Recursive integral time-extrapolation methods for waves: A comparative review

Geophysics ◽  
2014 ◽  
Vol 79 (1) ◽  
pp. T9-T26 ◽  
Author(s):  
Xiang Du ◽  
Paul J. Fowler ◽  
Robin P. Fletcher
Keyword(s):  
Wave Equations ◽  
Finite Difference Methods ◽  
Single Mode ◽  
Fourier Integral ◽  
Numerical Dispersion ◽  
Time Step ◽  
Integral Time ◽  
Comparative Review ◽  
Time Extrapolation ◽  
Accurate Wave

We compared several families of algorithms for recursive integral time-extrapolation (RITE) algorithms for waves in isotropic and anisotropic media. These methods allow simulating accurate wave extrapolation with little numerical dispersion even when using larger time steps than are usually possible for conventional finite-difference methods. These various RITE algorithms all share the use of mixed space/wavenumber-domain operators derived from Fourier integral solutions of single-mode wave equations. We evaluated a taxonomy for RITE methods based on how they approximated the influence of medium heterogeneity. One family of methods uses mixed-domain series expansions to provide accurate approximations to heterogeneous extrapolators even for large time steps. We compared several methods for deriving coefficients for such series approximations. Another family of methods uses interpolation between different homogeneous extrapolations to approximate heterogeneous time extrapolation. Such methods can be based on interpolating either the extrapolators themselves or interpolating between reference wavefields extrapolated using different homogeneous parameters. Interpolation methods work well for smooth media, but can suffer from oscillatory artifacts at large velocity discontinuities unless the time step is small. We tested numerical examples of the various families of RITE algorithms to determine their relative strengths and limitations.

Time-stepping wave-equation solution for seismic modeling using a multiple-angle formula and the Taylor expansion

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. T299-T311
Author(s):  
Edvaldo S. Araujo ◽  
Reynam C. Pestana
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Finite Difference Methods ◽  
Computational Cost ◽  
Rapid Expansion ◽  
Numerical Dispersion ◽  
Seismic Modeling ◽  
Time Step ◽  
Velocity Models ◽  
Time Migration

We have developed an analytical solution for wave equations using a multiple-angle formula. The new solution based on the multiple-angle expansion allows us to generate a family of solutions for the acoustic-wave equation, which may be combined with Taylor-series, Chebyshev, Hermite, and Legendre polynomial expansions or any other expansion for the cosine function and used for seismic modeling, reverse time migration, and inverse problems. Extension of this method to the solution of elastic and anisotropic wave equations is also straightforward. We also derive a criterion using the stability and dispersion relations to determine the order of the solution for a given time step and, thus, obtaining stable wavefields free of numerical dispersion. Afterward, numerical tests are performed using complex 2D velocity models to evaluate the effectiveness and robustness of our method, combined with second- or fourth-order Taylor approximations. Our multiple-angle approach is stable and provides reliable seismic modeling results for larger times steps than those usually used by conventional finite-difference methods. Moreover, multiple-angle schemes using a second-order Taylor approximation for each cosine term have a lower computational cost than the mixed wavenumber-space rapid expansion method.

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 Recursive integral time extrapolation of elastic waves using low-rank symbol approximation

2017 ◽  
Vol 211 (3) ◽  
pp. 1478-1493 ◽  
Author(s):  
Junzhe Sun ◽  
Sergey Fomel ◽  
Yanadet Sripanich ◽  
Paul Fowler
Keyword(s):  
Elastic Waves ◽  
Low Rank ◽  
Integral Time ◽  
Time Extrapolation

Numerical effects on the simulation of surfactant flooding for enhanced oil recovery

2021 ◽  
Author(s):  
Olaitan Akinyele ◽  
Karl D. Stephen
Keyword(s):  
Interfacial Tension ◽  
Cell Size ◽  
Oil Recovery ◽  
Flow In Porous Media ◽  
Numerical Dispersion ◽  
Sharp Change ◽  
Surfactant Flooding ◽  
Fractional Flow ◽  
Time Step ◽  
Scale Models

