Multiscale methods for the wave equation

Engquist, Björn; Holst, Henrik; Runborg, Olof

doi:10.1002/pamm.200700930

Cited by 13 publications

(23 citation statements)

References 4 publications

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“…The functionF and F are defined in (8) and (5) respectively and we note that here F (x, p) =Āp. The integer q depends on the smoothness of the kernel used to compute the weighted average of f in (8).…”

Section: Convergence Theorymentioning

confidence: 99%

Multi-scale methods for wave propagation in heterogeneous media

Engquist

Holst²,

Runborg³

2011

Communications in Mathematical Sciences

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100

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Multi-scale wave propagation problems are computationally costly to solve by traditional techniques because the smallest scales must be represented over a domain determined by the largest scales of the problem. We have developed and analyzed new numerical methods for multi-scale wave propagation in the framework of heterogeneous multi-scale method. The numerical methods couples simulations on macro-and micro-scales for problems with rapidly oscillating coefficients. We show that the complexity of the new method is significantly lower than that of traditional techniques with a computational cost that is essentially independent of the micro-scale. A convergence proof is given and numerical results are presented for periodic problems in one, two and three dimensions. The method is also successfully applied to non-periodic problems and for long time integration where dispersive effects occur.

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Section: Convergence Theorymentioning

confidence: 99%

Multi-scale methods for wave propagation in heterogeneous media

Engquist

Holst²,

Runborg³

2011

Communications in Mathematical Sciences

Self Cite

100

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show abstract

“…Owhadi and Zhang [80,81,82] proposed the multiscale method for the wave equation based on the global change of coordinates. E and Engquist [35,36] proposed the heterogeneous multiscale method (HMM) that was later developed in finitedifference and finite-element formulations [41,42,1]. The HMM also requires evaluations of local problem in each time step, which is expensive.…”

Section: Introductionmentioning

confidence: 99%

Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media

Gao

Gibson

et al. 2015

Journal of Computational Physics

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It is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs. However, the computational cost of conventional numerical modeling methods, such as finite-difference method and finite-element method, becomes prohibitively expensive when applied to very large models. We propose a Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media, where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element. The application of multiscale basis functions can capture the fine scale medium property variations, and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite-element method for wave equation, while restricting the error to low values. We formulate the continuous Galerkin and discontinuous Galerkin formulation of the multiscale method, both of which have pros and cons. Applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system.

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“…This problem has also been addressed by the so-called multiscale method for wave equations (Vdovina et al, 2005;Korostyshevskaya and Minkoff, 2006;Engquist et al, 2007;Owhadi and Zhang, 2008;Minkoff, 2008, 2011;Abdulle and Grote, 2011;Chung et al, 2011bChung et al, , 2013Fu et al, 2013;Gao et al, 2013;Gibson et al, 2014). These various approaches to the multiscale problem can be quite different in their underlying principles, but they tend to reach one specific goal, that is, to solve the wave equations on a set of coarsely discretized mesh to approximate the solutions of the wave equations on the finely discretized mesh, and each coarse element may contain finer elements with highly heterogeneous medium properties in space.…”

Section: Introductionmentioning

confidence: 99%

A numerical homogenization method for heterogeneous, anisotropic elastic media based on multiscale theory

et al. 2015

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The development of reliable methods for upscaling finescale models of elastic media has long been an important topic for rock physics and applied seismology. Several effective medium theories have been developed to provide elastic parameters for materials such as finely layered media or randomly oriented or aligned fractures. In such cases, the analytic solutions for upscaled properties can be used for accurate prediction of wave propagation. However, such theories cannot be applied directly to homogenize elastic media with more complex, arbitrary spatial heterogeneity. Therefore, we have proposed a numerical homogenization algorithm based on multiscale finite-element methods for simulating elastic wave propagation in heterogeneous, anisotropic elastic media. Specifically, our method used multiscale basis functions obtained from a local linear elasticity problem with appropriately defined boundary conditions. Homogenized, effective medium parameters were then computed using these basis functions, and the approach applied a numerical discretization that was similar to the rotated staggered-grid finite-difference scheme. Comparisons of the results from our method and from conventional, analytical approaches for finely layered media showed that the homogenization reliably estimated elastic parameters for this simple geometry. Additional tests examined anisotropic models with arbitrary spatial heterogeneity in which the average size of the heterogeneities ranged from several centimeters to several meters, and the ratio between the dominant wavelength and the average size of the arbitrary heterogeneities ranged from 10 to 100. Comparisons to finite-difference simulations proved that the numerical homogenization was equally accurate for these complex cases.

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Multiscale methods for the wave equation

Cited by 13 publications

References 4 publications

Multi-scale methods for wave propagation in heterogeneous media

Multi-scale methods for wave propagation in heterogeneous media

Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media

A numerical homogenization method for heterogeneous, anisotropic elastic media based on multiscale theory

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