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Year 2021, Volume: 9 Issue: 4, 158 - 169, 31.12.2021
https://doi.org/10.36753/mathenot.947552

Abstract

References

  • [1] Fletcher,C.A.J.,Generatingexactsolutionsofthetwo-dimensionalBurgers’equations,Int.J.forNumer.Meth. Fluids, 3 (1983),3, pp. 213-216. https://doi.org/10.1002/fld.1650030302
  • [2] C.A.J.Fletcher,Acomparisonoffiniteelementandfinitedifferencesolutionsoftheone-andtwo-dimensional Burgers’ equations, Journal of Computational Physics, 51(1), 159-188 (1983). https://doi.org/10.1016/0021- 9991(83)90085-2
  • [3] Goyon,O.,MultilevelSchemesforSolvingUnsteadyEquations,Int.J.Numer.Meth.Fluids,22(1996),10,pp. 937-959
  • [4] ArshedAli,Siraj-ul-IslamandSirajulHaq(2009)AComputationalMeshfreeTechniquefortheNumericalSo- lution of the Two-Dimensional Coupled Burgers’ Equations, International Journal for Computational Methods in Engineering Science and Mechanics, 10:5, 406-422. https://doi.org/10.1080/15502280903108016
  • [5] Jain,P.C.,Holla,D.N.,NumericalsolutionsofcoupledBurgers’equation,Int.J.Numer.Meth.,13(1978),4,pp. 213-222. https://doi.org/10.1016/0020-7462(78)90024-0
  • [6] Bahadır, A. R. A fully implicit finite-difference scheme for two-dimensional Burgers’ equations, Applied Mathematics and Computation, Applied Mathematics and Computation 137 (2003) 131–137. https://doi.org/10.1016/S0096-3003(02)00091-7
  • [7] A.H. Khater, R.S. Temsah, M.M. Hassan, A Chebyshev spectral collocation method for solving Burgers’-type equations, Journal of Computational and Applied Mathematics 222 (2008) 333–350. https://doi.org/10.1016/j.cam.2007.11.007
  • [8] R.C.MittalandRamJiwari,DifferentialQuadratureMethodforTwo-DimensionalBurgers’Equations,Inter- national Journal for Computational Methods in Engineering Science and Mechanics, 10:450–459, 2009. DOI: 10.1080/15502280903111424
  • [9] W.Liao,Afourth-orderfinite-differencemethodforsolvingthesystemoftwo-dimensionalBurgers’equations, Int. J. Numer. Meth. Fluids 2010; 64:565–590. https://doi.org/10.1002/fld.2163
  • [10] H. Zhu, H. Shu, M. Ding, Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method, Computers and Mathematics with Applications 60 (2010) 840-848. https://doi.org/10.1016/j.camwa.2010.05.031
  • [11] V.K.Srivastava,M.Tamsir,U.Bhardwaj,YVSSSanyasiraju,Crank-NicolsonSchemeforNumericalSolutions of Two-dimensional Coupled Burgers’ Equations, International Journal of Scientific & Engineering Research 2(5), pp1-6 May-2011
  • [12] MohammadTamsir,VineetKumarSrivastava,Asemi-implicitfinite-differenceapproachfortwo-dimensional coupled Burgers equations, International Journal of Scientific & Engineering Research, 2(6), pp. 46-51, June-2011, ISSN 2229-5518
  • [13] V. K. Srivastava, M. Tamsir, Crank-Nicolson Semi-Implicit Approach For Numerical Solutions of Two- Di- mensional Coupled Nonlinear Burgers Equations, Int. J. of Applied Mechanics and Engineering, 2012, 17(2), pp.571-581
  • [14] S. Thakar and S.Wani ,Linear Method For Two Dimensional Burgers Equation,Ultra Scientist Vol. 25(1)A, 156-168 (2013). Two dimensional coupled Burgers equations by Rubin-Graves Type Linearization 169
  • [15] VineetK.Srivastava,MukeshK.Awasthi,andSaritaSingh,Animplicitlogarithmicfinite-differencetechnique for two dimensional coupled viscous Burgers’ equation, AIP Advances 3, 122105 (2013); doi: 10.1063/1.4842595
  • [16] VineetK.Srivastava,SaritaSingh,andMukeshK.Awasthi,NumericalsolutionsofcoupledBurgersequations by an implicit finite difference scheme, AIP Advances 3(8), 082131 (2013); doi: 10.1063/1.4820355
  • [17] VineetK.SrivastavaandBrajeshKumarSingh,Arobustfinitedifferenceschemeforthenumericalsolutionsof two dimensional time dependent coupled nonlinear Burgers equations, Int. J. of Appl. Math and Mech. 10 (7): 28-39, 2014.
  • [18] L.Zhang,L.WangandX.Ding,Exactfinite-differenceschemeandnonstandardfinite-differenceschemefor coupled Burgers equation, Advances in Difference Equations 2014, 2014:122. DOI: 10.1186/1687-1847-2014-122
  • [19] R C Mittal Amit Tripathi , (2015), Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-spline finite elements, Engineering Computations, Vol. 32 Iss 5 pp. 1275 - 1306. https://doi.org/10.1108/EC-04-2014-0067
  • [20] Mohammad Tamsir, VineetK. Srivastava, Ram Jiwari, An algorithm based on exponential modified cubic B- spline differential quadrature method for nonlinear Burgers’ equation, Applied Mathematics and Computation, 290 (2016) 111–124. https://doi.org/10.1016/j.amc.2016.05.048
  • [21] T. Zhanlav, O. Chuluunbaatar, V. Ulziibayar, Higher-Order Numerical Solution of Two-Dimensional Coupled Burgers Equations, American Journal of Computational Mathematics, 2016, 6, 120-129. DOI: 10.4236/ajcm.2016.62013
  • [22] NgondiepE.Anefficientthree-levelexplicittime-splitschemeforsolvingtwo-dimensionalunsteadynonlinear coupled Burgers’ equations. Int J Numer Meth Fluids. 2020;92:266–284. https://doi.org/10.1002/fld.4783
  • [23] M. Saqib, S. Hasnain and D. S. Mashat, Highly Efficient Computtaional Methods for Two Dimensional Coupled Nonlinear Unsteady Convection-Diffusion Problems, IEEE Access, Vol.5, 2017. DOI: 10.1109/ACCESS.2017.2699320
  • [24] F.W.WubsandE.D.deGoede,Anexplicit-implicitmethodforaclassoftime-dependentpartialdifferentialequations, Appl. Numer. Math., 9 (1992) 157-181. https://doi.org/10.1016/0168-9274(92)90012-3
  • [25] Y.Chai,J.Ouyang,AppropriatestabilizedGalerkinapproachesforsolvingtwo-dimensionalcoupledBurgers’ equations at high Reynolds numbers, Computers and Mathematics with Applications 79 (2020) 1287–1301. https://doi.org/10.1016/j.camwa.2019.08.036
  • [26] S.G.RubinandR.A.Graves,ACubicSplineApproximationforProblemsinFluidMechanics,NASA,Washington, D.C., October, 1975.
  • [27] H.S.Shukla,M.Tamsir,V.K.Srivastava,J.Kumar,NumericalSolutionoftwodimensionalcoupledviscousBurgers’ Equation using the Modified Cubic B-Spline Differential Quadrature Method, ArciheX. DOI: 10.1063/1.4902507.

