A New Interface Method for Hyperbolic Problems with Discontinuous Coefficients: One-Dimensional Acoustic Example

Piraux, Joël; Lombard, Bruno

doi:10.1006/jcph.2001.6696

Cited by 44 publications

(63 citation statements)

References 12 publications

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“…If M = 0 and K → +∞, we get D 2k+2 = 0. Then conditions (3.6) and (3.9) recover conditions for perfect interfaces examined in [23].…”

Section: Results 3 Limit Values Ofmentioning

confidence: 99%

“…We recall the key points of the Explicit Simplified Interface Method (ESIM) [23,19]. Firstly, we define smooth extensions U * (x, t n ) of the exact solution U (x, t n ) on both sides of α at each time step t n ( Fig.…”

Section: The Interface Methodmentioning

confidence: 99%

“…where trunc(x) refers to the truncated value of x. Consequently to (4.29), LaxWendroff and finite volume with flux limiter schemes require k ≥ 2, and WENO 5 requires k ≥ 3. We do not demonstrate the Theorem 4.2, because the proof is very close to the one given in [23]. To see the key point, i.e.…”

Section: Local Truncation Analysismentioning

confidence: 99%

“…In [23], we analysed the local truncation error of the coupling (4.17) for the perfect conditions (3.1) and for k ≤ 3. To apply the same analysis for the spring-mass conditions (3.2) and for k ≤ 3, we need to take into account the new structure of block vectors and block matrices (4.7) and (4.13).…”

Section: Local Truncation Analysismentioning

confidence: 99%

“…To do so, we use the Explicit Simplified Interface Method (ESIM) [23] previously developed in the simple case of fluid/fluid perfect interfaces; the goal in [23] was only to maintain the order of accuracy with discontinuous coefficients. This method is an extension of the Immersed Interface Method (IIM) [17], [18], [30], [32], [33] developed by LeVeque and his collaborators, and of the Explicit Jump Immersed Interface Method (EJIIM) [29], [31] developed by Wiegmann and Bube.…”

mentioning

confidence: 99%

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How to Incorporate the Spring-Mass Conditions in Finite-Difference Schemes

Lombard¹,

Piraux²

2003

SIAM J. Sci. Comput.

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Abstract. The spring-mass conditions are an efficient way to model imperfect contacts between elastic media. These conditions link together the limit values of the elastic stress and of the elastic displacement on both sides of interfaces. To insert these spring-mass conditions in classical finitedifference schemes, we use an interface method, the Explicit Simplified Interface Method (ESIM). This insertion is automatic for a wide class of schemes. The interfaces do not need to coincide with the uniform cartesian grid. The local truncation error analysis and numerical experiments show that the ESIM maintains, with interfaces, properties of the schemes in homogeneous medium.

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“…If M = 0 and K → +∞, we get D 2k+2 = 0. Then conditions (3.6) and (3.9) recover conditions for perfect interfaces examined in [23].…”

Section: Results 3 Limit Values Ofmentioning

confidence: 99%

Section: The Interface Methodmentioning

confidence: 99%

Section: Local Truncation Analysismentioning

confidence: 99%

Section: Local Truncation Analysismentioning

confidence: 99%

mentioning

confidence: 99%

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How to Incorporate the Spring-Mass Conditions in Finite-Difference Schemes

Lombard¹,

Piraux²

2003

SIAM J. Sci. Comput.

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High Order Discontinuous Cut Finite Element Methods for Linear Hyperbolic Conservation Laws with an Interface

Frachon

Kreiss

et al. 2022

J Sci Comput

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We develop a family of cut finite element methods of different orders based on the discontinuous Galerkin framework, for hyperbolic conservation laws with stationary interfaces in both one and two space dimensions, and for moving interfaces in one space dimension. Interface conditions are imposed weakly and so that both conservation and stability are ensured. A CutFEM with discontinuous elements in space is developed and coupled to standard explicit time stepping schemes for linear advection problems and the acoustic wave problem with stationary interfaces. In the case of moving interfaces, we propose a space-time CutFEM based on discontinuous elements both in space and time for linear advection problems. We show that the proposed CutFEM are conservative and energy stable. For the stationary interface case an a priori error estimate is proven. Numerical computations in both one and two space dimensions support the analysis, and in addition demonstrate that the proposed methods have the expected accuracy.

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Numerical modeling of mechanical wave propagation

Seriani

Oliveira

2020

Riv. Nuovo Cim.

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The numerical modeling of mechanical waves is currently a fundamental tool for the study and investigation of their propagation in media with heterogeneous physical properties and/or complex geometry, as, in these cases, analytical methods are usually not applicable. These techniques are used in geophysics (geophysical interpretation, subsoil imaging, development of new methods of exploration), seismology (study of earthquakes, regional and global seismology, accurate calculation of synthetic seismograms), in the development of new methods for ultrasonic diagnostics in materials science (non-destructive methods) and medicine (acoustic tomography). In this paper we present a review of numerical methods that have been developed and are currently used. In particular we review the key concepts and pioneering ideas behind finite-difference methods, pseudospectral methods, finite-volume methods, Galerkin continuous and discontinuous finite-element methods (classical or based on spectral interpolation), and still others such as physics-compatible, and multiscale methods. We focus on their formulations in time domain along with the main temporal discretization schemes. We present the theory and implementation for some of these methods. Moreover, their computational characteristics are evaluated in order to aid the choice of the method for each practical situation.

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A New Interface Method for Hyperbolic Problems with Discontinuous Coefficients: One-Dimensional Acoustic Example

Cited by 44 publications

References 12 publications

How to Incorporate the Spring-Mass Conditions in Finite-Difference Schemes

How to Incorporate the Spring-Mass Conditions in Finite-Difference Schemes

High Order Discontinuous Cut Finite Element Methods for Linear Hyperbolic Conservation Laws with an Interface

Numerical modeling of mechanical wave propagation

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