Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation

Kamrul Hasan; Humaira Takia; Muhammad Masudur Rahaman; Mehedi Hasan Sikdar; Bellal Hossain; Khokon Hossen

doi:doi:10.11648/j.ajasr.20220801.13

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Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation

Kamrul Hasan, Humaira Takia, Muhammad Masudur Rahaman, Mehedi Hasan Sikdar, Bellal Hossain, Khokon Hossen

Published in American Journal of Applied Scientific Research (Volume 8, Issue 1)

Received: 21 March 2022 Accepted: 11 April 2022 Published: 26 April 2022

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Abstract

In everyday life human faces shock waves and rarefaction largely in their surroundings. Hence it’s necessary to know the behavior of these waves to protect destructive effects. The aim of this work to observe the propagation of shock and rarefaction waves in various dynamics due to solve non-linear hyperbolic inviscid Burgers’ equation numerically. The models adopted here two numerical schemes which enable us to solve non-linear hyperbolic Burgers’ equation numerically. The first order explicit upwind scheme (EUDS) and second order Lax-Wendroff schemes are used to solve this equation to improve our understanding of the numerical diffusion (smearing) and oscillations that can be present when using such schemes. In order to understand the behavior of the solution we use method of characteristics to find the exact solution of inviscid Burgers’ equation. Numerical solutions are studied for different initial conditions and the shock and rarefaction waves are investigated for Riemann problem. We present stability analysis of the schemes and establish stability condition which leads to determine time step selection in terms of spatial step size with maximum initial value. Numerical result for these schemes are compared with an exact solution of inviscid Burgers’ equation in terms of accuracy by error estimation. The numerical features of the rate of convergence are presented graphically. This analysis helps us to understand a wide range of physical phenomenon of the properties of wave as well as saves in several aspects in real life.

Published in	American Journal of Applied Scientific Research (Volume 8, Issue 1)
DOI	10.11648/j.ajasr.20220801.13
Page(s)	18-24
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Shock and Rarefaction, Burgers’ Equation, Explicit Upwind and Lax-Wendroff Schemes, Rankine-Hugoniot Jump Condition, Riemann Problem

References

[1]	Ben-Dor, G., Igra, O., & Elperin, T. (Eds.). (2001). Handbook of Shock Waves: Shock Wave Interactions and Propagation. Academic Press.
[2]	Olver, P. J. (2014). Introduction to partial differential equations (pp. 182-184). New York, NY, USA:: Springer.
[3]	J. Burgers. Proc. K. Ned. Akad., 1940.43 (37).
[4]	Burgers, J. M. (1948). A mathematical model illustrating the theory of turbulence. In Advances in applied mechanics (Vol. 1, pp. 171-199). Elsevier.
[5]	Cole, J. D. (1951). On a quasi-linear parabolic equation occurring in aerodynamics. Quarterly of applied mathematics, 9 (3), 225-236. Math., (9): 225, 236, 1951.
[6]	Abazari, R., & Borhanifar, A. (2010). Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method. Computers & Mathematics with Applications, 59 (8), 2711-2722.
[7]	Bateman, H. (1915). Some recent researches on the motion of fluids. Monthly Weather Review, 43 (4), 163-170.
[8]	Beylkin, G., & Keiser, J. M. (1997). On the adaptive numerical solution of nonlinear partial differential equations in wavelet bases. Journal of computational physics, 132 (2), 233-259.
[9]	Ali, A., & Andallah, L. S. (2016). Inflow outflow effect and shock wave analysis in a traffic flow simulation. American Journal of Computational Mathematics, 6 (02), 55.
[10]	Lax, P. D. (1973). Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Society for Industrial and Applied Mathematics.
[11]	LeVeque, R. J. (1998). Nonlinear conservation laws and finite volume methods. In Computational methods for astrophysical fluid flow (pp. 1-159). Springer, Berlin, Heidelberg.
[12]	Colombo, R. M. (2004). Wave front tracking in systems of conservation laws. Applications of mathematics, 49 (6), 501-537.
[13]	Smoller, J. A. (1983). Reaction-diffusion equations and shock waves, vol. 258.
[14]	Ben-Dor, G., Igra, O., & Elperin, T. (Eds.). (2000). Handbook of shock waves, three volume set. Elsevier.
[15]	Evans, L. C. (2010). Partial differential equations (Vol. 19). American Mathematical Soc..
[16]	LeFloch, P. G. (2002). Hyperbolic systems of conservation laws. Lectures in Mathematics ETH Zürich.

