scholarly journals Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization

2016 ◽  
Vol 26 (14) ◽  
pp. 2651-2684 ◽  
Author(s):  
Assyr Abdulle ◽  
Timothée Pouchon
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Heterogeneous Media ◽  
Multiscale Model ◽  
Bloch Wave ◽  
Time Interval ◽  
Numerical Homogenization ◽  
Effective Equation ◽  
Long Time ◽  
Effective Models

A family of effective equations that capture the long time dispersive effects of wave propagation in heterogeneous media in an arbitrary large periodic spatial domain [Formula: see text] is proposed and analyzed. For a wave equation with highly oscillatory periodic spatial tensors of characteristic length [Formula: see text], we prove that the solution of any member of our family of effective equations is [Formula: see text]-close to the true oscillatory wave over a time interval of length [Formula: see text] in a norm equivalent to the [Formula: see text] norm. We show that the previously derived effective equation in [T. Dohnal, A. Lamacz and B. Schweizer, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Model. Simulat. 12 (2014) 488–513] belongs to our family of effective equations. Moreover, while Bloch wave techniques were previously used, we show that asymptotic expansion techniques give an alternative way to derive such effective equations. An algorithm to compute the tensors involved in the dispersive equation and allowing for efficient numerical homogenization methods over long time is proposed.

scholarly journals Time-dependent attractor of wave equations with nonlinear damping and linear memory

2019 ◽  
Vol 17 (1) ◽  
pp. 89-103
Author(s):  
Qiaozhen Ma ◽  
Jing Wang ◽  
Tingting Liu
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Nonlinear Damping ◽  
Time Dependent ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time

Abstract In this article, we consider the long-time behavior of solutions for the wave equation with nonlinear damping and linear memory. Within the theory of process on time-dependent spaces, we verify the process is asymptotically compact by using the contractive functions method, and then obtain the existence of the time-dependent attractor in $\begin{array}{} H^{1}_0({\it\Omega})\times L^{2}({\it\Omega})\times L^{2}_{\mu}(\mathbb{R}^{+};H^{1}_0({\it\Omega})) \end{array}$.

Fractional Laplacians viscoacoustic wavefield modeling with k-space-based time-stepping error compensating scheme

Geophysics ◽  
2020 ◽  
Vol 85 (1) ◽  
pp. T1-T13 ◽  
Author(s):  
Ning Wang ◽  
Tieyuan Zhu ◽  
Hui Zhou ◽  
Hanming Chen ◽  
Xuebin Zhao ◽  
...  
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Time Derivative ◽  
Time Stepping ◽  
Long Time ◽  
Sampling Intervals ◽  
Spatial Derivatives ◽  
The Stability ◽  
Wavefield Extrapolation ◽  
Space Approach

The spatial derivatives in decoupled fractional Laplacian (DFL) viscoacoustic and viscoelastic wave equations are the mixed-domain Laplacian operators. Using the approximation of the mixed-domain operators, the spatial derivatives can be calculated by using the Fourier pseudospectral (PS) method with barely spatial numerical dispersions, whereas the time derivative is often computed with the finite-difference (FD) method in second-order accuracy (referred to as the FD-PS scheme). The time-stepping errors caused by the FD discretization inevitably introduce the accumulative temporal dispersion during the wavefield extrapolation, especially for a long-time simulation. To eliminate the time-stepping errors, here, we adopted the [Formula: see text]-space concept in the numerical discretization of the DFL viscoacoustic wave equation. Different from existing [Formula: see text]-space methods, our [Formula: see text]-space method for DFL viscoacoustic wave equation contains two correction terms, which were designed to compensate for the time-stepping errors in the dispersion-dominated operator and loss-dominated operator, respectively. Using theoretical analyses and numerical experiments, we determine that our [Formula: see text]-space approach is superior to the traditional FD-PS scheme mainly in three aspects. First, our approach can effectively compensate for the time-stepping errors. Second, the stability condition is more relaxed, which makes the selection of sampling intervals more flexible. Finally, the [Formula: see text]-space approach allows us to conduct high-accuracy wavefield extrapolation with larger time steps. These features make our scheme suitable for seismic modeling and imaging problems.

scholarly journals ASYMPTOTIC EXPANSION FOR DAMPED WAVE EQUATIONS WITH PERIODIC COEFFICIENTS

2001 ◽  
Vol 11 (07) ◽  
pp. 1285-1310 ◽  
Author(s):  
R. ORIVE ◽  
E. ZUAZUA ◽  
A. F. PAZOTO
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Heat Kernel ◽  
Wave Equations ◽  
Bloch Wave ◽  
Periodic Coefficients ◽  
First Approximation ◽  
Wave Decomposition ◽  
Dissipative Wave Equation

We consider a linear dissipative wave equation in ℝN with periodic coefficients. By means of Bloch wave decomposition, we obtain an expansion of solutions as t→∞ and conclude that, in a first approximation, the solutions behave as the homogenized heat kernel.

scholarly journals Reconstruction of Quasi-Local Numerical Effective Models from Low-Resolution Measurements

