A Compact Fourth Order Scheme for the Helmholtz Equation in Polar Coordinates

Britt, S.; Tsynkov, Semyon; Turkel, Eli

doi:10.1007/s10915-010-9348-3

Cited by 40 publications

(36 citation statements)

References 30 publications

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“…It is consistent with order O(τ 2 + h 2 + τ h 2 ) but only on the solutions of (1) rather than on all sufficiently smooth functions, since in the course of derivation we have used the equation-based differentiation (5). Accordingly, replacing the second derivative with respect to x in (6) by the central difference in (7) is referred to as equation-based differencing.…”

Section: Example: the Lax-wendroff Schemementioning

confidence: 97%

“…Higher order accuracy can be achieved without expanding the stencil in schemes known as Collatz "Mehrstellen" [10], equation-based and related compact schemes [2,3,5,6,20,31,47,48,51] and Trefftz-FLAME [54]. Such schemes rely on a targeted approximation of the class of solutions rather than of a much broader class of generic sufficiently smooth functions.…”

Section: Numerical Approximation Of Differential Equations: Standard mentioning

confidence: 99%

“…In [5], we constructed a similar compact fourth order scheme for the standard Helmholtz equation, but in polar coordinates, which entails variable coefficients. In [6], we have extended the analysis to the case of a more general Helmholtz equation, with variable coefficients under the derivatives in the Laplacian-like part.…”

Section: Compact High Order Scheme For the Variable Coefficient Helmhmentioning

confidence: 99%

“…Specific choices of the boundary conditions for the discrete AP that we made for our simulations are outlined in Sect. 5. Note, that if the continuous source function f (x, y) is sufficiently smooth on the entire square Ω 0 , and if f is the trace of this function on the grid K 0 , then u = G (h) B (h) f is a fourth order accurate approximation of the solution u to the differential equation Lu = f on Ω 0 subject to the chosen boundary condition at ∂Ω 0 .…”

Section: Definition 2 the Problemmentioning

confidence: 99%

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The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

2012

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The method of difference potentials was originally proposed by Ryaben'kii and can be interpreted as a generalized discrete version of the method of Calderon's operators in the theory of partial differential equations. It has a number of important advantages; it easily handles curvilinear boundaries, variable coefficients, and non-standard boundary conditions while keeping the complexity at the level of a finite-difference scheme on a regular structured grid. The method of difference potentials assembles the overall solution of the original boundary value problem by repeatedly solving an auxiliary problem. This auxiliary problem allows a considerable degree of flexibility in its formulation and can be chosen so that it is very efficient to solve.Compact finite difference schemes enable high order accuracy on small stencils at virtually no extra cost. The scheme attains consistency only on the solutions of the differential equation rather than on a wider class of sufficiently smooth functions. Unlike standard high order schemes, compact approximations require no additional boundary conditions beyond those needed for the differential equation itself. However, they exploit two stencils-one applies to the left-hand side of the equation and the other applies to the right-hand side of the equation.Dedicated to our friend Saul Abarbanel on the occasion of his 80th birthday. J Sci Comput (2012) 53:150-193 151 We shall show how to properly define and compute the difference potentials and boundary projections for compact schemes. The combination of the method of difference potentials and compact schemes yields an inexpensive numerical procedure that offers high order accuracy for non-conforming smooth curvilinear boundaries on regular grids. We demonstrate the capabilities of the resulting method by solving the inhomogeneous Helmholtz equation with a variable wavenumber with high order (4 and 6) accuracy on Cartesian grids for non-conforming boundaries such as circles and ellipses.

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Section: Example: the Lax-wendroff Schemementioning

confidence: 97%

Section: Numerical Approximation Of Differential Equations: Standard mentioning

confidence: 99%

Section: Compact High Order Scheme For the Variable Coefficient Helmhmentioning

confidence: 99%

Section: Definition 2 the Problemmentioning

confidence: 99%

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The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

2012

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“…The compact finite difference scheme has been extensively studied for solving the Helmholtz equation on the behalf of high order accuracy on small stencils. Starting from the noticeable work [1], a family of high-order compact schemes are derived for variety of Helmholtz equations based on the differencing of governing equation to eliminate higher order derivatives in the discretization error, such as, the compact finite difference scheme for one dimension problems [2], the compact fourth order accuracy finite difference approximation for two and three dimension problems on the Cartesian coordinate [1,3] and polar coordinates [4,5]. Other papers have extended the HOC scheme to Helmholtz equation with non-homogeneous materials [6], with nonstandard boundary conditions [7] and with singular solutions [8].…”

Section: Introductionmentioning

confidence: 99%

High Order Compact Scheme and the Moving Mesh Method for Helmholtz Equation with Variable Wave Numbers

Cao¹,

Yuan²,

Ge³

2017

dtcse

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Abstract. High order compact scheme has extensively been used for solving the Helmholtz equation on uniform mesh. Although uniform spacing is computationally advantageous for isotropic problems, the non-uniform mesh will be superior to the problems with the anisotropic properties or boundary layers. In this paper, we proposed a compact scheme on nonuniform mesh which has at least third order accuracy for the 2D Helmholtz equation with variable wave number. A novel adaptive mesh algorithm is also presented to generate the orthogonal nonuniform mesh. The 2D Helmholtz equation is solved by combining the HOC scheme on nonuniform mesh with the moving mesh method. The multigrid method is used to solve the discrete algebraic system. Numerical experiments are executed to verify the accuracy and efficiency of the presented method.

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Electromagnetic scattering from a cavity embedded in an impedance ground plane

Sun

et al. 2018

Math Methods in App Sciences

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This paper is concerned with time‐harmonic electromagnetic scattering from a cavity embedded in an impedance ground plane. The fillings (which may be inhomogeneous) do not protrude the cavity and the space above the ground plane is empty. This problem is obviously different from those considered in previous work where either perfectly conducting boundary conditions were used or the cavity was assumed to be empty. By employing the Green's function method, we reduce the scattering problem to a boundary‐value problem in a bounded domain (the cavity), with impedance boundary conditions on the cavity walls and an impedance‐to‐Dirichlet condition on the cavity aperture. Existence and uniqueness of the solution are proved for the weak formulation of the reduced problem. We also propose a numerical method to calculate the radar cross section (RCS), which is a parameter of physical interest. Numerical experiments show that the proposed model and numerical method are efficient for the calculation of RCS from cavities.

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A Compact Fourth Order Scheme for the Helmholtz Equation in Polar Coordinates

Cited by 40 publications

References 30 publications

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

High Order Compact Scheme and the Moving Mesh Method for Helmholtz Equation with Variable Wave Numbers

Electromagnetic scattering from a cavity embedded in an impedance ground plane

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