A modified numerical-flux-based discontinuous Galerkin method for 2D wave propagations in isotropic and anisotropic media

We have developed a new discontinuous Galerkin (DG) method to solve the 2D seismic wave equations in isotropic and anisotropic media. This method uses a modified numerical flux that is based on a linear combination of the Godunov and the centered fluxes. A weighting factor is introduced in this modified numerical flux that is expected to be optimized to some extent. Through the investigations on the considerations of numerical stability, numerical dispersion, and dissipation errors, we develop a possible choice of optimal weighting factor. Several numerical experiments confirm the effectiveness of the proposed method. We evaluate a convergence test based on cosine wave propagation without the source term, which shows that the numerical errors in the modified flux-based DG method and the Godunov-flux-based method are quite similar. However, the improved computational efficiency of the modified flux over the Godunov flux can be demonstrated only at a small sampling rate. Then, we apply the proposed method to simulate the wavefields in acoustic, elastic, and anisotropic media. The numerical results show that the modified DG method produces small numerical dispersion and obtains results in good agreement with the reference solutions. Numerical wavefield simulations of the Marmousi model show that the proposed method also is suitable for the heterogeneous case.

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A discontinuous Galerkin fast-sweeping eikonal solver for fast and accurate traveltime computation in 3D tilted anisotropic media

Geophysics ◽

10.1190/geo2018-0555.1 ◽

2019 ◽

Vol 84 (2) ◽

pp. C107-C118 ◽

Cited By ~ 5

Author(s):

Philippe Le Bouteiller ◽

Mondher Benjemaa ◽

Ludovic Métivier ◽

Jean Virieux

Keyword(s):

Discontinuous Galerkin ◽

Seismic Anisotropy ◽

Computational Cost ◽

Anisotropic Media ◽

Finite Difference Schemes ◽

Seismic Amplitude ◽

Fast Sweeping ◽

Complex Models ◽

Dg Method ◽

Sweeping Method

We tackle the challenging problem of efficient and accurate seismic traveltime computation in 3D anisotropic media by applying the fast-sweeping method to a discontinuous Galerkin (DG)-based eikonal solver. Using this method leads to a stable and highly accurate scheme, which is faster than finite-difference schemes for a given precision, and with a low computational cost compared to the standard Runge-Kutta DG formulation. The integral formulation of the DG method also makes it easy to handle seismic anisotropy and complex topographies. Several numerical tests on complex models, such as the 3D SEG advanced modeling model, are given as illustration, highlighting the efficiency and the accuracy of this new approach. In the near future, these results will be used together with accurate solvers for seismic amplitude and take-off angle computation to revisit asymptotic inversion (traveltime/slope tomography) and imaging approaches (quantitative migration involving amplitudes and angles).

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Numerical dispersion analysis of discontinuous Galerkin method with different basis functions for acoustic and elastic wave equations

Geophysics ◽

10.1190/geo2017-0485.1 ◽

2018 ◽

Vol 83 (3) ◽

pp. T87-T101 ◽

Cited By ~ 6

Author(s):

Weijuan Meng ◽

Li-Yun Fu

Keyword(s):

Galerkin Method ◽

Discontinuous Galerkin ◽

Discontinuous Galerkin Method ◽

Basis Function ◽

Wave Equations ◽

Basis Functions ◽

Numerical Dispersion ◽

Dispersion Property ◽

Lagrange Polynomial ◽

Equidistant Nodes

The discontinuous Galerkin method (DGM) has been applied to investigate seismic wave propagation recently. However, few studies have examined the dispersion property of DGM with different basis functions. Therefore, three common basis functions, Legendre polynomial, Lagrange polynomial with equidistant nodes, and Lagrange polynomial with Gauss-Lobatto-Legendre (GLL) nodes, are used for numerical approximation. The numerical dispersion and anisotropy numerical behavior of acoustic and elastic waves are compared, and the numerical errors of different order methods are analyzed. The result shows that the dispersion errors for all basis functions reduce generally with increasing interpolation orders, but with large differences in different directions. Specifically, the Legendre basis function and Lagrange basis function with GLL nodes have attractive advantages over the Lagrange polynomial with equidistant nodes for numerical computation. We verified the dispersion properties by theoretical and numerical analyses.

