Cascadic Newton’s method for the elliptic Monge–Ampère equation

2020 ◽  
Vol 18 (03) ◽  
pp. 2050018
Author(s):  
Qin Li ◽  
Zhiyong Liu
Keyword(s):  
Finite Difference ◽  
Newton’S Method ◽  
Newton's Method ◽  
Finite Difference Methods ◽  
Computation Time ◽  
Type Interpolation ◽  
Local Type ◽  
Cascadic Multigrid ◽  
Monge Ampere ◽  
Prolongation Operator

In this paper, a cascadic Newton’s method is designed to solve the Monge–Ampère equation. In the process of implementing the cascadic multigrid, we use the Full-Local type interpolation as prolongation operator and Newton iteration as smoother. In order to obtain Full-Local type interpolation, we provide several finite difference stencils. Especially, the skewed finite difference methods are first applied by us for the elliptic Monge–Ampère equation. Based on Full-Local interpolation techniques and cascade principle, the new algorithm can save a large amount of computation time. Some numerical experiments are provided to confirm the efficiency of our proposed method.

Reversed time migration in spatial frequency domain

Geophysics ◽  
1983 ◽  
Vol 48 (5) ◽  
pp. 627-635 ◽  
Author(s):  
Dan Loewenthal ◽  
Irshad R. Mufti
Keyword(s):  
Finite Difference ◽  
Acoustic Waves ◽  
Seismic Waves ◽  
Finite Difference Methods ◽  
Computation Time ◽  
Real Data ◽  
Coordinate Frame ◽  
Seismic Section ◽  
Time Migration ◽  
Similar Scheme

During the past decade, finite‐difference methods have become important tools for direct modeling of seismic data as well as for certain interpretation processes. One of the earliest applications of these methods to seismics is the pioneering contribution of Alterman who, in a series of papers (Alterman and Karal, 1968; Alterman and Aboudi, 1968; Alterman and Rotenberg, 1969; Alterman and Loewenthal, 1972) demonstrated the usefulness of such numerical computations for the propagation of seismic waves in elastic media. A clear exposition of these techniques, as well as a comparison of results obtained from them with the corresponding analytical solutions, can be found in Alterman and Karal (1968). This subject was further developed and extended to more complicated models by Boore (1970), Ottaviani (1971), and Kelly et al (1976). Claerbout introduced a somewhat different finite‐difference approach (Claerbout, 1970; Claerbout and Johnson, 1971) for modeling the acoustic waves which often dominate the reflection seismogram. In his approach, the original wave equation, which governs the propagation of the acoustic waves, is modified in such a way so as to allow the propagation of either only upcoming or only downgoing waves. By moving the coordinate frame with the downgoing waves, Claerbout showed that one could greatly reduce computation time. Using the same concepts, he showed (Claerbout and Doherty, 1972) how to use a similar scheme for migrating a seismic section by downward continuation of the upcoming waves. This migration method is an interesting extension of the ideas of Hagedoorn (1954) and was found to be extremely useful with real data (Larner and Hatton, 1976; Loewenthal et al, 1976).

scholarly journals Staggered-Grid Finite Difference Method with Variable-Order Accuracy for Porous Media

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Jinghuai Gao ◽  
Yijie Zhang
Keyword(s):  
Porous Media ◽  
Finite Difference ◽  
Finite Difference Methods ◽  
Computation Time ◽  
High Order ◽  
Staggered Grid ◽  
Difference Operators ◽  
Variable Order ◽  
Low Velocity ◽  
Order Accuracy

The numerical modeling of wave field in porous media generally requires more computation time than that of acoustic or elastic media. Usually used finite difference methods adopt finite difference operators with fixed-order accuracy to calculate space derivatives for a heterogeneous medium. A finite difference scheme with variable-order accuracy for acoustic wave equation has been proposed to reduce the computation time. In this paper, we develop this scheme for wave equations in porous media based on dispersion relation with high-order staggered-grid finite difference (SFD) method. High-order finite difference operators are adopted for low-velocity regions, and low-order finite difference operators are adopted for high-velocity regions. Dispersion analysis and modeling results demonstrate that the proposed SFD method can decrease computational costs without reducing accuracy.

