A high-order compact scheme for the one-dimensional Helmholtz equation with a discontinuous coefficient

Feng, Xiufang

doi:10.1080/00207160.2011.648184

Cited by 9 publications

(4 citation statements)

References 4 publications

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“…The interface problem (4.5) is a simple interface problem in the sense that discontinuous wave number is not involved in the equation, thus many different approaches have been proposed to solve it, such as immersed finite element method [29,36], matched interface and boundary method [70], Nitsche's method [72], immersed interface method [21,22,68]. Most of these approaches are second-order methods, whereas higher-order accuracy combined with the compactness of the difference stencils gives highly accurate numerical solutions on relatively coarser grids with greater computational efficiency [40,47,55].…”

Section: Discretization Of the Interface Problem On A Cartesian Gridmentioning

confidence: 99%

An Efficient Cartesian Grid-Based Method for Scattering Problems with Inhomogeneous Media

Hua¹,

null²,

Zhang³

2023

NMTMA

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Boundary integral equations provide a powerful tool for the solution of scattering problems. However, often a singular kernel arises, in which case the standard quadratures will give rise to unavoidable deteriorations in numerical precision, thus special treatment is needed to handle the singular behavior. Especially, for inhomogeneous media, it is difficult if not impossible to find out an analytical expression for Green's function. In this paper, an efficient fourth-order accurate Cartesian grid-based method is proposed for the two-dimensional Helmholtz scattering and transmission problems with inhomogeneous media. This method provides an alternative approach to indirect integral evaluation by solving equivalent interface problems on Cartesian grid with a modified fourth-order accurate compact finite difference scheme and a fast Fourier transform preconditioned conjugate gradient (FFT-PCG) solver. A remarkable point of this method is that there is no need to know analytical expressions for Green's function. Numerical experiments are provided to demonstrate the advantage of the current approach, including its simplicity in implementation, its high accuracy and efficiency.

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Section: Discretization Of the Interface Problem On A Cartesian Gridmentioning

confidence: 99%

An Efficient Cartesian Grid-Based Method for Scattering Problems with Inhomogeneous Media

Hua¹,

null²,

Zhang³

2023

NMTMA

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“…Existing literature of theoretical and numerical treatment to Helmholtz equation using finite difference methods (1)(2)(3) , finite element methods (4) and for existence uniqueness results can be found at (5)(6)(7) . 1D and 2D Helmholtz equation have been treated by Xiufang Feng (8) and Xiufang Feng et al (9) respectively by using high order compact finite difference methods.…”

Section: Introductionmentioning

confidence: 99%

A fourth order orthogonal spline collocation method Interface boundary value problem

Bhal¹,

Panda²

2022

IJST

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Objective: A higher order numerical scheme for two-point boundary interface problem with Dirichlet and Neumann boundary condition on two different sides is propounded. Methods: Orthogonal cubic spline collocation techniques have been used (OSC) for the two-point interface boundary value problem.To approximate the solution a piecewise Hermite cubic basis functions have been used. Findings: Remarkable features of the OSC are accounted for the numerous applications, theoretical clarity, and convenient execution. The stability and efficiency of orthogonal spline collocation methods over B-splines have made the former more preferable than the latter. As against finite element methods, determining the approximate solution and the coefficients of stiffness matrices and mass is relatively fast as the evaluation of integrals is not a requirement. The systematic incorporation of boundary and interface conditions in OSC adds to the list of advantages of preferring this method. Novelty: As against the existing methodologies it becomes clear from our findings that OSC is dominantly computationally superior. A computational treatment has been implemented on the two-point interface boundary value problem with super-convergent results of derivative at the nodal points, being the noteworthy finding of the study.

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“…A new design of sixth-order compact finite-difference method with a nine-point stencil is developed by Nabavi et al [9] to solve the Helmholtz equation in two-dimensional domain under the circumstance of Dirichlet and Neumann boundary. With the idea of the immersed interface method, third-and fourth-order compact finite difference schemes were proposed for solving the Helmholtz equations with discontinuous coefficient [10,11]. It is worth noting that the advert of [12,13] a new high-order finite difference discretization strategy, based on the Richardson extrapolation technique and an operator interpolation scheme, is explored to solve convection diffusion equations [14] which exploits an innovative adaptive scheme in terms of Adaptive Mesh Refinement (AMR) and Multigrid Algorithms to achieve a settlement of the fourthorder two-dimensional Poisson equation.…”

Section: Introductionmentioning

confidence: 99%

Higher-Order Compact Finite Difference for Certain PDEs in Arbitrary Dimensions

Gao

Liu

2020

Journal of Function Spaces

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In this paper, we first present the expression of a model of a fourth-order compact finite difference (CFD) scheme for the convection diffusion equation with variable convection coefficient. Then, we also obtain the fourth-order CFD schemes of the diffusion equation with variable diffusion coefficients. In addition, a fine description of the sixth-order CFD schemes is also developed for equations with constant coefficients, which is used to discuss certain partial differential equations (PDEs) with arbitrary dimensions. In this paper, various ways of numerical test calculations are prepared to evaluate performance of the fourth-order CFD and sixth-order CFD schemes, respectively, and the empirical results are proved to verify the effectiveness of the schemes in this paper.

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A high-order compact scheme for the one-dimensional Helmholtz equation with a discontinuous coefficient

Cited by 9 publications

References 4 publications

An Efficient Cartesian Grid-Based Method for Scattering Problems with Inhomogeneous Media

An Efficient Cartesian Grid-Based Method for Scattering Problems with Inhomogeneous Media

A fourth order orthogonal spline collocation method Interface boundary value problem

Higher-Order Compact Finite Difference for Certain PDEs in Arbitrary Dimensions

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