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An optimal nearly analytic discrete-weighted Runge-Kutta discontinuous Galerkin hybrid method for acoustic wavefield modeling

Geophysics ◽

10.1190/geo2015-0686.1 ◽

2016 ◽

Vol 81 (5) ◽

pp. T251-T263 ◽

Cited By ~ 10

Author(s):

Dinghui Yang ◽

Xijun He ◽

Xiao Ma ◽

Yanjie Zhou ◽

Jingshuang Li

Keyword(s):

Discontinuous Galerkin ◽

Hybrid Method ◽

Hybrid Algorithm ◽

Wave Equations ◽

Homogeneous Model ◽

Computational Cost ◽

Numerical Dispersion ◽

Computational Domain ◽

Geometrical Structures ◽

Runge Kutta

The newly developed optimal nearly analytic discrete (ONAD) and the weighted Runge-Kutta discontinuous Galerkin (WRKDG) methods can effectively suppress the numerical dispersion caused by discretizing wave equations, but it is difficult for ONAD to implement on flexible meshes, whereas the WRKDG has high computational cost for wavefield simulations. We have developed a new hybrid algorithm by combining the ONAD method with the WRKDG method. In this hybrid algorithm, the computational domain was split into several subdomains, in which the subdomain for the ONAD method used regular Cartesian grids, whereas the subdomain for the WRKDG method used triangular grids. The hybrid method was at least third-order spatially accurate. We have applied the proposed method to simulate the scalar wavefields for different models, including a homogeneous model, a rough topography model, a fracture model, and a cave model. The numerical results found that the hybrid method can deal with complicated geometrical structures, effectively suppress numerical dispersion, and provide accurate seismic wavefields. Numerical examples proved that our hybrid method can significantly reduce the CPU time and save storage requirement for the tested models. This implies that the hybrid method is especially suitable for the simulation of waves propagating in complex media.

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Dispersion-dissipation analysis of triangular numerical-flux-based discontinuous Galerkin method for elastic wave equations

Journal of Computational Physics ◽

10.1016/j.jcp.2020.109630 ◽

2020 ◽

Vol 418 ◽

pp. 109630

Author(s):

Xijun He ◽

Dinghui Yang ◽

Chujun Qiu

Keyword(s):

Elastic Wave ◽

Galerkin Method ◽

Discontinuous Galerkin ◽

Discontinuous Galerkin Method ◽

Wave Equations ◽

Numerical Flux ◽

Elastic Wave Equations

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Pure P- and S-Wave Equations in Anisotropic Media

10.3997/2214-4609.202010297 ◽

2020 ◽

Author(s):

A. Stovas ◽

T. Alkhalifah ◽

U. Bin Waheed

Keyword(s):

Wave Equations ◽

Anisotropic Media ◽

S Wave

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Consistent Weighted Average Flux of Well-Balanced TVD-RK Discontinuous Galerkin Method for Shallow Water Flows

Modelling and Simulation in Engineering ◽

10.1155/2015/591282 ◽

2015 ◽

Vol 2015 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Thida Pongsanguansin ◽

Montri Maleewong ◽

Khamron Mekchay

Keyword(s):

Shallow Water ◽

Discontinuous Galerkin ◽

Numerical Solutions ◽

Order Approximation ◽

Weighted Average ◽

Flat Bottom ◽

Flux Function ◽

Dg Method ◽

Average Flux ◽

Steady Solutions

A well-balanced scheme with total variation diminishing Runge-Kutta discontinuous Galerkin (TVD-RK DG) method for solving shallow water equations is presented. Generally, the flux function at cell interface in the TVD-RK DG scheme is approximated by using the Harten-Lax-van Leer (HLL) method. Here, we apply the weighted average flux (WAF) which is higher order approximation instead of using the HLL in the TVD-RK DG method. The consistency property is shown. The modified well-balanced technique for flux gradient and source terms under the WAF approximations is developed. The accuracy of numerical solutions is demonstrated by simulating dam-break flows with the flat bottom. The steady solutions with shock can be captured correctly without spurious oscillations near the shock front. This presents the other flux approximations in the TVD-RK DG method for shallow water simulations.

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Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations

Communications on Applied Mathematics and Computation ◽

10.1007/s42967-021-00149-y ◽

2021 ◽

Author(s):

Lu Zhang ◽

Daniel Appelö ◽

Thomas Hagstrom

Keyword(s):

Discontinuous Galerkin ◽

Wave Equations ◽

Second Order ◽

Difference Methods

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Rapid Determination of Sulfite by Flow Injection Analysis Based on Fuchsin′s Addition

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.295-298.950 ◽

2013 ◽

Vol 295-298 ◽

pp. 950-953 ◽

Cited By ~ 1

Author(s):

Dong Yuan ◽

Da You Fu ◽

Wen Yuan Tan

Keyword(s):

Flow Injection ◽

Addition Reaction ◽

Rapid Determination ◽

Sampling Rate ◽

Optimum Conditions ◽

Linear Calibration ◽

Relative Standard ◽

Replicate Measurements ◽

Good Agreement

A rapid spectrophotometric method for flow injection determination of sulfite in tan wastewater is described. The proposed method was based on the addition reaction of sulfite with fuchsin in Na2B4O7-NaOH medium. The optimum conditions allow a linear calibration range of 0.01-1.20 μg ml-1 SO32-. The detection limit is 0.0023μg ml-1 (S/N=3), and the relative standard deviation for night replicate measurements is 1.1% for 0.5μg ml-1 of sulfite. The sampling rate is 60 samples h-1. The procedure has been applied to the determination of sulfite in tan wastewater. The results were in good agreement with those obtained by pararosaniline method.

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