Numerical Simulation of Two Dimensional Coupled Burgers Equations by Rubin-Graves Type Linearization

Year 2021, Volume: 9 Issue: 4, 158 - 169, 31.12.2021
https://doi.org/10.36753/mathenot.947552

Abstract

In the present article, the numerical solution of the two-dimensional coupled Burgers equation has been sought by finite difference method based on Rubin-Graves type linearization. Three models with appropriate initial and boundary conditions are applied to the problem. In order to show the accuracy of the method, the error norms $L_{2}$, $L_{\infty}$ are computed. The error norms $L_{2}$, $L_{\infty}$ of the obtained numerical solutions are compared with the error norms of some of the numerical solutions in the literature.

References

  • [1] Fletcher,C.A.J.,Generatingexactsolutionsofthetwo-dimensionalBurgers’equations,Int.J.forNumer.Meth. Fluids, 3 (1983),3, pp. 213-216. https://doi.org/10.1002/fld.1650030302
  • [2] C.A.J.Fletcher,Acomparisonoffiniteelementandfinitedifferencesolutionsoftheone-andtwo-dimensional Burgers’ equations, Journal of Computational Physics, 51(1), 159-188 (1983). https://doi.org/10.1016/0021- 9991(83)90085-2
  • [3] Goyon,O.,MultilevelSchemesforSolvingUnsteadyEquations,Int.J.Numer.Meth.Fluids,22(1996),10,pp. 937-959
  • [4] ArshedAli,Siraj-ul-IslamandSirajulHaq(2009)AComputationalMeshfreeTechniquefortheNumericalSo- lution of the Two-Dimensional Coupled Burgers’ Equations, International Journal for Computational Methods in Engineering Science and Mechanics, 10:5, 406-422. https://doi.org/10.1080/15502280903108016
  • [5] Jain,P.C.,Holla,D.N.,NumericalsolutionsofcoupledBurgers’equation,Int.J.Numer.Meth.,13(1978),4,pp. 213-222. https://doi.org/10.1016/0020-7462(78)90024-0
  • [6] Bahadır, A. R. A fully implicit finite-difference scheme for two-dimensional Burgers’ equations, Applied Mathematics and Computation, Applied Mathematics and Computation 137 (2003) 131–137. https://doi.org/10.1016/S0096-3003(02)00091-7
  • [7] A.H. Khater, R.S. Temsah, M.M. Hassan, A Chebyshev spectral collocation method for solving Burgers’-type equations, Journal of Computational and Applied Mathematics 222 (2008) 333–350. https://doi.org/10.1016/j.cam.2007.11.007
  • [8] R.C.MittalandRamJiwari,DifferentialQuadratureMethodforTwo-DimensionalBurgers’Equations,Inter- national Journal for Computational Methods in Engineering Science and Mechanics, 10:450–459, 2009. DOI: 10.1080/15502280903111424
  • [9] W.Liao,Afourth-orderfinite-differencemethodforsolvingthesystemoftwo-dimensionalBurgers’equations, Int. J. Numer. Meth. Fluids 2010; 64:565–590. https://doi.org/10.1002/fld.2163
  • [10] H. Zhu, H. Shu, M. Ding, Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method, Computers and Mathematics with Applications 60 (2010) 840-848. https://doi.org/10.1016/j.camwa.2010.05.031
  • [11] V.K.Srivastava,M.Tamsir,U.Bhardwaj,YVSSSanyasiraju,Crank-NicolsonSchemeforNumericalSolutions of Two-dimensional Coupled Burgers’ Equations, International Journal of Scientific & Engineering Research 2(5), pp1-6 May-2011
  • [12] MohammadTamsir,VineetKumarSrivastava,Asemi-implicitfinite-differenceapproachfortwo-dimensional coupled Burgers equations, International Journal of Scientific & Engineering Research, 2(6), pp. 46-51, June-2011, ISSN 2229-5518
