Fourth-Order Compact Schemes for Helmholtz Equations with Piecewise Wave Numbers in the Polar Coordinates

Su, Xiaolu; Feng, Xiufang; Li, Zhilin

doi:10.4208/jcm.1604-m2015-0290

Cited by 5 publications

(4 citation statements)

References 5 publications

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“…Example [3, Example 3.2] We consider the following problem in an annulus:

\frac{1}{r} {((), {ru}_{r} ((),, r, θ))}_{r} + \frac{1}{r^{2}} u_{θ θ} (r, θ) + w^{2} u (r, θ) = f (r, θ), (r, θ) \in Ω \equiv (1, 2) \times (0, 2 π),

with the boundary conditions

u (1, θ) = \cos (θ), u (2, θ) = 2 \cos (θ), θ \in [0, 2 π],

and

u (r, 0) = u (r, 2 π), u_{θ} (r, 0) = u_{θ} (r, 2 π), r \in [1, 2],

where

w^{2} (x) = \{\begin{matrix} center \\ w_{-}^{2}, & (r, θ) \in [1,1.5] \times (0, 2 π), \\ w_{+}^{2}, & (r, θ) \in (1.5,2] \times (0, 2 π <...>) \end{matrix}

…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The results of numerical experiments clearly demonstrate the efficacy of OSC employing an MDA for the solution of Helmholtz problems with interfaces. In comparison, no linear system solvers are discussed in [2, 3, 5, 11]. It should be noted that the boundary conditions on the sides y = 0, 1, of the unit square can be Dirichlet, as shown here, Neumann or mixed; that is, Neumann or Dirichlet on y = 0 and Dirichlet or Neumann on y = 1.…”

Section: Discussionmentioning

confidence: 99%

“…Such a problem arises in many physical applications, and various numerical techniques have been developed for its approximate solution particularly for the case in which ω 2 , the wave number, is a constant. Recently, high order (i.e., order three or four) compact finite difference methods have been developed for the solution of (1.1) with Dirichlet boundary conditions and a vertical interface at x = x ℓ , which we call Γ ℓ , such that

ω^{2} = \{\begin{matrix} left \\ ω_{-}^{2}, & (x, y) \in [0, x_{ℓ}] \times [0, 1], \\ ω_{+}^{2}, & (x, y) \in (x_{ℓ}, 1] \times [0, 1] . \end{matrix}

In the derivation of these methods, special attention is devoted to the approximation of the interface conditions since, even though u and u x may be continuous across this interface, that is,

{([], u)}_{| x = x_{ℓ}} = {([], u_{x})}_{| x = x_{ℓ}} = 0,

where

{([], u)}_{| x = x_{ℓ}} = \underset{x \to x_{ℓ}^{+}}{normallim} u (x, y) - \underset{x \to x_{ℓ}^{-}}{normallim} u (x, y), y \in (0, 1),

the second or higher order partial derivatives with respect to x and the source function f ( x , y ) may be discontinuous across the interface; see, for example, [2, 3]. Moreover, frequently only Dirichlet boundary conditions are considered avoiding the added complication of deriving high‐order approximations to Neumann boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A fourth–order orthogonal spline collocation method for two‐dimensional Helmholtz problems with interfaces

Bhal¹,

Danumjaya²,

Fairweather

2020

Numerical Methods Partial

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Orthogonal spline collocation is implemented for the numerical solution of two-dimensional Helmholtz problems with discontinuous coefficients in the unit square. A matrix decomposition algorithm is used to solve the collocation matrix system at a cost of O(N 2 log N) on an N × N partition of the unit square. The results of numerical experiments demonstrate the efficacy of this approach, exhibiting optimal global estimates in various norms and superconvergence phenomena for a broad spectrum of wave numbers.

show abstract

“…Example [3, Example 3.2] We consider the following problem in an annulus:

\frac{1}{r} {((), {ru}_{r} ((),, r, θ))}_{r} + \frac{1}{r^{2}} u_{θ θ} (r, θ) + w^{2} u (r, θ) = f (r, θ), (r, θ) \in Ω \equiv (1, 2) \times (0, 2 π),

with the boundary conditions

u (1, θ) = \cos (θ), u (2, θ) = 2 \cos (θ), θ \in [0, 2 π],

and

u (r, 0) = u (r, 2 π), u_{θ} (r, 0) = u_{θ} (r, 2 π), r \in [1, 2],

where

w^{2} (x) = \{\begin{matrix} center \\ w_{-}^{2}, & (r, θ) \in [1,1.5] \times (0, 2 π), \\ w_{+}^{2}, & (r, θ) \in (1.5,2] \times (0, 2 π <...>) \end{matrix}

…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

ω^{2} = \{\begin{matrix} left \\ ω_{-}^{2}, & (x, y) \in [0, x_{ℓ}] \times [0, 1], \\ ω_{+}^{2}, & (x, y) \in (x_{ℓ}, 1] \times [0, 1] . \end{matrix}

In the derivation of these methods, special attention is devoted to the approximation of the interface conditions since, even though u and u x may be continuous across this interface, that is,

{([], u)}_{| x = x_{ℓ}} = {([], u_{x})}_{| x = x_{ℓ}} = 0,

where

{([], u)}_{| x = x_{ℓ}} = \underset{x \to x_{ℓ}^{+}}{normallim} u (x, y) - \underset{x \to x_{ℓ}^{-}}{normallim} u (x, y), y \in (0, 1),

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A fourth–order orthogonal spline collocation method for two‐dimensional Helmholtz problems with interfaces

Bhal¹,

Danumjaya²,

Fairweather

2020

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

“…Britt et al [11] constructed a high-order compact difference scheme for the Helmholtz equation. Su et al [12] proposed a fourth-order method for solving Helmholtz equation with discontinuous wave number and Dirichlet boundary condition. A novel compact scheme based on finite difference discretizations and geometric grid has been developed to solve two-dimensional mildly non-linear elliptic equations in polar co-ordinate constituting singular terms [13].…”

Section: Introductionmentioning

confidence: 99%

High-Order Finite Difference Method for Helmholtz Equation in Polar Coordinates

Zhu¹,

Zhao²

2019

AJCM

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We present a fourth-order finite difference scheme for the Helmholtz equation in polar coordinates. We employ the finite difference format in the interior of the region and derive a nine-point fourth-order scheme. Specially, ghost points outside the region are applied to obtain the approximation for the Neumann boundary condition. We obtain the matrix form of the linear system and the sparsity of the coefficient matrix is favorable for the computation of the Helmholtz equation. The feasibility and accuracy of the method are validated by two test examples which have exact solutions.

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Fourth order orthogonal spline collocation methods for two-point boundary value problems with interfaces

Bhal

Mohanty

Nandi

2022

J Anal

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Fourth-Order Compact Schemes for Helmholtz Equations with Piecewise Wave Numbers in the Polar Coordinates

Cited by 5 publications

References 5 publications

A fourth–order orthogonal spline collocation method for two‐dimensional Helmholtz problems with interfaces

A fourth–order orthogonal spline collocation method for two‐dimensional Helmholtz problems with interfaces

High-Order Finite Difference Method for Helmholtz Equation in Polar Coordinates

Fourth order orthogonal spline collocation methods for two-point boundary value problems with interfaces

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