This paper is devoted to numerically solving the Burgers’ equation in unbounded region, which describes the nonlinear wave propagation and diffusion effect. How to numerically and efficiently solve this problem remains a challenge due to the principal difficulties of not only the unboundedness but also nonlinearity. This paper aims to design the discrete transparent boundary conditions of Burgers’ equation to overcome the unboundedness. The presence of a nonlinear term in the equation makes it challenging to derive suitable artificial boundary conditions. To deal with the nonlinear term, a linear parabolic equation is obtained by applying the well-known Cole-Hopf transformation to the Burgers’ equation. Employing the
-transformation, we then establish the discrete transparent boundary conditions for the linearized problem. Subsequently, the original equation is reduced to an initial boundary value problem, that can be efficiently solved by the finite difference method. The numerical analysis is reported rigorously. Some numerical results are given to demonstrate the accuracy and feasibility of the proposed method.
References
[1]
Burgers, J.M. (1940) Application of a Model System to Illustrate Some Points of the Statistical Theory of Free Turbulence. Proc. Roy. Neth. Acad. Sci. Amsterdam, 43, 2-12.
[2]
Zabusky, N.J. and Kruskal, M.D. (1965) Interaction of “Solitons” in a Collisionless Plasma and the Recurrence of Initial States. Physical Review Letters, 15, 240-243. https://doi.org/10.1103/physrevlett.15.240
[3]
Chen, C., Zhang, X. and Liu, Z. (2020) A High-Order Compact Finite Difference Scheme and Precise Integration Method Based on Modified Hopf-Cole Transformation for Numerical Simulation of N-Dimensional Burgers’ System. Applied Mathematics and Computation, 372, Article ID: 125009. https://doi.org/10.1016/j.amc.2019.125009
[4]
Hopf, E. (1950) The Partial Differential Equation . Communications on Pure and Applied Mathematics, 3, 201-230. https://doi.org/10.1002/cpa.3160030302
[5]
Cole, J.D. (1951) On a Quasi-Linear Parabolic Equation Occurring in Aerodynamics. Quarterly of Applied Mathematics, 9, 225-236. https://doi.org/10.1090/qam/42889
[6]
Sun, H. and Sun, Z. (2014) On Two Linearized Difference Schemes for Burgers’ Equation. International Journal of Computer Mathematics, 92, 1160-1179. https://doi.org/10.1080/00207160.2014.927059
[7]
Xu, P. and Sun, Z. (2008) A Second-Order Accurate Difference Scheme for the Two-Dimensional Burgers’ System. Numerical Methods for Partial Differential Equations, 25, 172-194. https://doi.org/10.1002/num.20335
[8]
Tamsir, M., Srivastava, V.K. and Jiwari, R. (2016) An Algorithm Based on Exponential Modified Cubic B-Spline Differential Quadrature Method for Nonlinear Burgers’ Equation. Applied Mathematics and Computation, 290, 111-124. https://doi.org/10.1016/j.amc.2016.05.048
[9]
Abdullah, M., Yaseen, M. and De la Sen, M. (2022) An Efficient Collocation Method Based on Hermite Formula and Cubic B-Splines for Numerical Solution of the Burgers’ Equation. Mathematics and Computers in Simulation, 197, 166-184. https://doi.org/10.1016/j.matcom.2022.02.013
[10]
Egidi, N., Maponi, P. and Quadrini, M. (2018) An Integral Equation Method for the Numerical Solution of the Burgers Equation. Computers & Mathematics with Applications, 76, 35-44. https://doi.org/10.1016/j.camwa.2018.04.002
[11]
Li, L. and Li, D. (2020) Exact Solutions and Numerical Study of Time Fractional Burgers’ Equations. Applied Mathematics Letters, 100, Article ID: 106011. https://doi.org/10.1016/j.aml.2019.106011
[12]
Zhang, Q., Qin, Y., Wang, X. and Sun, Z. (2021) The Study of Exact and Numerical Solutions of the Generalized Viscous Burgers’ Equation. Applied Mathematics Letters, 112, Article ID: 106719. https://doi.org/10.1016/j.aml.2020.106719
