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A comparison of Fourier pseudospectral method and finite volume method used to solve the Burgers equation

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Abstract

The Burger’s equation serves as a useful mathematical model to be applied in fluid dynamic problems. Besides, this equation is often used to test and compare numerical techniques. In the present paper, the numerical solutions determined by both the finite volume method (FVM) and the Fourier pseudospectral method (FPM), and the results are compared with the analytical, exact solution. Furthermore, performance in obtaining the results is properly reported. The analysis includes periodic and non-periodic domains, for both methods. The non-linear term in the FVM, is discretion by using upwind and central-differencing schemes, showing a rate of convergence for the second-order accurate. Due requirement of the domains in FPM to be periodic, in the present work, the immersed boundary methodology is used, to solve the equation at non-periodic domains. It will also be shown that this approach damages the accuracy and the rate convergence of the FPM.

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Abbreviations

U :

Velocity at axis x (m/s)

t :

Time (s)

x :

Axis coordinate x (m)

u b :

Velocity of diffusion (m/s)

NLT:

Non-linear term

Δx :

Dimension of volume cell at direction x

W:

West cell

P:

Principal cell

E:

East cell

u P :

Velocity at the main cell

u e, u w :

Velocity at that volume face

δ :

Distance between the central of cell

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Acknowledgments

The authors would like to thank the CAPES, CNPq, FAPEMIG, FAPEG and PETROBRAS for the financial support. To the School of Mechanical Engineering of the Federal University of Uberlândia and to the School of Electrical, Mechanical and Computation Engineering of the Federal University of Goiás.

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Authors and Affiliations

  1. Escola de Engenharia Elétrica, Mecânica e Computação, Universidade Federal de Goiás, Av. Universitária, 1488, Bloco: A, Piso: 3, Setor Leste Universitário, Goiânia, GO, 74605-010, Brazil

    A. A. Nascimento & F. P. Mariano

  2. Faculdade de Engenharia Mecânica, Universidade Federal de Uberlândia, Av. João Naves de Ávilla, 2121, Bloco: 5P, Santa Mônica, Uberlândia, MG, 38400-902, Brazil

    A. Silveria-Neto & E. L. M. Padilla

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  1. A. A. Nascimento
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  2. F. P. Mariano
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  3. A. Silveria-Neto
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  4. E. L. M. Padilla
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Correspondence to A. A. Nascimento.

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Technical Editor: Francisco Ricardo Cunha.

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Nascimento, A.A., Mariano, F.P., Silveria-Neto, A. et al. A comparison of Fourier pseudospectral method and finite volume method used to solve the Burgers equation. J Braz. Soc. Mech. Sci. Eng. 36, 737–742 (2014). https://doi.org/10.1007/s40430-013-0124-9

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  • DOI: https://doi.org/10.1007/s40430-013-0124-9

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