Stabilized leapfrog based local time-stepping method for the wave equation

Grote, Marcus J.; Michel, Simone; Sauter, Stéfan A.

doi:10.1090/mcom/3650

Cited by 16 publications

(16 citation statements)

References 38 publications

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“…Thus, subtract (2. 19)-(2.20) from (2.8)-(2.9) and take ϕ h = Π h u − u h , ψ h = P h q − q h , we obtain by (2.13) that…”

Section: The Semi-discrete Unfitted Finite Element Methodsmentioning

confidence: 98%

“…We remark, however, since the minimum size of the mesh h min is smaller for a well resolved mesh, the computation is more expensive as the result of smaller time step due to the CFL condition. One possible remedy, which deserves further investigation, is the methods of local time stepping for which we refer to the recent works Carle et al [7], Grote et al [19] and the references therein.…”

Section: Numerical Examplesmentioning

confidence: 99%

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A high order explicit time finite element method for the acoustic wave equation with discontinuous coefficients

Chen¹,

Liu²,

Xiang³

2021

Preprint

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In this paper, we propose a novel high order explicit time discretization method for the acoustic wave equation with discontinuous coefficients. The space discretization is based on the unfitted finite element method in the discontinuous Galerkin framework which allows us to treat problems with complex interface geometry on Cartesian meshes. The strong stability and optimal hp-version error estimates both in time and space are established. Numerical examples confirm our theoretical results.

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“…Thus, subtract (2. 19)-(2.20) from (2.8)-(2.9) and take ϕ h = Π h u − u h , ψ h = P h q − q h , we obtain by (2.13) that…”

Section: The Semi-discrete Unfitted Finite Element Methodsmentioning

confidence: 98%

Section: Numerical Examplesmentioning

confidence: 99%

A high order explicit time finite element method for the acoustic wave equation with discontinuous coefficients

Chen¹,

Liu²,

Xiang³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Our main contribution is to rigorously analyze the whole class of methods, although we are mostly interested in methods based on Chebyshev polynomials. Such methods have also been analyzed in [7] using discrete semigroup techniques. By using generating functions as a main tool we are able to weaken their CFL condition and regularity assumption on the exact solution (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…By using generating functions as a main tool we are able to weaken their CFL condition and regularity assumption on the exact solution (1.1). Moreover, our analysis includes the case of positive semidefinite matrices L, to which the technique in [7] does not apply.…”

Section: Introductionmentioning

confidence: 99%

“…We thus propose a general class of multirate leapfrog-type methods which allows to use step sizes which are independent on the stiff part of the equation and also very efficient to implement. This class comprises local time-stepping schemes [5,7] but also locally implicit or locally trigonometric integrators. Our main contribution is a rigorous error and stability analysis with special emphasis on explicit multirate methods, which are based on stabilized leapfrog-Chebyshev polynomials introduced in [4].…”

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confidence: 99%

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Error Analysis of Multirate Leapfrog-Type Methods for Second-Order Semilinear Odes

Carle¹,

Hochbruck²

2022

SIAM J. Numer. Anal.

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In this paper we consider the numerical solution of second-order semilinear differential equations, for which the stiffness is induced by only a few components of the linear part. For such problems, the leapfrog scheme suffers from severe restrictions on the step size to ensure stability. We thus propose a general class of multirate leapfrog-type methods which allows to use step sizes which are independent on the stiff part of the equation and also very efficient to implement. This class comprises local time-stepping schemes [5,7] but also locally implicit or locally trigonometric integrators. Our main contribution is a rigorous error and stability analysis with special emphasis on explicit multirate methods, which are based on stabilized leapfrog-Chebyshev polynomials introduced in [4].

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An energy preserving time scheme based on the mortar element method for effective transmission conditions between fluid and solid domains in transient wave propagation problems

Impériale

2023

Numerical Meth Engineering

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We are interested in providing an efficient numerical scheme to represent wave propagation in time domain for configurations where a fluid domain and a solid domain are separated by a thin coating layer. Various computational challenges and bottlenecks occur in this context: incorporating fluid–solid coupling, enabling non‐conform space discretization when wavelengths greatly differ from the fluid and the solid domain, and rendering robust time discretization w.r.t. the thin layer thickness. In order to address these issues, the proposed approach combines the mortar element method with so‐called effective transmission conditions, where the effect of the coating layer is incorporated through spring—mass coefficients at the interface between the fluid and solid domain. Using discrete energy arguments, we are able to prove that the associated fully discrete scheme is stable upon a stability condition independent of the thin coating layer. Moreover, we provide an efficient time marching algorithm leading to significantly improved performances, illustrated in 2D and 3D configurations linked to ultrasonic testing experiments modeling.

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Stabilized leapfrog based local time-stepping method for the wave equation

Cited by 16 publications

References 38 publications

A high order explicit time finite element method for the acoustic wave equation with discontinuous coefficients

A high order explicit time finite element method for the acoustic wave equation with discontinuous coefficients

Error Analysis of Multirate Leapfrog-Type Methods for Second-Order Semilinear Odes

An energy preserving time scheme based on the mortar element method for effective transmission conditions between fluid and solid domains in transient wave propagation problems

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