  • [13] V. K. Srivastava, M. Tamsir, Crank-Nicolson Semi-Implicit Approach For Numerical Solutions of Two- Di- mensional Coupled Nonlinear Burgers Equations, Int. J. of Applied Mechanics and Engineering, 2012, 17(2), pp.571-581
  • [14] S. Thakar and S.Wani ,Linear Method For Two Dimensional Burgers Equation,Ultra Scientist Vol. 25(1)A, 156-168 (2013). Two dimensional coupled Burgers equations by Rubin-Graves Type Linearization 169
  • [15] VineetK.Srivastava,MukeshK.Awasthi,andSaritaSingh,Animplicitlogarithmicfinite-differencetechnique for two dimensional coupled viscous Burgers’ equation, AIP Advances 3, 122105 (2013); doi: 10.1063/1.4842595
  • [16] VineetK.Srivastava,SaritaSingh,andMukeshK.Awasthi,NumericalsolutionsofcoupledBurgersequations by an implicit finite difference scheme, AIP Advances 3(8), 082131 (2013); doi: 10.1063/1.4820355
  • [17] VineetK.SrivastavaandBrajeshKumarSingh,Arobustfinitedifferenceschemeforthenumericalsolutionsof two dimensional time dependent coupled nonlinear Burgers equations, Int. J. of Appl. Math and Mech. 10 (7): 28-39, 2014.
  • [18] L.Zhang,L.WangandX.Ding,Exactfinite-differenceschemeandnonstandardfinite-differenceschemefor coupled Burgers equation, Advances in Difference Equations 2014, 2014:122. DOI: 10.1186/1687-1847-2014-122
  • [19] R C Mittal Amit Tripathi , (2015), Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-spline finite elements, Engineering Computations, Vol. 32 Iss 5 pp. 1275 - 1306. https://doi.org/10.1108/EC-04-2014-0067
  • [20] Mohammad Tamsir, VineetK. Srivastava, Ram Jiwari, An algorithm based on exponential modified cubic B- spline differential quadrature method for nonlinear Burgers’ equation, Applied Mathematics and Computation, 290 (2016) 111–124. https://doi.org/10.1016/j.amc.2016.05.048
  • [21] T. Zhanlav, O. Chuluunbaatar, V. Ulziibayar, Higher-Order Numerical Solution of Two-Dimensional Coupled Burgers Equations, American Journal of Computational Mathematics, 2016, 6, 120-129. DOI: 10.4236/ajcm.2016.62013
  • [22] NgondiepE.Anefficientthree-levelexplicittime-splitschemeforsolvingtwo-dimensionalunsteadynonlinear coupled Burgers’ equations. Int J Numer Meth Fluids. 2020;92:266–284. https://doi.org/10.1002/fld.4783
  • [23] M. Saqib, S. Hasnain and D. S. Mashat, Highly Efficient Computtaional Methods for Two Dimensional Coupled Nonlinear Unsteady Convection-Diffusion Problems, IEEE Access, Vol.5, 2017. DOI: 10.1109/ACCESS.2017.2699320
  • [24] F.W.WubsandE.D.deGoede,Anexplicit-implicitmethodforaclassoftime-dependentpartialdifferentialequations, Appl. Numer. Math., 9 (1992) 157-181. https://doi.org/10.1016/0168-9274(92)90012-3
  • [25] Y.Chai,J.Ouyang,AppropriatestabilizedGalerkinapproachesforsolvingtwo-dimensionalcoupledBurgers’ equations at high Reynolds numbers, Computers and Mathematics with Applications 79 (2020) 1287–1301. https://doi.org/10.1016/j.camwa.2019.08.036
  • [26] S.G.RubinandR.A.Graves,ACubicSplineApproximationforProblemsinFluidMechanics,NASA,Washington, D.C., October, 1975.
  • [27] H.S.Shukla,M.Tamsir,V.K.Srivastava,J.Kumar,NumericalSolutionoftwodimensionalcoupledviscousBurgers’ Equation using the Modified Cubic B-Spline Differential Quadrature Method, ArciheX. DOI: 10.1063/1.4902507.
There are 27 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Murat Yağmurlu 0000-0003-1593-0254