Abstract Numerical simulation of surfactant flooding using conventional reservoir simulation models can lead to unreliable forecasts and bad decisions due to the appearance of numerical effects. The simulations give approximate solutions to systems of nonlinear partial differential equations describing the physical behavior of surfactant flooding by combining multiphase flow in porous media with surfactant transport. The approximations are made by discretization of time and space which can lead to spurious pulses or deviations in the model outcome. In this work, the black oil model was simulated using the decoupled implicit method for various conditions of reservoir scale models to investigate behaviour in comparison with the analytical solution obtained from fractional flow theory. We investigated changes to cell size and time step as well as the properties of the surfactant and how it affects miscibility and flow. The main aim of this study was to understand pulse like behavior that has been observed in the water bank to identify cause and associated conditions. We report for the first time that the pulses occur in association with the simulated surfactant water flood front and are induced by a sharp change in relative permeability as the interfacial tension changes. Pulses are diminished when the adsorption rate was within the value of 0.0002kg/kg to 0.0005kg/kg. The pulses are absent for high resolution model of 5000 cells in x direction with a typical cell size as used in well-scale models. The growth or damping of these pulses may vary from case to case but in this instance was a result of the combined impact of relative mobility, numerical dispersion, interfacial tension and miscibility. Oil recovery under the numerical problems reduced the performance of the flood, due to large amounts of pulses produced. Thus, it is important to improve existing models and use appropriate guidelines to stop oscillations and remove errors.

A New Unconditionally Stable Time Integration Method for Analysis of Nonlinear Structural Dynamics

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
Ali Akbar Gholampour ◽  
Mehdi Ghassemieh ◽  
Mahdi Karimi-Rad
Keyword(s):  
Time Integration ◽  
Second Order ◽  
Numerical Dispersion ◽  
Computational Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Similar Time ◽  
Unconditionally Stable ◽  
Average Acceleration

A new time integration scheme is presented for solving the differential equation of motion with nonlinear stiffness. In this new implicit method, it is assumed that the acceleration varies quadratically within each time step. By increasing the order of acceleration, more terms of the Taylor series are used, which are expected to have responses with better accuracy than the classical methods. By considering this assumption and employing two parameters δ and α, a new family of unconditionally stable schemes is obtained. The order of accuracy, numerical dissipation, and numerical dispersion are used to measure the accuracy of the proposed method. Second order accuracy is achieved for all values of δ and α. The proposed method presents less dissipation at the lower modes in comparison with Newmark's average acceleration, Wilson-θ, and generalized-α methods. Moreover, this second order accurate method can control numerical damping in the higher modes. The numerical dispersion of the proposed method is compared with three unconditionally stable methods, namely, Newmark's average acceleration, Wilson-θ, and generalized-α methods. Furthermore, the overshooting effect of the proposed method is compared with these methods. By evaluating the computational time for analysis with similar time step duration, the proposed method is shown to be faster in comparison with the other methods.

Numerical simulation of higher-order diffusion-wave equations of variable coefficients using the meshless spectral method

2020 ◽  
pp. 2150010
Author(s):  
Manzoor Hussain ◽  
Sirajul Haq
Keyword(s):  
Wave Equations ◽  
Interpolation Method ◽  
Variable Coefficients ◽  
Higher Order ◽  
Implicit Time ◽  
Test Problems ◽  
Time Step ◽  
Diffusion Wave ◽  
Time Step Size ◽  
Time Stepping

In this paper, meshless spectral interpolation technique using implicit time stepping scheme is proposed for the numerical simulations of time-fractional higher-order diffusion wave equations (TFHODWEs) of variable coefficients. Meshless shape functions, obtained from radial basis functions (RBFs) and point interpolation method (PIM), are used for spatial approximation. Central differences coupled with quadrature rule of [Formula: see text] are employed for fractional temporal approximation. For advancement of solution, an implicit time stepping scheme is then invoked. Simulations performed for different benchmark test problems feature good agreement with exact solutions. Stability analysis of the proposed method is theoretically discussed and computationally validated to support the analysis. Accuracy and efficiency of the proposed method are assessed via [Formula: see text], [Formula: see text] and [Formula: see text] error norms as well as number of nodes [Formula: see text] and time step-size [Formula: see text].