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APA Style

Kamrul Hasan, Humaira Takia, Muhammad Masudur Rahaman, Mehedi Hasan Sikdar, Bellal Hossain, et al. (2022). Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation. American Journal of Applied Scientific Research, 8(1), 18-24. https://doi.org/10.11648/j.ajasr.20220801.13

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ACS Style

Kamrul Hasan; Humaira Takia; Muhammad Masudur Rahaman; Mehedi Hasan Sikdar; Bellal Hossain, et al. Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation. Am. J. Appl. Sci. Res. 2022, 8(1), 18-24. doi: 10.11648/j.ajasr.20220801.13

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AMA Style

Kamrul Hasan, Humaira Takia, Muhammad Masudur Rahaman, Mehedi Hasan Sikdar, Bellal Hossain, et al. Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation. Am J Appl Sci Res. 2022;8(1):18-24. doi: 10.11648/j.ajasr.20220801.13

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@article{10.11648/j.ajasr.20220801.13,
  author = {Kamrul Hasan and Humaira Takia and Muhammad Masudur Rahaman and Mehedi Hasan Sikdar and Bellal Hossain and Khokon Hossen},
  title = {Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation},
  journal = {American Journal of Applied Scientific Research},
  volume = {8},
  number = {1},
  pages = {18-24},
  doi = {10.11648/j.ajasr.20220801.13},
  url = {https://doi.org/10.11648/j.ajasr.20220801.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajasr.20220801.13},
  abstract = {In everyday life human faces shock waves and rarefaction largely in their surroundings. Hence it’s necessary to know the behavior of these waves to protect destructive effects. The aim of this work to observe the propagation of shock and rarefaction waves in various dynamics due to solve non-linear hyperbolic inviscid Burgers’ equation numerically. The models adopted here two numerical schemes which enable us to solve non-linear hyperbolic Burgers’ equation numerically. The first order explicit upwind scheme (EUDS) and second order Lax-Wendroff schemes are used to solve this equation to improve our understanding of the numerical diffusion (smearing) and oscillations that can be present when using such schemes. In order to understand the behavior of the solution we use method of characteristics to find the exact solution of inviscid Burgers’ equation. Numerical solutions are studied for different initial conditions and the shock and rarefaction waves are investigated for Riemann problem. We present stability analysis of the schemes and establish stability condition which leads to determine time step selection in terms of spatial step size with maximum initial value. Numerical result for these schemes are compared with an exact solution of inviscid Burgers’ equation in terms of accuracy by error estimation. The numerical features of the rate of convergence are presented graphically. This analysis helps us to understand a wide range of physical phenomenon of the properties of wave as well as saves in several aspects in real life.},
 year = {2022}
}

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TY  - JOUR
T1  - Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation
AU  - Kamrul Hasan
AU  - Humaira Takia
AU  - Muhammad Masudur Rahaman
AU  - Mehedi Hasan Sikdar
AU  - Bellal Hossain
AU  - Khokon Hossen
Y1  - 2022/04/26
PY  - 2022
N1  - https://doi.org/10.11648/j.ajasr.20220801.13
DO  - 10.11648/j.ajasr.20220801.13
T2  - American Journal of Applied Scientific Research
JF  - American Journal of Applied Scientific Research
JO  - American Journal of Applied Scientific Research
SP  - 18
EP  - 24
PB  - Science Publishing Group
SN  - 2471-9730
UR  - https://doi.org/10.11648/j.ajasr.20220801.13
AB  - In everyday life human faces shock waves and rarefaction largely in their surroundings. Hence it’s necessary to know the behavior of these waves to protect destructive effects. The aim of this work to observe the propagation of shock and rarefaction waves in various dynamics due to solve non-linear hyperbolic inviscid Burgers’ equation numerically. The models adopted here two numerical schemes which enable us to solve non-linear hyperbolic Burgers’ equation numerically. The first order explicit upwind scheme (EUDS) and second order Lax-Wendroff schemes are used to solve this equation to improve our understanding of the numerical diffusion (smearing) and oscillations that can be present when using such schemes. In order to understand the behavior of the solution we use method of characteristics to find the exact solution of inviscid Burgers’ equation. Numerical solutions are studied for different initial conditions and the shock and rarefaction waves are investigated for Riemann problem. We present stability analysis of the schemes and establish stability condition which leads to determine time step selection in terms of spatial step size with maximum initial value. Numerical result for these schemes are compared with an exact solution of inviscid Burgers’ equation in terms of accuracy by error estimation. The numerical features of the rate of convergence are presented graphically. This analysis helps us to understand a wide range of physical phenomenon of the properties of wave as well as saves in several aspects in real life.
VL  - 8
IS  - 1
ER  -

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Author Information

Kamrul Hasan

Department of Civil Engineering, Uttara University, Dhaka, Bangladesh
Humaira Takia

Department of Physics and Mechanical Engineering, Patuakhali Science and Technology University, Patuakhali, Bangladesh
Muhammad Masudur Rahaman

Department of Mathematics, Patuakhali Science and Technology University, Patuakhali, Bangladesh
Mehedi Hasan Sikdar

Department of Statistics, Patuakhali Science and Technology University, Patuakhali, Bangladesh
Bellal Hossain

