite-difference methods. For the examples solved the numerical results obtained by the present method are in closer agreement with the exact solution than are those obtained by the other methods.

An implicit finite-difference operator for the Helmholtz equation

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. T97-T107 ◽  
Author(s):  
Chunlei Chu ◽  
Paul L. Stoffa
Keyword(s):  
Finite Difference ◽  
Helmholtz Equation ◽  
Frequency Domain ◽  
Difference Operator ◽  
Computational Cost ◽  
Finite Difference Discretization ◽  
Implicit Finite Difference ◽  
Explicit Finite Difference ◽  
The One ◽  
Difference Discretization

We have developed an implicit finite-difference operator for the Laplacian and applied it to solving the Helmholtz equation for computing the seismic responses in the frequency domain. This implicit operator can greatly improve the accuracy of the simulation results without adding significant extra computational cost, compared with the corresponding conventional explicit finite-difference scheme. We achieved this by taking advantage of the inherently implicit nature of the Helmholtz equation and merging together the two linear systems: one from the implicit finite-difference discretization of the Laplacian and the other from the discretization of the Helmholtz equation itself. The end result of this simple yet important merging manipulation is a single linear system, similar to the one resulting from the conventional explicit finite-difference discretizations, without involving any differentiation matrix inversions. We analyzed grid dispersions of the discrete Helmholtz equation to show the accuracy of this implicit finite-difference operator and used two numerical examples to demonstrate its efficiency. Our method can be extended to solve other frequency domain wave simulation problems straightforwardly.

Optimizing the finite-difference implementation of three-dimensional free-surface boundary in frequency-domain modeling of elastic waves

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. T363-T379
Author(s):  
Jian Cao ◽  
Jing-Bo Chen
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Frequency Domain ◽  
Computational Cost ◽  
Good Choice ◽  
Weighted Averaging ◽  
Seismic Exploration ◽  
Surface Boundary ◽  
Domain Modeling ◽  
Near Surface

The problem of modeling seismic wave propagation for multiple sources, such as in the solution of gradient-based elastic full-waveform inversion, is an important topic in seismic exploration. The frequency-domain finite-difference (FD) method is a good choice for this purpose, mainly because of its simple discretization and high computational efficiency. However, when it comes to modeling the complete elastic wavefields, this approach has limited surface-wave accuracy because, when modeling with the strong form of the wave equation, it is not always easy to implement an accurate stress-free boundary condition. Although a denser spatial sampling is helpful for overcoming this problem, the additional discrete points will significantly increase the computational cost in the resolution of its resulting discrete system, especially in 3D problems. Furthermore, sometimes, when modeling with optimized schemes, an inconsistency in the computation precision between the regions at the free surface and inside the model volume would happen and introduce numerical artifacts. To overcome these issues, we have considered optimizing the FD implementation of the free-surface boundary. In our method, the problem was formulated in terms of a novel system of partial differential equations satisfied at the free surface, and the weighted-averaging strategy was introduced to optimize its discretization. With this approach, we can impose FD schemes for the free surface and internal region consistently and improve their discretization precision simultaneously. Benchmark tests for Lamb’s problem indicate that the proposed free-surface implementation contributes to improving the simulation accuracy on surface waves, without increasing the number of grid points per wavelength. This reveals the potential of developing optimized schemes in the free-surface implementation. In particular, through the successful introduction of weighting coefficients, this free-surface FD implementation enables adaptation to the variation of Poisson’s ratio, which is very useful for modeling in heterogeneous near-surface weathered zones.

A new method for the analysis of plasmin and plasminogen. Application of high-performance affinity chromatography.

1985 ◽  
Vol 16 (1) ◽  
pp. 23-25
Author(s):  
Kenichi KASAI ◽  
Kiyohito SHIMURA ◽  
Naofumi ITO ◽  
Kohji NOGUCHI ◽  
Mutsuyoshi KAZAMA
Keyword(s):  
Affinity Chromatography ◽  
High Performance ◽  
New Method ◽  
High Performance Affinity Chromatography

Influence of Tooth Errors and Truing Principles of Grinding Wheel Abrasion for Machining Arc Enveloping Worm

2013 ◽  
Vol 652-654 ◽  
pp. 2153-2158
Author(s):  
Wu Ji Jiang ◽  
Jing Wei
Keyword(s):  
Mathematical Model ◽  
Case Studies ◽  
High Precision ◽  
High Performance ◽  
New Method ◽  
Grinding Wheel ◽  
Grinding Process ◽  
The Mathematical Model

Controlling the tooth errors induced by the variation of diameter of grinding wheel is the key problem in the process of ZC1 worm grinding. In this paper, the influence of tooth errors by d1, m and z1 as the grinding wheel diameter changes are analyzed based on the mathematical model of the grinding process. A new mathematical model and truing principle for the grinding wheel of ZC1 worm is presented. The shape grinding wheel truing of ZC1 worm is carried out according to the model. The validity and feasibility of the mathematical model is proved by case studies. The mathematical model presented in this paper provides a new method for reducing the tooth errors of ZC1 worm and it can meet the high-performance and high-precision requirements of ZC1 worm grinding.

Thermal Modeling of Unlooped and Looped Pulsating Heat Pipes

2001 ◽  
Vol 123 (6) ◽  
pp. 1159-1172 ◽  
Author(s):  
Mohammad B. Shafii ◽  
Amir Faghri ◽  
Yuwen Zhang
Keyword(s):  
Heat Transfer ◽  
Finite Difference ◽  
Heat Pipes ◽  
Analytical Models ◽  
Charge Ratio ◽  
Sensible Heat ◽  
Governing Equations ◽  
Heating Wall ◽  
Heat Mode ◽  
Explicit Finite Difference

Analytical models for both unlooped and looped Pulsating Heat Pipes (PHPs) with multiple liquid slugs and vapor plugs are presented in this study. The governing equations are solved using an explicit finite difference scheme to predict the behavior of vapor plugs and liquid slugs. The results show that the effect of gravity on the performance of top heat mode unlooped PHP is insignificant. The effects of diameter, charge ratio, and heating wall temperature on the performance of looped and unlooped PHPs are also investigated. The results also show that heat transfer in both looped and unlooped PHPs is due mainly to the exchange of sensible heat.

Sign in / Sign up

Export Citation Format

Share Document