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

Diaz, Julien; Grote, Marcus J.

doi:10.1016/j.cma.2015.03.027

Cited by 29 publications

(20 citation statements)

References 41 publications

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“…with κ as in (20). The CFL condition (19), together with the continuity and the coercivity of a and p ≥ 2, implies κ ∈ 0, 4p 2 . Thus, Lemma 19 (Appendix) implies…”

Section: Remark 13mentioning

confidence: 99%

“…The CFL condition (19) implies (∆t) 2 λ max p < 4 so that the eigenvalues are different and S p is diagonalizable. From [45, Satz (6.9.2)(2)] we conclude that there is a norm |||·||| in R 2 such that the associated matrix norm |||S p ||| is bounded from above by the spectral radius:…”

Section: Stabilitymentioning

confidence: 99%

“…Theorem 17 can be combined with known error estimates for the semidiscrete error e (n+1) S to obtain an error estimate of the total error. Theorem 18 Let the bilinear form a (·, ·) satisfy (1) and let the CFL condition (19) hold. Assume that the exact solution satisfies u ∈ W 1,∞ [0, T ] ; H m+1 (Ω) ∩ W 5,∞ [0, T ] ; L 2 (Ω) .…”

Section: This Yieldsmentioning

confidence: 99%

See 2 more Smart Citations

Convergence Analysis of Energy Conserving Explicit Local Time-Stepping Methods for the Wave Equation

Grote¹,

Mehlin²,

Sauter

2018

SIAM J. Numer. Anal.

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Local adaptivity and mesh refinement are key to the efficient simulation of wave phenomena in heterogeneous media or complex geometry. Locally refined meshes, however, dictate a small time-step everywhere with a crippling effect on any explicit time-marching method. In [18] a leap-frog (LF) based explicit local time-stepping (LTS) method was proposed, which overcomes the severe bottleneck due to a few small elements by taking small time-steps in the locally refined region and larger steps elsewhere. Here a rigorous convergence proof is presented for the fullydiscrete LTS-LF method when combined with a standard conforming finite element method (FEM) in space. Numerical results further illustrate the usefulness of the LTS-LF Galerkin FEM in the presence of corner singularities.

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“…with κ as in (20). The CFL condition (19), together with the continuity and the coercivity of a and p ≥ 2, implies κ ∈ 0, 4p 2 . Thus, Lemma 19 (Appendix) implies…”

Section: Remark 13mentioning

confidence: 99%

Section: Stabilitymentioning

confidence: 99%

See 1 more Smart Citation

Convergence Analysis of Energy Conserving Explicit Local Time-Stepping Methods for the Wave Equation

Grote¹,

Mehlin²,

Sauter

2018

SIAM J. Numer. Anal.

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“…To simplify the development of an LTS variant of the Newmark time-stepping scheme for a SEM, we embrace the framework developed by Diaz and Grote [7]. They were able to prove and demonstrate optimal convergence and stability properties for second and fourth order leapfrog methods, with recent extension to multiple refinement levels [8]. We derive an LTS variant of the Newmark time-stepping scheme with additional considerations for the SEM, absorbing boundary conditions, and multiple refinement levels.…”

Section: Introductionmentioning

confidence: 99%

Newmark local time stepping on high-performance computing architectures

Rietmann

Grote

Peter

et al. 2017

Journal of Computational Physics

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Please cite this article in press as: M. Rietmann et al., Newmark local time stepping on high-performance computing architectures, J. Comput. Phys. (2016), http://dx. AbstractIn multi-scale complex media, finite element meshes often require areas of local refinement, creating small elements that can dramatically reduce the global time-step for wave-propagation problems due to the CFL condition. Local time stepping (LTS) algorithms allow an explicit time-stepping scheme to adapt the time-step to the element size, allowing near-optimal time-steps everywhere in the mesh. We develop an efficient multilevel LTS-Newmark scheme and implement it in a widely used continuous finite element seismic wave-propagation package. In particular, we extend the standard LTS formulation with adaptations to continuous finite element methods that can be implemented very efficiently with very strong element-size contrasts (more than 100x). Capable of running on large CPU and GPU clusters, we present both synthetic validation examples and large scale, realistic application examples to demonstrate the performance and applicability of the method and implementation on thousands of CPU cores and hundreds of GPUs.

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“…Here, we summarize methods of direct time integration suitable for usage in dynamic FE computations [13]: explicit methods [2,3,5,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], implicit methods [32][33][34][35][36][37][38], implicit-explicit methods [39][40][41], multi-time step and time sub-cycling methods [31,42,43], heterogeneous and asynchronous time integrators [44][45][46], variational time integrators [47,48], various local stepping approaches in time [49][50][51][52][53], time schemes for higher-order FEM and isogeometric analysis [2,54], and methods based on binary partitioning using a variable time step size…”

Section: Introductionmentioning

confidence: 99%

Efficient implementation of an explicit partitioned shear and longitudinal wave propagation algorithm

Kolman

Cho

Park

2015

Int. J. Numer. Meth. Engng

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SUMMARYThe paper complements and extends the previous works on partitioned explicit wave propagation analysis methods, which were presented for discontinuous wave propagation problems in solids. An efficient implementation of the partitioned explicit wave propagation analysis methods is introduced. The present implementation achieves about 25% overall computational effort compared with the previous implementation with the same accuracy. The present algorithm tracks, with different integration time step sizes in accordance with their different wave speeds, the propagation fronts of longitudinal and shear waves. This is accomplished by integrating separately the element-by-element partitioned longitud inal and shear equations of motion. The state vectors (displacements, velocity and accelerations) of the longitudinal and shear components are reconciled at the end of each time step. The reconciliation procedure does not require any system parameters such as material properties, density, unlike conventional artificial viscosity methods. Numerical examples are presented as applied to linear and non-linear wave propagation problems, which demonstrate high-fidelity wavefront tracking ability of the present method, and compared with existing conventional wave propagation analysis methods.

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

Cited by 29 publications

References 41 publications

Convergence Analysis of Energy Conserving Explicit Local Time-Stepping Methods for the Wave Equation

Convergence Analysis of Energy Conserving Explicit Local Time-Stepping Methods for the Wave Equation

Newmark local time stepping on high-performance computing architectures

Efficient implementation of an explicit partitioned shear and longitudinal wave propagation algorithm

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