Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping

Basabe, Jonás D. De; Sen, Mrinal K.

doi:10.1111/j.1365-246x.2010.04536.x

Cited by 99 publications

(53 citation statements)

References 50 publications

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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: 75%

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: 75%

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

Mulder¹

2013

PIER

View full text Add to dashboard Cite

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“…The time variation of the source is the second derivative of a Gaussian with a dominant period of 50 s, centered on time t = 60 s. The mesh is composed of 256 Â 256 spectral elements at the surface and contains a total of 655,360 spectral elements and 43,666,752 independent grid points. We use a time step of 0.20 s to honor the CFL stability condition [70,22,77] and propagate the waves for a total of 10,000 time step, i.e., 2000 s. We record the time variation (due to the propagation of seismic waves across the mesh) of the three components of the displacement vector (a so-called 'seismogram') at latitude 37.42°, longitude 139.22°and a depth of 22.78 km. Fig.…”

Section: Comparison To a Reference Solution In C + Mpimentioning

confidence: 99%

“…Time integration of this system is usually performed based on a second-order centered finite-difference Newmark time scheme (e.g. [75,70]), although higher-order time schemes can be used if necessary [76,77].…”

Section: Serial Algorithmmentioning

confidence: 99%

High-order finite-element seismic wave propagation modeling with MPI on a large GPU cluster

Komatitsch

Erlebacher

Göddeke³

et al. 2010

Journal of Computational Physics

257

123

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“…Currently, however, the set of additional nodes needed for mass lumping for tetrahedral elements beyond polynomial degree three (cubic shape function) is unknown for 3D problems. Discontinuous Galerkin methods (DGMs) offer an attractive alternative for the spatial discretization of time-dependent hyperbolic problems (Arnold et al, 2002) and geophysical applications de la Puente et al, 2007;Etienne et al, 2010;De Basabe and Sen, 2010;Baldassari et al, 2011).…”

Section: Introductionmentioning

confidence: 99%

Local time stepping with the discontinuous Galerkin method for wave propagation in 3D heterogeneous media

et al. 2013

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Modeling and imaging techniques for geophysics are extremely demanding in terms of computational resources. Seismic data attempt to resolve smaller scales and deeper targets in increasingly more complex geologic settings. Finite elements enable accurate simulation of time-dependent wave propagation in heterogeneous media. They are more costly than finite-difference methods, but this is compensated by their superior accuracy if the finite-element mesh follows the sharp impedance contrasts and by their improved efficiency if the element size scales with wavelength, hence with the local wave velocity. However, 3D complex geologic settings often contain details on a very small scale compared to the dominant wavelength, requiring the mesh to contain elements that are smaller than dictated by the wavelength. Also, limitations of the mesh generation software may produce regions where the elements are much smaller than desired. In both cases, this leads to a reduction of the time step required to solve the wave propagation and significantly increases the computational cost. Local time stepping (LTS) can improve the computational efficiency and speed up the simulation. We evaluated a local formulation of an LTS scheme with second-order accuracy for the discontinuous Galerkin finite-element discretization of the wave equation. We tested the benefits of the scheme by considering a geologic model for a North-Sea-type example.

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Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping

Cited by 99 publications

References 50 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

High-order finite-element seismic wave propagation modeling with MPI on a large GPU cluster

Local time stepping with the discontinuous Galerkin method for wave propagation in 3D heterogeneous media

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