2020 ◽  
Vol 85 (1) ◽  
Author(s):  
A. Caiazzo ◽  
R. Maier ◽  
D. Peterseim
Keyword(s):  
Numerical Experiments ◽  
Heterogeneous Media ◽  
Matrix Representation ◽  
Numerical Homogenization ◽  
Scale Separation ◽  
The Matrix ◽  
Inversion Procedure ◽  
The One ◽  
Effective Models ◽  
Non Locality

Abstract We consider the inverse problem of reconstructing an effective model for a prototypical diffusion process in strongly heterogeneous media based on coarse measurements. The approach is motivated by quasi-local numerical effective forward models that are provably reliable beyond periodicity assumptions and scale separation. The goal of this work is to show that an identification of the matrix representation related to these effective models is possible. On the one hand, this provides a reasonable surrogate in cases where a direct reconstruction is unfeasible due to a mismatch between the coarse data scale and the microscopic quantities to be reconstructed. On the other hand, the approach allows us to investigate the requirement for a certain non-locality in the context of numerical homogenization. Algorithmic aspects of the inversion procedure and its performance are illustrated in a series of numerical experiments.

scholarly journals DISPERSIVE EFFECTIVE MODELS FOR WAVES IN HETEROGENEOUS MEDIA

2011 ◽  
Vol 21 (09) ◽  
pp. 1871-1899 ◽  
Author(s):  
AGNES LAMACZ
Keyword(s):  
Heterogeneous Media ◽  
Variable Coefficients ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Smooth Functions ◽  
Long Time ◽  
Constant Coefficients ◽  
The One ◽  
Effective Models ◽  
Dispersive Models

We study the long-time behavior of waves in a strongly heterogeneous medium, starting from the one-dimensional scalar wave equation with variable coefficients. We assume that the coefficients are periodic with period ε and ε > 0 is a small length parameter. Our main result concerns homogenization and consists in the rigorous derivation of two different dispersive models. The first is a fourth-order equation with constant coefficients including powers of ε. In the second model, the ε-dependence is completely avoided by considering a third-order linearized Korteweg–de Vries equation. Our result is that both simplified models describe the long-time behavior well. An essential tool in our analysis is an adaption operator which modifies smooth functions according to the periodic structure of the medium.

True-amplitude, angle-domain, common-image gathers from one-way wave-equation migrations

Geophysics ◽  
2007 ◽  
Vol 72 (1) ◽  
pp. S49-S58 ◽  
Author(s):  
Yu Zhang ◽  
Sheng Xu ◽  
Norman Bleistein ◽  
Guanquan Zhang
Keyword(s):  
Wave Equation ◽  
Seismic Data ◽  
Wave Equations ◽  
Heterogeneous Media ◽  
Image Point ◽  
Amplitude Wave ◽  
Amplitude Variation ◽  
Imaging Condition ◽  
New Methods ◽  
Amplitude Variation With Angle

True-amplitude wave-equation migration provides a quality migrated image of the earth’s interior. In addition, the amplitude of the output provides an estimate of the angular-dependent reflection coefficient, similar to the output of Kirchhoff inversion. Recently, true-amplitude wave-equation migration for common-shot data has been proposed to generate amplitude-reliable, shot-domain, common-image gathers in heterogeneous media. We present a method to directly produce angle-domain common-image gathers from both common-shot and shot-receiver wave-equation migration. Generating true-amplitude, shot-domain, common-image gathers requires a deconvolution-type imaging condition using the ratio of the upgoing and downgoing wavefield, each downward-projected to the image point. Producing true-amplitude, angle-domain, common-image gathers requires, instead, the product of the upgoing wavefield and the complexconjugate of the downgoing wavefield in the imaging condition. Since multiplication is a more stable computational process than division, the new methods proposed provide more stable ways of inverting seismic data. Furthermore, the resulting common-image gathers can be directly used for migrated amplitude-variation-with angle analysis and tomography-based velocity analysis. Shot-receiver wave-equation migration requires new true-amplitude, one-way wave equations with one depth variable and transverse variables for the coordinates corresponding to sources and receivers, hence, two transverse coordinates in 2D and four transverse coordinates in 3D. We propose a modified double-square-root one-way wave equation to produce true amplitude common-image angle gathers. We also demonstrate the new methods with some synthetic examples. Some numerical examples show that the new methods we propose give better amplitude performance on the migrated angle gathers.

DISSIPATIVE QUASI-PARTICLES: THE GENERALIZED WAVE EQUATION APPROACH

2002 ◽  
Vol 12 (11) ◽  
pp. 2435-2444 ◽  
Author(s):  
C. I. CHRISTOV
Keyword(s):  
Wave Equation ◽  
Difference Scheme ◽  
Energy Production ◽  
Wave Equations ◽  
Equation Approach ◽  
Quasi Particle ◽  
Localized Solutions ◽  
Long Time ◽  
Localized Waves ◽  
Generalized Wave Equation

