Compact optimal quadratic spline collocation methods for the Helmholtz equation

Fairweather, Graeme; Karageorghis, Andréas; Maack, Jon

doi:10.1016/j.jcp.2010.12.041

Cited by 27 publications

(10 citation statements)

References 32 publications

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“…Let θ s (x, t) ∈ S 2 be the biquadratic spline interpolant of the function θ(x, t), which was introduced in [6,8], so that…”

Section: Preliminariesmentioning

confidence: 99%

“…In 2008, Bialecki et al [3] developed the QSC methods and presented a new QSC method for the Helmholtz equation with homogeneous Dirichlet boundary conditions. Later, Fairweather et al [8] extended this method to the solutions of Helmholtz equation with non-homogeneous Dirichlet, Neumann and mixed boundary conditions. Luo et al [17] studied the QSC method and efficient preconditioner for the Helmholtz equation with Robbins boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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Numerical solution of fractional bioheat equation by quadratic spline collocation method

Qin¹,

Wu²

2016

J. Nonlinear Sci. Appl.

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“…Let θ s (x, t) ∈ S 2 be the biquadratic spline interpolant of the function θ(x, t), which was introduced in [6,8], so that…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical solution of fractional bioheat equation by quadratic spline collocation method

Qin¹,

Wu²

2016

J. Nonlinear Sci. Appl.

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“…Consider the Helmholtz equation [6]     Table 1 indicates the number collocation points. In this table we have also calculated the experimental convergence rate c of the error at the collocation points which is defined as ch th…”

Section: Examplementioning

confidence: 99%

“…Analytical solution of PDEs , however , either does not exist or is difficult to find . Recent contribution in this regard includes meshless methods [3], finite-difference methods [4], Alternating-Direction Sinc-Galerkin method (ADSG) [5] , quadratic spline collocation method (QSCM) [6] , Liu and Lin method [7] and so on .…”

Section: Introductionmentioning

confidence: 99%

New Implementation of Legendre Polynomials for Solving Partial Differential Equations

Davari¹,

Ahmadi²

2013

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In this paper we present a proposal using Legendre polynomials approximation for the solution of the second order linear partial differential equations . Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients . The performance of presented method has been compared with other methods , namely Sinc-Galerkin , quadratic spline collocation and LiuLin method . Numerical examples show better accuracy of the proposed method . Moreover , the computation cost decreases at least by a factor of 6 in this method .

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“…In [91], this method is extended to Helmholtz problems subject to Dirichlet, Neumann, mixed and periodic boundary conditions. Maack [151] also formulated a superconvergent QSC method for the biharmonic Dirichlet problem (3.23)-(3.26) which involves the Schur complement approach and an MDA.…”

Section: Quadratic Spline Collocationmentioning

confidence: 99%

Matrix decomposition algorithms for elliptic boundary value problems: a survey

2010

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We provide an overview of matrix decomposition algorithms (MDAs) for the solution of systems of linear equations arising when various discretization techniques are applied in the numerical solution of certain separable elliptic boundary value problems in the unit square. An MDA is a direct method which reduces the algebraic problem to one of solving a set of independent one-dimensional problems which are generally banded, block tridiagonal, or almost block diagonal. Often, fast Fourier transforms (FFTs) can be employed in an MDA with a resulting computational cost of O(N 2 log N) on an N × N uniform partition of the unit square. To formulate MDAs, we require knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two-point boundary value problems in one space dimension. In many important cases, these eigensystems are known explicitly, while in others, they must be computed. The first MDAs were formulated almost fifty years ago, for finite difference methods. Herein, we discuss more recent developments in the formulation and application of MDAs in spline collocation, finite element Galerkin and spectral methods, and the method

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Compact optimal quadratic spline collocation methods for the Helmholtz equation

Cited by 27 publications

References 32 publications

Numerical solution of fractional bioheat equation by quadratic spline collocation method

Numerical solution of fractional bioheat equation by quadratic spline collocation method

New Implementation of Legendre Polynomials for Solving Partial Differential Equations

Matrix decomposition algorithms for elliptic boundary value problems: a survey

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