A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers’ equation

Jiwari, Ram; Mittal, R. C.; Sharma, Kapil K.

doi:10.1016/j.amc.2012.12.035

Cited by 76 publications

(53 citation statements)

References 20 publications

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“…In seeking an efficient discretization technique to obtain accurate numerical solution using a considerably small number of grid points, [1] and [14] introduced the method of differential quadrature (DQ).where a partial derivative of a function with respect to a coordinate direction is expressed as a linear weighted sum of all the functional values at all mesh points along that direction. The key to DQ is to determine the weighting coefficient for the discretization of a derivative of any order.…”

Section: Differential Quadrature Methodsmentioning

confidence: 99%

Numerical Solution of the Coupled Viscous Burgers’ Equation Using Differential Quadrature Method Based on Fourier Expansion Basis

Jima¹,

Shiferaw²,

Tsegaye³

2018

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The differential quadrature method based on Fourier expansion basis is applied in this work to solve coupled viscous Burgers' equation with appropriate initial and boundary conditions. In the first step for the given problem we have discretized the interval and replaced the differential equation by the Differential quadrature method based on Fourier expansion basis to obtain a system of ordinary differential equation (ODE) then we implement the numerical scheme by computer programing and perform numerical solution. Finally the validation of the present scheme is demonstrated by numerical example and compared with some existing numerical methods in literature. The method is analyzed for stability and convergence. It is found that the proposed numerical scheme produces a good result as compared to other researcher's result and even generates a value at the nodes or mesh points that the results have not seen yet. 822Applied Mathematics proximate solution for problems which are difficult to solve analytically. The de-

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Section: Differential Quadrature Methodsmentioning

confidence: 99%

Numerical Solution of the Coupled Viscous Burgers’ Equation Using Differential Quadrature Method Based on Fourier Expansion Basis

Jima¹,

Shiferaw²,

Tsegaye³

2018

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“…Additionally, the numerical convergence rates of the time and space are calculated by:

{italicrate}_{t} ≔ {normallog}_{2} ({italicErr}_{Δ t}^{n} / {italicErr}_{\frac{Δ t}{2}}^{n}), {italicrate}_{s} ≔ {normallog}_{2} ({italicErr}_{Δ x}^{n} / {italicErr}_{\frac{Δ x}{2}}^{n}),

respectively, where

{italicErr}_{Δ t}^{n}

and

{italicErr}_{Δ x}^{n}

represent the L ∞ ( t n ) and L R 2 ( t n ) errors which are obtained for the temporal step size Δ t , and the spatial grid size Δ x , respectively. All numerical simulations are performed using MATLAB 2014a (8.3.0.532; MatheWorks, Seoul, Korea) on Windows 10.Example We consider the viscous Burgers' equation (1) over a domain Ω = [0, 1] with initial and boundary conditions

\{\begin{matrix} left \\ u (0, x) = \frac{2 ν π \sin (π x)}{σ + \cos (π x)}, x \in Ω, \\ u (t, 0) = u (t, 1) = 0, t > 0, \end{matrix}

and the analytic solution

u (t, x) = \frac{2 ν π \exp (- π^{2} ν italict) \sin (π x)}{σ + \exp ()...</p>}

…”

Section: Accuracy Validation and Robustnessmentioning

confidence: 99%

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

Bak

2019

Numerical Methods Partial

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In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L‐stable and third‐order L(α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero.

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“…The authors of [6] examined different exact solutions of the one-dimensional Burgers' equation. Many researchers approximate the solutions of the Burgers' equation by various numerical schemes such as finite difference, finite element, exponentially fitted, Haar wavelet, differential quadrature [7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

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

Seydaoğlu

2019

Mathematics

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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.

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A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers’ equation

Cited by 76 publications

References 20 publications

Numerical Solution of the Coupled Viscous Burgers’ Equation Using Differential Quadrature Method Based on Fourier Expansion Basis

Numerical Solution of the Coupled Viscous Burgers’ Equation Using Differential Quadrature Method Based on Fourier Expansion Basis

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

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

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