[13]
Verma, A.K., Rawani, M.K. and Cattani, C. (2021) A Numerical Scheme for a Class of Generalized Burgers’ Equation Based on Haar Wavelet Nonstandard Finite Difference Method. Applied Numerical Mathematics, 168, 41-54. https://doi.org/10.1016/j.apnum.2021.05.019
[14]
Dağ, İ., Irk, D. and Saka, B. (2005) A Numerical Solution of the Burgers’ Equation Using Cubic B-Splines. Applied Mathematics and Computation, 163, 199-211. https://doi.org/10.1016/j.amc.2004.01.028
[15]
Yaseen, M. and Abbas, M. (2019) An Efficient Computational Technique Based on Cubic Trigonometric B-Splines for Time Fractional Burgers’ Equation. International Journal of Computer Mathematics, 97, 725-738. https://doi.org/10.1080/00207160.2019.1612053
[16]
Jiwari, R. (2012) A Haar Wavelet Quasilinearization Approach for Numerical Simulation of Burgers’ Equation. Computer Physics Communications, 183, 2413-2423. https://doi.org/10.1016/j.cpc.2012.06.009
[17]
Han, H. and Huang, Z. (2004) Exact Artificial Boundary Conditions for the Schrödinger Equation in . Communications in Mathematical Sciences, 2, 79-94. https://doi.org/10.4310/cms.2004.v2.n1.a5
[18]
Ehrhardt, M. and Zheng, C. (2008) Exact Artificial Boundary Conditions for Problems with Periodic Structures. Journal of Computational Physics, 227, 6877-6894. https://doi.org/10.1016/j.jcp.2008.03.042
[19]
Zheng, C. (2007) Numerical Solution to the Sine-Gordon Equation Defined on the Whole Real Axis. SIAM Journal on Scientific Computing, 29, 2494-2506. https://doi.org/10.1137/050640643
[20]
Zhang, J., Xu, Z. and Wu, X. (2008) Unified Approach to Split Absorbing Boundary Conditions for Nonlinear Schrödinger Equations. Physical Review E, 78, Article ID: 026709. https://doi.org/10.1103/physreve.78.026709
[21]
Zhang, J., Xu, Z. and Wu, X. (2009) Unified Approach to Split Absorbing Boundary Conditions for Nonlinear Schrödinger Equations: Two-Dimensional Case. Physical Review E, 79, Article ID: 046711. https://doi.org/10.1103/physreve.79.046711
[22]
Li, H., Wu, X. and Zhang, J. (2011) Local Absorbing Boundary Conditions for Nonlinear Wave Equation on Unbounded Domain. Physical Review E, 84, Article ID: 036707. https://doi.org/10.1103/physreve.84.036707
[23]
Arnold, A. and Ehrhardt, M. (1998) Discrete Transparent Boundary Conditions for Wide Angle Parabolic Equations in Underwater Acoustics. Journal of Computational Physics, 145, 611-638. https://doi.org/10.1006/jcph.1998.6043
[24]
Ehrhardt, M. (2001) Discrete Artificial Boundary Conditions. Ph.D. Thesis, Technische Universität Berlin.
[25]
Arnold, A. and Ehrhardt, M. (2001) Discrete Transparent Boundary Conditions for the Schrödinger Equation. Revista di Mathematica della Universita di Parma, 6, 57-108.
[26]
Tsynkov, S.V. (1998) Numerical Solution of Problems on Unbounded Domains. a Review. Applied Numerical Mathematics, 27, 465-532. https://doi.org/10.1016/s0168-9274(98)00025-7
[27]
Givoli, D. (2004) High-Order Local Non-Reflecting Boundary Conditions: A Review. Wave Motion, 39, 319-326. https://doi.org/10.1016/j.wavemoti.2003.12.004
[28]
Antoine, X., Arnold, A., Besse, C., Ehrhardt, M. and Schädle, A. (2008) A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations. Communications in Computational Physics, 4, 729-796.
[29]
Han, H., Xu, Z. and Wu, X. (2006) Artificial Boundary Method for Burgers’ Equation Using Non-Linear Boundary Conditions. Journal of Computational Mathematics, 24, 295-304.
[30]
Wu, X. and Zhang, J. (2008) Artificial Boundary Method for Two-Dimensional Burgers’ Equation. Computers & Mathematics with Applications, 56, 242-256. https://doi.org/10.1016/j.camwa.2007.11.047
