A Conservative Space-time Mesh Refinement Method for the 1-D Wave Equation. Part II: Analysis

Collino, Francis; Fouquet, Thierry; Joly, Patrick

doi:10.1007/s00211-002-0447-4

Cited by 50 publications

(50 citation statements)

References 6 publications

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“…Our construction of a stable time discretization of the continuous equations will be based on this conservation property (as for instance in the work of [13], [15], [3]). …”

Section: Energy and A Priori Estimatesmentioning

confidence: 99%

Stability and dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string

Chabassier¹,

Imperiale²

2013

Wave Motion

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We study the implicit time discretization of piano strings governing equations within the Timoshenko prestressed beam model. Such model features two different waves, namely the flexural and shear waves, that propagate with very different velocities. We present a novel implicit time discretization that reduces the numerical dispersion while allowing the use of a large time step in the numerical computations. After analyzing the continuous system and the two branches of eigenfrequencies associated with the propagating modes, the classical θ-scheme is studied. We present complete new proofs of stability using energy-based approaches that provide uniform results with respect to the featured time step. A dispersion analysis confirms that θ = 1/12 reduces the numerical dispersion, but yields a severely constrained stability condition for the application considered. Therefore we propose a new θ-like scheme, which allows to reduce the numerical dispersion while relaxing this stability condition. Stability proofs are also provided for this new scheme. Theoretical results are illustrated with numerical experiments corresponding to the simulation of a realistic piano string.

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“…Our construction of a stable time discretization of the continuous equations will be based on this conservation property (as for instance in the work of [13], [15], [3]). …”

Section: Energy and A Priori Estimatesmentioning

confidence: 99%

Stability and dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string

Chabassier¹,

Imperiale²

2013

Wave Motion

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“…Highly e cient in practice, centered time discretizations also display remarkably high accuracy over long times and remain even nowadays probably the most popular methods for the time integration of wave equations. In [13] Collino, Fouquet and Joly proposed an LTS method for the wave equation in first-order form, which conserves a discrete energy yet requires every time-step the solution of a linear system on the interface between the coarse and the fine mesh. It was analyzed in [14,15] and later extended to elastodynamics [16] and Maxwell's equations [17].…”

Section: Introductionmentioning

confidence: 99%

“…In [13] Collino, Fouquet and Joly proposed an LTS method for the wave equation in first-order form, which conserves a discrete energy yet requires every time-step the solution of a linear system on the interface between the coarse and the fine mesh. It was analyzed in [14,15] and later extended to elastodynamics [16] and Maxwell's equations [17]. By combining a symplectic integrator with a DG discretization of Maxwell's equations in first-order form, Piperno [18] proposed a second-order explicit local time-stepping scheme, which also conserves a discrete energy.…”

Section: Introductionmentioning

confidence: 99%

Multi-level explicit local time-stepping methods for second-order wave equations

Diaz

Grote

2015

Computer Methods in Applied Mechanics and Engineering

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To cite this version:Julien Diaz, Marcus Grote. Multi-level explicit local time-stepping methods for second-order wave equations. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2015, 291, pp.240-265. 10 Abstract Local mesh refinement severly impedes the e ciency of explicit time-stepping methods for numerical wave propagation. Local time-stepping (LTS) methods overcome the bottleneck due to a few small elements by allowing smaller time-steps precisely where those elements are located.Yet when the region of local mesh refinement itself contains a sub-region of even smaller elements, any local time-step again will be overly restricted. To remedy the repeated bottleneck

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“…To overcome the overly restrictive stability constraint, various LTS schemes (Collino et al, 2003;Piperno, 2006) have been developed, which use either implicit time-stepping or explicit but smaller time steps, but only where the smallest elements in the mesh are located. Because DGMs allow for a local formulation, they are particularly well suited for the development of explicit LTS schemes.…”

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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A Conservative Space-time Mesh Refinement Method for the 1-D Wave Equation. Part II: Analysis

Cited by 50 publications

References 6 publications

Stability and dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string

Stability and dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string

Multi-level explicit local time-stepping methods for second-order wave equations

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

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