Abstract
In this paper, one-dimensional Burgers equation and one-dimensional Laval nozzle flow Euler equation are numerically solved, and the stability of the numerical solutions is analyzed theoretically. In order to satisfy the stability of the numerical solution, the explicit MacCormack scheme is used to obtain the stable solution of the computer numerical simulation. In addition, the numerical and analytical solutions of the Euler equation for one-dimensional Laval nozzle flow are compared, and the results are completely consistent, which verifies the correctness of the numerical solution.
Similar content being viewed by others
References
Choudhary, M.K., Karki, K.C., Patankar, S.V.: Mathematical modeling of heat transfer, condensation, and cappilary flow in porous insulation on a cold pipe. Int. J. Heat Mass Transfer 47, 5629–5638 (2004)
Gibsona, P.W., Charmchib, M.: Modeling convection/diffusion processes in porous textiles with inclusion of humidity-dependent air permeability. Int. Commun. Heat Mass Transfer 24, 709–724 (1997)
Huang, C.M., Vandewalle, S.: Unconditionally stable difference methods for delay partial differential equations. Numer. Math. 3(122), 579–601 (2012)
Gess, B., Liu, W., Schenke, A.: Random attractors for locally monotone stochastic partial differential equations. J. Diff. Eq. 269, 3414–3455 (2020)
Zhou, H., Huang, D.Q., Wang, W.S.: Some new difference inequalities and an application To discrete-time control systems. J. Appl. Mathematics: 214609 (2012)
Moutsinga, O.: Burgers’ equation and the sticky particles model. J. Math. Phys. 6(53), 063709 (2012)
Adams, D.M.: Application of the hydraulic analogy to axisymmetric nonideal compressible gas systems. J. Spacecr. Rockets https://doi.org/10.2514/3.28866 (2015)
Elazar, M., Fleurov, V., BarAd, S.: An all-optical event horizon in an optical analogue of a laval nozzle. Analogue Gravity Phenomenology, https://doi.org/10.1007/978-3-319-00266-8-12 (2013)
Marino, F., Maitland, C., Vocke, D.: Emergent geometries and nonlinear-wave dynamics in photon fluids. Sci. Rep. https://doi.org/10.1038/srep23282 (2016)
Burgers, J.M.A.: mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1, 171–199 (1984)
Eymard, R., Fuhrmann, J., Grtner, K.: A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems. Numer. Math. 102, 463–495 (2006)
Wang, J.Y.: Further analysis on stability of delayed neural networks via inequality technique. J. Inequal. Appl. https://doi.org/10.1186/1029-242X-2011-103 (2011)
Hoff, D.: Stability and convergence of finite difference methods for systems of nonlinear reactiondiffusion equations. SIAM J. Numer. Anal. 15, 1161–1177 (1978)
Aksan, E.N.A.: numerical solution of Burgers equation by finite element method constructed on the method of discretization in time. Appl. Math. Comput 170, 895–904 (2005)
Hesameddini, E., Gholampour, R.: Soliton and numerical solutions of the Burgers equation and comparing them. Int. J. Math. Anal. 4, 2547–2564 (2010)
Liao, W., Zhu, J.: Efficient and accurate finite difference schemes for one-dimensional Burgers equation. Int. J. Comput. Math. 88 (12), 2575–2590 (2011)
Rodionov, A.V.: Artificial viscosity to cure the carbuncle phenomenon: the three-dimensional case. J. Comput. Phys. https://doi.org/10.1016/j.jcp.2018.02.001 (2018)
Yu, W.L., Feng, Y.D., Zhou, H., Cao, S.Z., Zhang, X.Y.: Fluid simulation analysis and optimization of micro Laval nozzle. Vacuum and Low Temperature 24(04), 246–250 (2018)
Rodionov, A.V.: Artificial viscosity in godunov-type schemes to cure the carbuncle phenomenon. J. Comput. Phys. https://doi.org/10.1016/j.jcp.2017.05.024 (2017)
Wood, W.L.: An exact solution for Burgers’s equation. Commun. Numer. Methods Eng. 22, 797–798 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, Y., Wang, Y. Numerical Solution and Stability Analysis for a Class of Nonlinear Differential Equations. Int J Theor Phys 60, 2573–2582 (2021). https://doi.org/10.1007/s10773-020-04679-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-020-04679-8