Stability analysis for boundary element methods for the diffusion equation

Sharp, S.

doi:10.1016/0307-904x(86)90007-7

Cited by 11 publications

(5 citation statements)

References 2 publications

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“…Discretizing the boundary and the domain by taking boundary and interior points, we get the solution vector = where = Ψ ( ), 1 ≤ , ≤ and the unknown vector , depending on time, is the solution of the linear system Ca = d obtained from the collocation of the equations ( 10)- (11). The nonsingular coefficient matrix Cn×n [15] and the right hand side vector dn×1 are given as The application of the RBF approximation to unsteady MHD duct flow equations ( 5)-( 6) gives…”

Section: Radial Basis Function Approximation (Rbf)mentioning

confidence: 99%

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Zamana bağımlı MHD kanal akışının nümerik çözümü ve kararlılık analizi

Tezer‐Sezgin¹,

Gürbüz²

2018

Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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This paper simulates the 2D transient magnetohydrodynamic (MHD) flow in a rectangular duct in terms of the velocity of the fluid and the induced magnetic field by using the radial basis function (RBF) approximation. The inhomogeneities in the Poisson's type MHD equations are approximated using the polynomial functions (1+r) and the particular solution is found satisfying both the equations and the boundary conditions (no-slip and insulated walls). The Euler scheme is used for advancing the solution to steady-state with a time increment and a relaxation parameter which are determined for achieving stable solution. It is shown that, as Hartmann number increases, the fluid becomes stagnant at the center of the duct, the flow is flattened and boundary layers are developed on the Hartmann and side walls. These are the wellknown characteristics of the MHD duct flow. The stability analysis is also carried in terms of the spectral radius of the coefficient matrix of the discretized coupled system. Stable solutions are obtained with RBF by using quite large time increment and suitable relaxation parameters on the expense of explicit Euler time-integration scheme used.

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Section: Radial Basis Function Approximation (Rbf)mentioning

confidence: 99%

“…The stability analysis of the BEM solution of the Diffusion equation is studied by Sharp in [11]. It is observed that as the time step decreases, the quality of the approximation deteriorates showing the state of the instability.…”

Section: Introductionmentioning

confidence: 99%

Zamana bağımlı MHD kanal akışının nümerik çözümü ve kararlılık analizi

Tezer‐Sezgin¹,

Gürbüz²

2018

Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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“…We may call these integrals as pseudo-singular integrals. The difficulty of numerically integrating a function with such behavior can introduce numerical unstable problems into the solution, as reported in [11][12][13][14]. Thus accurate calculation of the domain integrals is important to the successful implementation of the pseudo-initial condition method.…”

Section: Introductionmentioning

confidence: 99%

A general algorithm for the numerical evaluation of domain integrals in 3D boundary element method for transient heat conduction

Dong

Zhang

Xie

et al. 2015

Engineering Analysis with Boundary Elements

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“…This is because as the time step approaches zero, the integrand in the domain integral (the time-dependent fundamental solution) is close to singular as the source point is on the integration element. The difficulty of numerically integrating a function with such behavior can introduce numerical unstable problems into the solution, as reported in [11][12][13][14]. Thus accurate calculation of the domain integrals is important for the successful implementation of the pseudo-initial condition method.…”

Section: Introductionmentioning

confidence: 99%

“…Sharp investigated that if the time step is chosen to be too small, the accuracy of the approximation deteriorates. And a stability condition [11] was derived to avoid the problem. Peirce et al investigated that the size of the spatial mesh relative to the time step affected the accumulation of errors in the one-step recursion scheme.…”

Section: Introductionmentioning

confidence: 99%

A new method for the numerical evaluation of domain integrals in a 3D boundary element method for transient heat conduction

Dong¹,

Zhang²,

Xie³

et al. 2014

WIT Transactions on Modelling and Simulation

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A new method is proposed for the numerical evaluation of domain integrals in a 3D boundary element method. These integrals arise in the solution of the transient heat conduction problems, using a time-dependent boundary integral equation method named as pseudo-initial condition method. As the time-dependent kernel in the domain integral is close to singular when small time step is used, a straightforward application of Gaussian quadrature may produce large errors, and thus lead to instability of the analysis. In this paper, a coordinate transformation coupled with an element subdivision technique is presented. The coordinate transformation is denoted as   , ,    transformation, while the element subdivision technique considers the position of the source point, the property of the time-dependent fundamental solution and the relations between the size of the element and the time step. With the coordinate transformation and the element subdivision technique, more Gaussian points are shifted towards the source point, thus more accurate results can be obtained. Numerical examples have demonstrated the accuracy and efficiency of the proposed method.

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Stability analysis for boundary element methods for the diffusion equation

Cited by 11 publications

References 2 publications

Zamana bağımlı MHD kanal akışının nümerik çözümü ve kararlılık analizi

Zamana bağımlı MHD kanal akışının nümerik çözümü ve kararlılık analizi

A general algorithm for the numerical evaluation of domain integrals in 3D boundary element method for transient heat conduction

A new method for the numerical evaluation of domain integrals in a 3D boundary element method for transient heat conduction

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