An Improved High Order Finite Difference Method for Non-conforming Grid Interfaces for the Wave Equation

Wang, Siyang

doi:10.1007/s10915-018-0723-9

Cited by 19 publications

(14 citation statements)

References 39 publications

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“…The new schemes are similar to those derived in [26], but with truncation errors of order p−1 instead of p−2. Since numerical experiments in [26] showed global convergence rates of order p + 1, one may expect the new schemes to converge with rate p + 2, which is supported by numerical experiments in section 7. Figure 1: Non-zero elements for an example pair of interpolation operators (p = 2).…”

Section: The Second Order Wave Equationmentioning

confidence: 91%

“…The approach has since been extended to the Schrödinger equation [20], the second order wave equation [29], and the advection-diffusion equation [12]. For equations with second derivatives in space, it has been observed that the SATs at the non-conforming interfaces worsen the largest local truncation error, and hence the global convergence rate, by one order, as compared to conforming interfaces [12,19,26]. The obvious remedy would have been to increase the order of accuracy of the interpolation operators, but [13] showed that this is impossible because the order of the interpolation operators is bounded from above by the order of the quadrature rule associated with every SBP operator [9].…”

Section: Introductionmentioning

confidence: 99%

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Order-Preserving Interpolation for Summation-by-Parts Operators at Nonconforming Grid Interfaces

Almquist¹,

Wang²,

Werpers³

2019

SIAM J. Sci. Comput.

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We study non-conforming grid interfaces for summation-by-parts finite difference methods applied to partial differential equations with second derivatives in space. To maintain energy stability, previous efforts have been forced to accept a reduction of the global convergence rate by one order, due to large truncation errors at the non-conforming interface. We avoid the order reduction by generalizing the interface treatment and introducing order preserving interpolation operators. We prove that, given two diagonal-norm summation-by-parts schemes, order preserving interpolation operators with the necessary properties are guaranteed to exist, regardless of the grid-point distributions along the interface. The new methods retain the stability and global accuracy properties of the underlying schemes for conforming interfaces.

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Section: The Second Order Wave Equationmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Order-Preserving Interpolation for Summation-by-Parts Operators at Nonconforming Grid Interfaces

Almquist¹,

Wang²,

Werpers³

2019

SIAM J. Sci. Comput.

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“…The second choice of SATs uses four penalty terms [28], which has a better stability property for problems with curved interfaces. The method was improved further in [1] from the accuracy perspective when non-periodic boundary conditions are used in the x-direction.…”

Section: The Sbp-sat Methodsmentioning

confidence: 99%

Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces

Wang¹,

Petersson²

2019

SIAM J. Sci. Comput.

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We analyze two types of summation-by-parts finite difference operators for approximating the second derivative with variable coefficient. The first type uses ghost points, while the second type does not use any ghost points. A previously unexplored relation between the two types of summation-by-parts operators is investigated. By combining them we develop a new fourth order accurate finite difference discretization with hanging nodes on the mesh refinement interface. We take the model problem as the two-dimensional acoustic wave equation in second order form in terms of acoustic pressure, and prove energy stability for the proposed method. Compared to previous approaches using ghost points, the proposed method leads to a smaller system of linear equations that needs to be solved for the ghost point values. Another attractive feature of the proposed method is that the explicit time step does not need to be reduced relative to the corresponding periodic problem. Numerical experiments, both for smoothly varying and discontinuous material properties, demonstrate that the proposed method converges to fourth order accuracy. A detailed comparison of the accuracy and the time-step restriction with the simultaneous-approximation-term penalty method is also presented.

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“…However, the scheme is not accurate when dealing with complex geometries such as the characterization of the coastlines during the simulation [18]. To deal with these limitations, refined mesh approaches are applied to increase the resolution of certain areas of interest [34]. Nevertheless, there are still some problems facing this approach when the waves are reflected from the coastline outside the finer mesh as such waves are not well resolved.…”

Section: Introductionmentioning

confidence: 99%

A Computational Model for Simulation of Shallow Water Waves by Elastic Deformations in the Topography

Al-Ghosoun¹

2021

CICP

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We propose a coupled model to simulate shallow water waves induced by elastic deformations in the bed topography. The governing equations consist of the depth-averaged shallow water equations including friction terms for the water freesurface and the well-known second-order elastostatics formulation for the bed deformation. The perturbation on the free-surface is assumed to be caused by a sudden change in the bottom beds. At the interface between the water flow and the bed topography, transfer conditions are implemented. Here, the hydrostatic pressure and friction forces are considered for the elastostatic equations whereas bathymetric forces are accounted for in the shallow water equations. The focus in the present study is on the development of a simple and accurate representation of the interaction between water waves and bed deformations in order to simulate practical shallow water flows without relying on complex partial differential equations with free boundary conditions. The effects of location and magnitude of the deformation on the flow fields and free-surface waves are investigated in details. Numerical simulations are carried out for several test examples on shallow water waves induced by sudden changes in the bed. The proposed computational model has been found to be feasible and satisfactory.

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An Improved High Order Finite Difference Method for Non-conforming Grid Interfaces for the Wave Equation

Cited by 19 publications

References 39 publications

Order-Preserving Interpolation for Summation-by-Parts Operators at Nonconforming Grid Interfaces

Order-Preserving Interpolation for Summation-by-Parts Operators at Nonconforming Grid Interfaces

Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces

A Computational Model for Simulation of Shallow Water Waves by Elastic Deformations in the Topography

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