Department of Mathematics, Patuakhali Science and Technology University, Patuakhali, Bangladesh
Khokon Hossen

Department of Physics and Mechanical Engineering, Patuakhali Science and Technology University, Patuakhali, Bangladesh

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APA Style

Kamrul Hasan, Humaira Takia, Muhammad Masudur Rahaman, Mehedi Hasan Sikdar, Bellal Hossain, et al. (2022). Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation. American Journal of Applied Scientific Research, 8(1), 18-24. https://doi.org/10.11648/j.ajasr.20220801.13

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ACS Style

Kamrul Hasan; Humaira Takia; Muhammad Masudur Rahaman; Mehedi Hasan Sikdar; Bellal Hossain, et al. Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation. Am. J. Appl. Sci. Res. 2022, 8(1), 18-24. doi: 10.11648/j.ajasr.20220801.13

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AMA Style

Kamrul Hasan, Humaira Takia, Muhammad Masudur Rahaman, Mehedi Hasan Sikdar, Bellal Hossain, et al. Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation. Am J Appl Sci Res. 2022;8(1):18-24. doi: 10.11648/j.ajasr.20220801.13

Copy | Download

@article{10.11648/j.ajasr.20220801.13,
  author = {Kamrul Hasan and Humaira Takia and Muhammad Masudur Rahaman and Mehedi Hasan Sikdar and Bellal Hossain and Khokon Hossen},
  title = {Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation},
  journal = {American Journal of Applied Scientific Research},
  volume = {8},
  number = {1},
  pages = {18-24},
  doi = {10.11648/j.ajasr.20220801.13},
  url = {https://doi.org/10.11648/j.ajasr.20220801.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajasr.20220801.13},
  abstract = {In everyday life human faces shock waves and rarefaction largely in their surroundings. Hence it’s necessary to know the behavior of these waves to protect destructive effects. The aim of this work to observe the propagation of shock and rarefaction waves in various dynamics due to solve non-linear hyperbolic inviscid Burgers’ equation numerically. The models adopted here two numerical schemes which enable us to solve non-linear hyperbolic Burgers’ equation numerically. The first order explicit upwind scheme (EUDS) and second order Lax-Wendroff schemes are used to solve this equation to improve our understanding of the numerical diffusion (smearing) and oscillations that can be present when using such schemes. In order to understand the behavior of the solution we use method of characteristics to find the exact solution of inviscid Burgers’ equation. Numerical solutions are studied for different initial conditions and the shock and rarefaction waves are investigated for Riemann problem. We present stability analysis of the schemes and establish stability condition which leads to determine time step selection in terms of spatial step size with maximum initial value. Numerical result for these schemes are compared with an exact solution of inviscid Burgers’ equation in terms of accuracy by error estimation. The numerical features of the rate of convergence are presented graphically. This analysis helps us to understand a wide range of physical phenomenon of the properties of wave as well as saves in several aspects in real life.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation
AU  - Kamrul Hasan
AU  - Humaira Takia
AU  - Muhammad Masudur Rahaman
AU  - Mehedi Hasan Sikdar
AU  - Bellal Hossain
AU  - Khokon Hossen
Y1  - 2022/04/26
PY  - 2022
N1  - https://doi.org/10.11648/j.ajasr.20220801.13
DO  - 10.11648/j.ajasr.20220801.13
T2  - American Journal of Applied Scientific Research
JF  - American Journal of Applied Scientific Research
JO  - American Journal of Applied Scientific Research
SP  - 18
EP  - 24
PB  - Science Publishing Group
SN  - 2471-9730
UR  - https://doi.org/10.11648/j.ajasr.20220801.13
AB  - In everyday life human faces shock waves and rarefaction largely in their surroundings. Hence it’s necessary to know the behavior of these waves to protect destructive effects. The aim of this work to observe the propagation of shock and rarefaction waves in various dynamics due to solve non-linear hyperbolic inviscid Burgers’ equation numerically. The models adopted here two numerical schemes which enable us to solve non-linear hyperbolic Burgers’ equation numerically. The first order explicit upwind scheme (EUDS) and second order Lax-Wendroff schemes are used to solve this equation to improve our understanding of the numerical diffusion (smearing) and oscillations that can be present when using such schemes. In order to understand the behavior of the solution we use method of characteristics to find the exact solution of inviscid Burgers’ equation. Numerical solutions are studied for different initial conditions and the shock and rarefaction waves are investigated for Riemann problem. We present stability analysis of the schemes and establish stability condition which leads to determine time step selection in terms of spatial step size with maximum initial value. Numerical result for these schemes are compared with an exact solution of inviscid Burgers’ equation in terms of accuracy by error estimation. The numerical features of the rate of convergence are presented graphically. This analysis helps us to understand a wide range of physical phenomenon of the properties of wave as well as saves in several aspects in real life.
VL  - 8
IS  - 1
ER  -

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