Finite difference automatically generated non-uniform grids in the numerical solution of the Reynolds' compressible one-dimensional squeeze-film equation

2003 ◽  
Vol 217 (3) ◽  
pp. 243-249 ◽  
Author(s):  
E. A. M. Almas ◽  
F. A. P. De Silva
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Computation Time ◽  
Squeeze Film ◽  
Uniform Grid ◽  
Efficient Computation ◽  
Uniform Grids ◽  
Uniform Model ◽  
Squeeze Number ◽  
Solution Accuracy

With the Reynolds equation, for compressible squeeze-film thrust bearings, the use of a finite difference discretization with a grid of nodes having a non-uniform spacing can result in a more efficient computation of the solution. Comparative tests of two variable spatial grid models against a uniform model were conducted with the classical finite difference methods for chosen combinations of squeeze number and excursion ratio values. For problems with a high value of σ, one of the non-uniform grid models has shown some advantages over the uniform model, requiring a smaller number of nodes and less computation time with the same solution accuracy.

Application of Backus average to simulate acoustic logs in thinly bedded formations

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. D93-D104
Author(s):  
Elsa Maalouf ◽  
Carlos Torres-Verdín ◽  
Jingxuan Li
Keyword(s):  
Finite Difference ◽  
Numerical Simulations ◽  
Finite Difference Methods ◽  
Homogeneous Medium ◽  
Computation Time ◽  
Transversely Isotropic ◽  
Simulation Method ◽  
Thin Layers ◽  
Homogeneous Layer ◽  
Spatial Averaging

Slowness logs acquired in layered formations are not only affected by spatial averaging associated with the borehole acoustic tool. Layers with thicknesses smaller than the acoustic wavelength can cause measurable effects on the associated wave propagation phenomena. While spatial averaging functions can be used to model tool averaging effects, computer-intensive numerical methods such as finite differences must be used to simulate slowness logs across formations with thin layers. We adopted Backus averaging as a faster alternative to model borehole slownesses when layer thicknesses are smaller than the acoustic wavelength (i.e., in the long-wavelength limit). Using synthetic models and numerical simulations via finite-element and finite-difference methods, we have determined that borehole slownesses of a stack of horizontal layers first approach the average slowness of the individual layers. However, as the layer thickness decreases, sonic slownesses approach the slowness of a homogeneous medium with elastic properties obtained from the Backus average. Therefore, to model acoustic logs acquired in layered formations, we first approximate thin layers as a single homogeneous layer with stiffness coefficients calculated using the Backus average. Next, we apply a spatial averaging function to reproduce the spatial averaging effect inherent to borehole acoustic tools. Results indicate that the latter method is accurate and efficient for fast modeling borehole slownesses of formations with thin layers that are isotropic and intrinsically vertical transversely isotropic. The fast simulation method decreases computation time by at least a factor of 10 and yields slowness logs with a relative error below 2% compared with finite-difference numerical simulations. We also determine that the moving Backus average that is typically applied to upscale acoustic logs for seismic applications is not accurate to model borehole acoustic logs acquired across thinly layered formations.

Application of Direct Newton’s Methods for Efficient Solution to Convective-Diffusive Transport Processes

2005 ◽  
Author(s):  
Mandhapati P. Raju
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Simple Procedure ◽  
Linear Equations ◽  
Computation Time ◽  
Transport Processes ◽  
Diffusive Transport ◽  
Direct Solution ◽  
Solution Methods ◽  
Direct Solution Method

Newton’s iterative technique is commonly used in solving a system of non-linear equations. The advantage of using Newton’s method is that it gives local quadratic convergence leading to high computational efficiency. Specifically, Newton’s method has been applied to finite volume formulation for convective-diffusive transport processes. A direct solution method is adopted. Development of sparse direct solvers has significantly reduced the computation time of direct solution methods. Here UMFPACK (Unsymmetric Multi-Frontal method), has been used to solve the resulting linear system obtained from Newton’s step. A simple damping strategy is applied to ensure the global convergence of the system of equations during the first few iterations. The efficiency of this method is compared to that of Picard’s iterative procedure and the SIMPLE procedure for convective-diffusive transport processes. A modified Newton technique is also analyzed which lead to significant reduction in total CPU time.

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