Approximations for Burgers equations with C-N scheme and RBF collocation methods

Jiang, Ziwu; Huantian(, Xie; Guo, Xiaoyi; Zhou, Jianwei

doi:10.22436/jnsa.009.06.23

Cited by 7 publications

(5 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…and ζ n can be computed from Equation (15). One can write Equations (5), (17), (18) and boundary conditions (2) in the matrix form as follows…”

Section: Lie-group Methodsmentioning

confidence: 99%

“…Bouhamidi et al [17] presented RBFs interpolation technique for spatial discretization and implicit Runge-Kutta (IRK) schemes for temporal discretization of the unsteady coupled Burgers'-type equations. Xie et al [18] approximated the solution of the Burgers' equation spatially by the multiquadric MQ-RBF and used C-N finite difference scheme as temporal discretization technique.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator

Seydaoğlu

2019

Mathematics

View full text Add to dashboard Cite

An efficient technique is proposed to solve the one-dimensional Burgers’ equation based on multiquadric radial basis function (MQ-RBF) for space approximation and a Lie-Group scheme for time integration. The comparisons of the numerical results obtained for different values of kinematic viscosity are made with the exact solutions and the reported results to demonstrate the efficiency and accuracy of the algorithm. It is shown that the numerical solutions concur with existing results and the proposed algorithm is efficient and can be easily implemented.

show abstract

“…and ζ n can be computed from Equation (15). One can write Equations (5), (17), (18) and boundary conditions (2) in the matrix form as follows…”

Section: Lie-group Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator

Seydaoğlu

2019

Mathematics

View full text Add to dashboard Cite

show abstract

“…A comparison with the method of Mittal and Jain 57 at different times is shown in Table 7, for x ∈ [0.5, 1.5] and ∈ {0.002, 0.000666666}. A further comparison with the mesh-free method of Xie et al 61 at different times is shown in Table 8, for x ∈ [ −3, 3] and = 0.0001. Figure B shows the approximate solution on the same region obtained using N = 12, for the parameters = 0.002, and = −0.4.…”

Section: Tablementioning

confidence: 99%

High‐order numerical solution of viscous Burgers' equation using a Cole‐Hopf barycentric Gegenbauer integral pseudospectral method

Elgindy

Dahy

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

We present a novel high-order numerical method to solve viscous Burgers' equation with smooth initial and boundary data. The proposed method combines Cole-Hopf transformation with well-conditioned integral reformulations to reduce the problem into either a single easy-to-solve integral equation with no constraints or an integral equation provided by a single integral boundary condition. Fully exponential convergence rates are established in both spatial and temporal directions by embracing a full Gegenbauer collocation scheme based on Gegenbauer-Gauss mesh grids using apt Gegenbauer parameter values and the latest technology of barycentric Gegenbauer differentiation and integration matrices. The global collocation matrices of the reduced algebraic linear systems were derived allowing for direct linear system solvers to be used. Rigorous error and convergence analyses are presented in addition to two easy-to-implement pseudocodes of the proposed computational algorithms. We further show three numerical tests to support the theoretical investigations and demonstrate the superior accuracy of the method even when the viscosity parameter → 0, in the absence of any adaptive strategies typically required for adaptive refinements.

show abstract

“…Sarboland and Aminataei (2014) presented two meshfree methods for solving the one-dimensional nonlinear non-homogeneous Burgers' equation using the MQ quasi-interpolation operator and direct and indirect radial basis function network schemes. Xie et al (2016) suggested a meshfree scheme by utilizing the finite difference and the MQ-RBF method, for solving the Burgers' equation numerically. Seydaoglu et al (2016) obtained the numerical solutions of Burgers' equation using highorder splitting methods combined with spectral methods, finite difference, and Weighted Essentially Nonoscillatory (WENO) schemes.…”

Section: Introductionmentioning

confidence: 99%

An efficient Strang splitting technique combined with the multiquadric-radial basis function for the Burgers’ equation

2021

View full text Add to dashboard Cite

In the present paper, two effective numerical schemes depending on a second-order Strang splitting technique are presented to obtain approximate solutions of the one-dimensional Burgers' equation utilizing the collocation technique and approximating directly the solution by multiquadric-radial basis function (MQ-RBF) method. To show the performance of both schemes, we have considered two examples of Burgers' equation. The obtained numerical results are compared with the available exact values and also those of other published methods. Moreover, the computed L 2 and L ∞ error norms have been given. It is found that the presented schemes produce better results as compared to those obtained almost all the schemes present in the literature.

show abstract

Approximations for Burgers equations with C-N scheme and RBF collocation methods

Cited by 7 publications

References 18 publications

A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator

A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator

High‐order numerical solution of viscous Burgers' equation using a Cole‐Hopf barycentric Gegenbauer integral pseudospectral method

An efficient Strang splitting technique combined with the multiquadric-radial basis function for the Burgers’ equation

Contact Info

Product

Resources

About