Generalized Wave Equations containing dispersion, dissipation and energy-production (GDWE) are considered in lieu of dissipative NEE as more suitable models for two-way interaction of localized waves. The quasi-particle behavior and the long-time evolution of localized solutions upon take-over and head-on collisions are investigated numerically by means of an adequate difference scheme which represents faithfully the balance/conservation laws. It is shown that in most cases the balance between energy production/dissipation and nonlinearity plays a similar role to the classical Boussinesq balance between dispersion and nonlinearity, namely it can create and support localized solutions which behave as quasi-particles upon collisions and for a reasonably long time after that.

scholarly journals A symplectic method for structure-preserving modelling of damped acoustic waves

2015 ◽  
Vol 471 (2183) ◽  
pp. 20150105 ◽  
Author(s):  
Xiaofan Li ◽  
Mingwen Lu ◽  
Shaolin Liu ◽  
Shizhong Chen ◽  
Huan Zhang ◽  
...  
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Numerical Simulations ◽  
Seismic Wave ◽  
Wave Equations ◽  
Superior Performance ◽  
Symplectic Method ◽  
Structure Preserving ◽  
Long Time

In this paper, a symplectic method for structure-preserving modelling of the damped acoustic wave equation is introduced. The equation is traditionally solved using non-symplectic schemes. However, these schemes corrupt some intrinsic properties of the equation such as the conservation of both precision and the damping property in long-term calculations. In the method presented, an explicit second-order symplectic scheme is used for the time discretization, whereas physical space is discretized by the discrete singular convolution differentiator. The performance of the proposed scheme has been tested and verified using numerical simulations of the attenuating scalar seismic-wave equation. Scalar seismic wave-field modelling experiments on a heterogeneous medium with both damping and high-parameter contrasts demonstrate the superior performance of the approach presented for suppression of numerical dispersion. Long-term computational experiments display the remarkable capability of the approach presented for long-time simulations of damped acoustic wave equations. Promising numerical results suggest that the approach is suitable for high-precision and long-time numerical simulations of wave equations with damping terms, as it has a structure-preserving property for the damping term.

scholarly journals Effective Models and Numerical Homogenization for Wave Propagation in Heterogeneous Media on Arbitrary Timescales

2020 ◽  
Vol 20 (6) ◽  
pp. 1505-1547
Author(s):  
Assyr Abdulle ◽  
Timothée Pouchon
Keyword(s):  
Wave Propagation ◽  
Wave Equations ◽  
Heterogeneous Media ◽  
Arbitrary Order ◽  
Ballistic Transport ◽  
Computational Cost ◽  
Numerical Procedure ◽  
Link Type ◽  
The Family ◽  
Effective Models

AbstractA family of effective equations for wave propagation in periodic media for arbitrary timescales $$\mathcal {O}(\varepsilon ^{-\alpha })$$ O ( ε - α ) , where $$\varepsilon \ll 1$$ ε ≪ 1 is the period of the tensor describing the medium, is proposed. The well-posedness of the effective equations of the family is ensured without requiring a regularization process as in previous models (Benoit and Gloria in Long-time homogenization and asymptotic ballistic transport of classical waves, 2017, arXiv:1701.08600; Allaire et al. in Crime pays; homogenized wave equations for long times, 2018, arXiv:1803.09455). The effective solutions in the family are proved to be $$\varepsilon $$ ε close to the original wave in a norm equivalent to the $${\mathrm {L}^{\infty }}(0,\varepsilon ^{-\alpha }T;{{\mathrm {L}^{2}}(\varOmega )})$$ L ∞ ( 0 , ε - α T ; L 2 ( Ω ) ) norm. In addition, a numerical procedure for the computation of the effective tensors of arbitrary order is provided. In particular, we present a new relation between the correctors of arbitrary order, which allows to substantially reduce the computational cost of the effective tensors of arbitrary order. This relation is not limited to the effective equations presented in this paper and can be used to compute the effective tensors of alternative effective models.

True amplitude corrections for a narrow-angle one-way elastic wave equation

Geophysics ◽  
2007 ◽  
Vol 72 (2) ◽  
pp. T19-T26 ◽  
Author(s):  
D. A. Angus
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Energy Flux ◽  
Wave Equations ◽  
Heterogeneous Media ◽  
Leading Order ◽  
Computationally Efficient ◽  
Elastic Wave Equation ◽  
Narrow Angle ◽  
Correction Terms

Wavefield extrapolators using one-way wave equations are computationally efficient methods for accurate traveltime modeling in laterally heterogeneous media, and have been used extensively in many seismic forward modeling and migration problems. However, most leading-order, one-way wave equations do not simulate waveform amplitudes accurately and this is primarily because energy flux is not accounted for correctly. I review the derivation of a leading-order, narrow-angle, one-way elastic wave equation for 3D media. I derive correction terms that enable energy-flux normalization and introduce a new higher-order, narrow-angle, one-way elastic wave extrapolator. By implementing these correction terms, the new true amplitude wave extrapolator allows accurate amplitude estimates in the presence of strong gradients. I present numerical examples for 1D velocity transition models to show that (1) the leading-order, narrow-angle propagator accurately models traveltimes, but overestimates transmitted- or primary-wave amplitudes and (2) the new amplitude corrected narrow-angle propagator accurately models both the traveltimes and amplitudes of all forward-traveling waves.

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