Abdulnasır Gagir This is me 0000-0003-1029-4447

Publication Date December 31, 2021
Submission Date June 3, 2021
Acceptance Date October 29, 2021
Published in Issue Year 2021 Volume: 9 Issue: 4

Cite

APA Yağmurlu, M., & Gagir, A. (2021). Numerical Simulation of Two Dimensional Coupled Burgers Equations by Rubin-Graves Type Linearization. Mathematical Sciences and Applications E-Notes, 9(4), 158-169. https://doi.org/10.36753/mathenot.947552
AMA Yağmurlu M, Gagir A. Numerical Simulation of Two Dimensional Coupled Burgers Equations by Rubin-Graves Type Linearization. Math. Sci. Appl. E-Notes. December 2021;9(4):158-169. doi:10.36753/mathenot.947552
Chicago Yağmurlu, Murat, and Abdulnasır Gagir. “Numerical Simulation of Two Dimensional Coupled Burgers Equations by Rubin-Graves Type Linearization”. Mathematical Sciences and Applications E-Notes 9, no. 4 (December 2021): 158-69. https://doi.org/10.36753/mathenot.947552.
EndNote Yağmurlu M, Gagir A (December 1, 2021) Numerical Simulation of Two Dimensional Coupled Burgers Equations by Rubin-Graves Type Linearization. Mathematical Sciences and Applications E-Notes 9 4 158–169.
IEEE M. Yağmurlu and A. Gagir, “Numerical Simulation of Two Dimensional Coupled Burgers Equations by Rubin-Graves Type Linearization”, Math. Sci. Appl. E-Notes, vol. 9, no. 4, pp. 158–169, 2021, doi: 10.36753/mathenot.947552.
ISNAD Yağmurlu, Murat - Gagir, Abdulnasır. “Numerical Simulation of Two Dimensional Coupled Burgers Equations by Rubin-Graves Type Linearization”. Mathematical Sciences and Applications E-Notes 9/4 (December 2021), 158-169. https://doi.org/10.36753/mathenot.947552.
JAMA Yağmurlu M, Gagir A. Numerical Simulation of Two Dimensional Coupled Burgers Equations by Rubin-Graves Type Linearization. Math. Sci. Appl. E-Notes. 2021;9:158–169.
MLA Yağmurlu, Murat and Abdulnasır Gagir. “Numerical Simulation of Two Dimensional Coupled Burgers Equations by Rubin-Graves Type Linearization”. Mathematical Sciences and Applications E-Notes, vol. 9, no. 4, 2021, pp. 158-69, doi:10.36753/mathenot.947552.
Vancouver Yağmurlu M, Gagir A. Numerical Simulation of Two Dimensional Coupled Burgers Equations by Rubin-Graves Type Linearization. Math. Sci. Appl. E-Notes. 2021;9(4):158-69.

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