Lax‐Wendroff and Nyström methods for seismic modelling

Chen, Jing‐Bo

doi:10.1111/j.1365-2478.2009.00802.x

Cited by 55 publications

(30 citation statements)

References 26 publications

(36 reference statements)

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“…[20] has 1.38307 for M t = 8, whereas [21] has 2.317, the same value as found here. For M t = 10, they have 1.38243 and 2.783, respectively, both different from the results listed above.…”

Section: Time-stepping Stabilitysupporting

confidence: 74%

See 1 more Smart Citation

New Triangular Mass-Lumped Finite Elements of Degree Six for Wave Propagation

Mulder¹

2013

PIER

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Abstract-Mass-lumped continuous finite elements allow for explicit time stepping with the second-order wave equation if the resulting integration weights are positive and provide sufficient accuracy. To meet these requirements on triangular and tetrahedral meshes, the construction of continuous finite elements for a given polynomial degree on the edges involves polynomials of higher degree in the interior. The parameters describing the supporting nodes of the Lagrange interpolating polynomials and the integration weights are the unknowns of a polynomial system of equations, which is linear in the integration weights. To find candidate sets for the nodes, it is usually required that the number of equations equals the number of unknowns, although this may be neither necessary nor sufficient. Here, this condition is relaxed by requiring that the number of equations does not exceed the number of unknowns. This resulted in two new types elements of degree 6 for symmetrically placed nodes. Unfortunately, the first type is not unisolvent. There are many different elements of the second type with a large range in their associated time-stepping stability limit. To assess the efficiency of the elements of various degrees, numerical tests on a simple problem with an exact solution were performed. Efficiency was measured by the computational time required to obtain a solution at a given accuracy. For the chosen example, elements of degree 4 with fourth-order time stepping appear to be the most efficient.

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“…[20] has 1.38307 for M t = 8, whereas [21] has 2.317, the same value as found here. For M t = 10, they have 1.38243 and 2.783, respectively, both different from the results listed above.…”

Section: Time-stepping Stabilitysupporting

confidence: 74%

“…Here, η can be identified with any of the eigenvalues of Table 1 of [20], in the row for 1D, multiplied by 2/π because there, a pseudo-spectral method is considered. [20] has 1.38307 for M t = 8, whereas [21] has 2.317, the same value as found here.…”

Section: Time-stepping Stabilitymentioning

confidence: 99%

New Triangular Mass-Lumped Finite Elements of Degree Six for Wave Propagation

Mulder¹

2013

PIER

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“…Chen [2] and Gao [3] solved ray-tracing using symplectic geometry algorithms, of which the accuracy is comparable with the fourorder Runge-Kutta method, and superior in speed. In terms of seismic wave numerical modeling based on wave equations, Chen [4] combined the symplectic scheme with the pseudo-spectral method to solve scalar wave equations, which showed smaller error growth compared to the nonsymplectic scheme.…”

Section: Introductionmentioning

confidence: 99%

Modeling of the Seismic Scalar Wave Field Using the Symplectic Scheme and Singular Kernel Convolutional Differentiator

Zhu

2011

Chinese J of Geophysics

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This work adopts both the symplectic difference scheme and singular kernel convolutional differentiator for modeling of the seismic scalar wave field. In this method the computational accuracy and stability have been greatly improved with a slight increase of calculation amount. Compared with other non‐symplectic numerical methods, major advantages of the method presented in this paper are the structure‐preserving property and a strong capability of long‐time tracing. This approach provides a new and effective choice for high‐accuracy modeling of large‐scale and long‐time history seismic wave fields.

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“…We first focus on the stability condition for Ψ t based on the standard Fourier analysis [11,33]. Substituting a plane wave solution…”

Section: Stability Analysismentioning

confidence: 99%

Modelling damped acoustic waves by a dissipation-preserving conformal symplectic method

2017

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We propose a novel stable and efficient dissipationpreserving method for acoustic wave propagations in attenuating media with both correct phase and amplitude. Through introducing the conformal multisymplectic structure, the intrinsic dissipation law and the conformal symplectic conservation law are revealed for the damped acoustic wave equation. The proposed algorithm is exactly designed to preserve a discrete version of the conformal symplectic conservation law. More specifically, two subsystems in conjunction with the original damped wave equation are derived. One is actually the conservative Hamiltonian wave equation and the other is a dissipative linear ordinary differential equation (ODE) system. Standard symplectic method is devoted to the conservative system, whereas the analytical solution is obtained for the ODE system. An explicit conformal symplectic scheme is constructed by concatenating these two parts of solutions by the Strang splitting technique. Stability analysis and convergence tests are given thereafter. A benchmark model in homogeneous media is presented to demonstrate the effectiveness and advantage of our method in suppressing numerical dispersion and preserving the energy dissipation. Further numerical tests show that our proposed method can efficiently capture the dissipation in heterogeneous media.

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Lax‐Wendroff and Nyström methods for seismic modelling

Cited by 55 publications

References 26 publications

New Triangular Mass-Lumped Finite Elements of Degree Six for Wave Propagation

New Triangular Mass-Lumped Finite Elements of Degree Six for Wave Propagation

Modeling of the Seismic Scalar Wave Field Using the Symplectic Scheme and Singular Kernel Convolutional Differentiator

Modelling damped acoustic waves by a dissipation-preserving conformal symplectic method

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