Transient Two-Dimensional Problems Utilizing The Least Squares Algorithm

1975 ◽  
Vol 97 (3) ◽  
pp. 467-469 ◽  
Author(s):  
J. C. Bruch ◽  
R. W. Lewis
Keyword(s):  
Least Squares ◽  
Finite Difference Methods ◽  
Computer Time ◽  
Two Dimensional ◽  
Time Step ◽  
Time Stepping ◽  
Large Time Step ◽  
Least Squares Algorithm ◽  
Methods Analytical ◽  
And Storage

The least squares time-stepping algorithm, which has previously been shown by the authors to be competitive for one-dimensional problems, is applied to the solution of several two-dimensional examples having constant material properties. The results are compared against answers obtained using recurrence relationships based on the finite element and finite difference methods. Analytical results for one of the examples are also used for comparison. The least squares algorithm proved to be more accurate for equal values of time step especially in the large time step cases. It, however, requires more computer time and storage than the other methods used. Several other limitations of the scheme are also presented.

scholarly journals Computationally efficient methods for modelling laser wakefield acceleration in the blowout regime

2012 ◽  
Vol 78 (4) ◽  
pp. 469-482 ◽  
Author(s):  
B. M. COWAN ◽  
S. Y. KALMYKOV ◽  
A. BECK ◽  
X. DAVOINE ◽  
K. BUNKERS ◽  
...  
Keyword(s):  
Initial Conditions ◽  
Cylindrical Symmetry ◽  
Small Time ◽  
Numerical Dispersion ◽  
Computationally Efficient ◽  
Plasma Bubble ◽  
Time Step ◽  
Simulation Code ◽  
Particle In Cell ◽  
Comput Phys

AbstractElectron self-injection and acceleration until dephasing in the blowout regime is studied for a set of initial conditions typical of recent experiments with 100-terawatt-class lasers. Two different approaches to computationally efficient, fully explicit, 3D particle-in-cell modelling are examined. First, the Cartesian code vorpal (Nieter, C. and Cary, J. R. 2004 VORPAL: a versatile plasma simulation code. J. Comput. Phys.196, 538) using a perfect-dispersion electromagnetic solver precisely describes the laser pulse and bubble dynamics, taking advantage of coarser resolution in the propagation direction, with a proportionally larger time step. Using third-order splines for macroparticles helps suppress the sampling noise while keeping the usage of computational resources modest. The second way to reduce the simulation load is using reduced-geometry codes. In our case, the quasi-cylindrical code calder-circ (Lifschitz, A. F. et al. 2009 Particle-in-cell modelling of laser-plasma interaction using Fourier decomposition. J. Comput. Phys.228(5), 1803–1814) uses decomposition of fields and currents into a set of poloidal modes, while the macroparticles move in the Cartesian 3D space. Cylindrical symmetry of the interaction allows using just two modes, reducing the computational load to roughly that of a planar Cartesian simulation while preserving the 3D nature of the interaction. This significant economy of resources allows using fine resolution in the direction of propagation and a small time step, making numerical dispersion vanishingly small, together with a large number of particles per cell, enabling good particle statistics. Quantitative agreement of two simulations indicates that these are free of numerical artefacts. Both approaches thus retrieve the physically correct evolution of the plasma bubble, recovering the intrinsic connection of electron self-injection to the nonlinear optical evolution of the driver.

Recursive integral time extrapolation methods for scalar waves

2010 ◽  
Author(s):  
Paul J. Fowler ◽  
Xiang Du ◽  
Robin P. Fletcher
Keyword(s):  
Extrapolation Methods ◽  
Integral Time ◽  
Scalar Waves ◽  
Time Extrapolation
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