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 ENG  Vol.13 No.6 , June 2021
Numerical Analysis of Upwind Difference Schemes for Two-Dimensional First-Order Hyperbolic Equations with Variable Coefficients
Abstract: In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.
Cite this paper: Sun, Y. and Yang, Q. (2021) Numerical Analysis of Upwind Difference Schemes for Two-Dimensional First-Order Hyperbolic Equations with Variable Coefficients. Engineering, 13, 306-329. doi: 10.4236/eng.2021.136023.
References

[1]   Li, R.H., Chen, Z.Y., Wu, W., et al. (2000) Generalized Difference Methods for Differential Equations. Marcel Dekker Inc., New York.
https://doi.org/10.1201/9781482270211

[2]   Evans, L.C. (2010) Partial Differential Equations. American Mathematical Society, Providence, Rhode Island.
https://doi.org/10.1090/gsm/019

[3]   Thomas, J.W. (1997) Numerical Partial Differential Equations: Finite Difference Methods. Springer-Verlag, Berlin.

[4]   Li, R.H. and Liu, B. (2009) Numerical Solution of Differential Equation. Higher Education Press, Beijing.

[5]   Garg, N.K., Kurganov, A., Liu, Y., et al. (2021) Semi-Discrete Central-Upwind Rankine-Hugoniot Schemes for Hyperbolic Systems of Conservation Laws. Journal of Computational Physics, 428, 110078.
https://doi.org/10.1016/j.jcp.2020.110078

[6]   Zhao, Z., Zhu, J., Chen, Y.B., Qiu, J.X., et al. (2019) A New Hybrid WENO Scheme for Hyperbolic Conservation Laws. Computers and Fluids, 179, 422-436.
https://doi.org/10.1016/j.compfluid.2018.10.024

[7]   Zhao, Z., Chen, Y.B., Qiu, J.X., et al. (2020) A Hybrid Hermite WENO Scheme for Hyperbolic Conservation Laws. Journal of Computational Physics, 405, 109175.
https://doi.org/10.1016/j.jcp.2019.109175

[8]   Kurganov, A., Petrova, G., Popov, B., et al. (2007) Adaptive Semidiscrete Central-Upwind Schemes for Nonconvex Hyperbolic Conservation Laws. Siam Journal on Scientific Computing, 29, 2381-2401.
https://doi.org/10.1137/040614189

[9]   Kurganov, A. and Petrova, G. (2009) Central-Upwind Schemes for Two-Layer Shallow Water Equations. Siam Journal on Scientific Computing, 31, 1742-1773.
https://doi.org/10.1137/080719091

[10]   Schncke, G., Krais, N., Bolemann, T., Gassner, G.J., et al. (2020) