Numerical Investigation on Convergence Rate of Singular Boundary Method

Li, Junpu; Chen, Wen; Fu, Zhuojia

doi:10.1155/2016/3564632

Cited by 10 publications

(13 citation statements)

References 20 publications

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“…(3). With this idea in mind, the SBM formulation is given by

\overset{true¯}{u} true(x_{i} true) = \sum_{j = 1, j \neq i}^{N} β_{j} G_{0} true(x_{i}, s_{j} true) + β_{i} U_{0}^{i}, x_{i} \in Γ_{D},

where

U_{0}^{i}

is the OIF on the Dirichlet boundary. Substituting the OIF and boundary conditions

\overset{true¯}{u} (x_{i})

into Eq.…”

Section: Numerical Methodologymentioning

confidence: 99%

Error bounds of singular boundary method for potential problems

Chen

2017

Numerical Methods Partial

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The singular boundary method (SBM) is a recent strong‐form boundary collocation method free of integration, mesh, and fictitious boundary. Although an extensive study has been reported in the literature on improving its accuracy and stability as well as its applications to diverse problems, little, however, has been done to analyze its convergence mathematically. The main purpose of this paper is to derive the explicit error bounds of the SBM for potential problems as well as to explain the essential difference between the origin intensity factor (OIF) in the SBM and the singular integration in the boundary element method (BEM). In the process of derivation, we also illustrate the physical meaning of OIF and explain the reason why the OIF has the function to correct the discretization error on the boundary. Finally, several benchmark examples are given to verify the effectiveness of the conclusions obtained from this article, as well as to investigate the different convergence behaviors between the SBM and BEM. It can be found that the SBM has the explicit error bound and is mathematically a stable technique.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1987–2004, 2017

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“…(3). With this idea in mind, the SBM formulation is given by

\overset{true¯}{u} true(x_{i} true) = \sum_{j = 1, j \neq i}^{N} β_{j} G_{0} true(x_{i}, s_{j} true) + β_{i} U_{0}^{i}, x_{i} \in Γ_{D},

where

U_{0}^{i}

is the OIF on the Dirichlet boundary. Substituting the OIF and boundary conditions

\overset{true¯}{u} (x_{i})

into Eq.…”

Section: Numerical Methodologymentioning

confidence: 99%

Error bounds of singular boundary method for potential problems

Chen

2017

Numerical Methods Partial

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“…For the finite-difference scheme (42), some results presented in the following theorems. Suppose that U k is the solution of the (42) with the initial condition u(x, 0) = U 0 , and the boundary condition U k | Ω = u(x, t k ), then the stability resultant can be expressed as follows: Theorem 4.1 Let u k ∈ H 1 (Ω) and G(.)…”

Section: Analysis Of Stability and Convergencementioning

confidence: 99%

“…▪ Now we perform an error estimate analysis for the approximated solution of the time-discretized problem (42). ▪ Now we perform an error estimate analysis for the approximated solution of the time-discretized problem (42).…”

Section: Theorem 43 the Time Discrete Numerical Solution Equationmentioning

confidence: 99%

“…Also Δ is Laplacian operator and ∇ is gradient operator. Recently, Chen and his coworkers proposed the SBM as a meshfree BCM [40][41][42][43][44][45][46][47]. In special case, if = 0, = 1 and = −1 then the problem acquired to the regularized long-wave equation (RLWE) that was studied as an improvement of the Korteweg-de Vries (KdV) equation for modeling long surface gravity waves.…”

Section: Introductionmentioning

confidence: 99%

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Meshless formulation to two‐dimensional nonlinear problem of generalized Benjamin–Bona–Mahony–Burgers through singular boundary method: Analysis of stability and convergence

Aslefallah

Abbasbandy

Shivanian

2019

Numerical Methods Partial

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In this study, the singular boundary method (SBM) is employed for the simulation of nonlinear generalized Benjamin–Bona–Mahony–Burgers problem with initial and Dirichlet‐type boundary conditions. The θ‐weighted finite difference method is used to discretize the time derivatives. Then the original equations are split into a system of partial differential equations. A splitting scheme is applied to split the solution of the inhomogeneous governing equation into homogeneous solution and particular solution. To solve this system, the method of particular solution (MPS) in combination with the SBM is used where the SBM is used for homogeneous solution and MPS is used for particular solution. Furthermore, the stability and convergence of the proposed method is conducted. Finally, several numerical examples with different domains are provided and compared with the exact analytical solutions to show the accuracy and efficiency in comparison with existing other methods.

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“…Since the solution points used in such method are arbitrary in term of spatial distribution, it is unnecessary to spend too much time in the arrangement of solution points even for the research object with complex geometries. Some classic meshfree methods include the method of fundamental solutions (MFS) [9,10], the reproducing kernel particle method (RKPM) [11,12], the element-free Galerkin method (EFGM) [13,14], the equivalent source method (ESM) [15,16], the singular boundary method (SBM) [17,18], and other meshfree methods [19,20].…”

mentioning

confidence: 99%

Lagrangian Meshfree Particle-based Computational Acoustics for Two-dimensional Sound Propagation and Scattering Problems

Zhang¹,

Li²,

Zhang³

2017

Preprint

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Meshfree particle method, which is always regarded as a pure Lagrangian approach, is easily represented complicated domain topologies, moving boundaries, and multiphase media. Solving acoustic problems with the mesfree particle method forms a branch of the acoustic wave modeling field, namely, particle-based computational acoustics (PCA). The aim of this paper is to improve the accuracy of using the PCA method to solve two-dimensional acoustic problems, and realize the particle representation with a hybrid meshfree and finite-difference time-domain (FDTD) method for acoustic boundary conditions at both the plane and curved surface. As a widely used Lagrangian meshfree method, the smoothed particle hydrodynamics (SPH) based on the support domain and the kernel function has developed rapidly in recent years. The traditional SPH method is easily implements parallel processing and has been applied in sound wave simulation. As a corrective method with higher accuracy than SPH, the acoustic propagation and scattering in the time domain is simulated with the corrective smoothed particle method (CSPM). Moreover, a hybrid meshfree-FDTD boundary treatment technique is utilized to represent different acoustic boundaries in the Lagrangian approach.In this boundary treatment technique, the parameter value of virtual particles is obtained with the FDTD method, which concerns truncation errors based on the Tayler series expansion. Soft, rigid, and Mur's absorbing boundary conditions are developed to simulate sound waves in finite and infinite domain. Results of modeling acoustic propagation and scattering show that CSPM is accurate and convergence with exact solutions, and different acoustic boundaries are validated to be effective in the computation.

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Numerical Investigation on Convergence Rate of Singular Boundary Method

Cited by 10 publications

References 20 publications

Error bounds of singular boundary method for potential problems

Error bounds of singular boundary method for potential problems

Meshless formulation to two‐dimensional nonlinear problem of generalized Benjamin–Bona–Mahony–Burgers through singular boundary method: Analysis of stability and convergence

Lagrangian Meshfree Particle-based Computational Acoustics for Two-dimensional Sound Propagation and Scattering Problems

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