The immersed interface method for acoustic wave equations with discontinuous coefficients

Zhang, Chaoming; LeVeque, Randall J.

doi:10.1016/s0165-2125(97)00046-2

Cited by 105 publications

(79 citation statements)

References 20 publications

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“…These methods are also termed embedded interface methods [46]. To overcome the staircasing problems and to impose the physically correct jump conditions, which are often encountered in the modeling of complex geometries, appropriate local modifications of the differentiation scheme close to boundaries and interfaces in a preprocessing stage are essential.…”

Section: Introductionmentioning

confidence: 99%

High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces

Zhao

Wang

2004

Journal of Computational Physics

196

164

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This paper introduces a series of novel hierarchical implicit derivative matching methods to restore the accuracy of high-order finite difference time-domain (FDTD) schemes of computational electromagnetics (CEM) with material interfaces in one (1D) and two spatial dimensions (2D). By making use of fictitious points, systematic approaches are proposed to locally enforce the physical jump conditions at material interfaces in a preprocessing stage, to arbitrarily high orders of accuracy in principle. While often limited by numerical instability, orders up to 16 in 1D and 2D are achieved. Detailed stability analyses are presented for the present approach to examine the up limit in constructing embedded FDTD methods. As natural generalizations of the high-order FDTD schemes, the proposed derivative matching methods automatically reduce to the standard FDTD schemes when the material interfaces are absent. An interesting feature of the present approach is that it encompasses a variety of schemes of different orders in a single code. Another feature of the present approach is that it can be robustly implemented with other high accuracy time-domain approaches, such as the multiresolution time-domain (MRTD) method and the local spectral time-domain (LSTD) method, to cope 1 with material interfaces. Numerical experiments on both 1D and 2D problems are carried out to test the convergence, examine the stability, access the efficiency, and explore the limitation of the proposed methods. It is found that operating at their best capacity, the proposed high order schemes are at least a million times more efficient than their fourth order versions in both 1D and 2D. Therefore, it is believed that the proposed hierarchical derivative matching methods are highly accurate, efficient, and robust for CEM.

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Section: Introductionmentioning

confidence: 99%

High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces

Zhao

Wang

2004

Journal of Computational Physics

196

164

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“…See [34] for more detailed discussions. A natural and successful approach for computing hyperbolic equations with singular coefficients is to build the interface condition into the numerical scheme.…”

Section: 4)mentioning

confidence: 99%

“…Many efficient numerical methods have been designed using this technique. For example, we mention the immersed interface methods by LeVeque and Li [20,34].…”

Section: 4)mentioning

confidence: 99%

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Convergence of an Immersed Interface Upwind Scheme for Linear Advection Equations with Piecewise Constant Coefficients II: Some Related Binomial Coefficients Inequalities

Wen¹

2009

JCM

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We study the L 1 -error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into the upwind scheme. We prove that, for initial data with a bounded variation, the numerical solution of the immersed interface upwind scheme converges in L 1 -norm to the differential equation with the corresponding interface condition. We derive the one-halfth order L 1 -error bounds with explicit coefficients following a technique used in [25]. We also use some inequalities on binomial coefficients proved in a consecutive paper [32].Mathematics subject classification: 65M06, 65M12, 65M25, 35F10.

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“…This type of methods is called "immersed interface method" [26]. It was originally developed with second-order accuracy [26,56] and then further extended to the fourth-order accuracy for both acoustic and elastic wave equations [54]. The immersed interface method requires knowledge about the jumps in the model parameters and their one-sided derivatives [42].…”

Section: Introductionmentioning

confidence: 99%

Accurate seabed modeling using finite difference methods

et al. 2017

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Finite difference is the most widely used method for seismic wavefield modeling. However, most finitedifference implementations discretize the Earth model over a fixed grid interval. This can lead to irregular model geometries being represented by 'staircase' discretization, and potentially causes mispositioning of interfaces within the media. This misrepresentation is a major disadvantage to finite difference methods, especially if there exist strong and sharp contrasts in the physical properties along an interface. The discretization of undulated seabed bathymetry is a common example of such misrepresentation of the physical properties in finite-difference grids, as the seabed is often a particularly sharp interface owing to the rapid and considerable change in material properties between fluid seawater and solid rock. There are two issues typically involved with seabed modeling using finite difference methods: firstly, the travel times of reflections from the seabed are inaccurate as a consequence of its spatial mispositioning; secondly, artificial diffractions are generated by the staircase representation of dipping seabed bathymetry. In this paper, we propose a new method that provides a solution to these two issues by positioning sharp interfaces at fractional grid locations. To achieve this, the velocity The Unconventional Natural Gas Institute, China University of Petroleum (Beijing), Beijing 102249, China model is first sampled in a model grid that allows the center of the seabed to be positioned at grid points, before being interpolated vertically onto a regular modeling grid using the windowed sinc function. This procedure allows undulated seabed bathymetry to be represented with improved accuracy during modeling. Numerical tests demonstrate that this method generates reflections with accurate travel times and effectively suppresses artificial diffractions.

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The immersed interface method for acoustic wave equations with discontinuous coefficients

Cited by 105 publications

References 20 publications

High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces

High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces

Convergence of an Immersed Interface Upwind Scheme for Linear Advection Equations with Piecewise Constant Coefficients II: Some Related Binomial Coefficients Inequalities

Accurate seabed modeling using finite difference methods

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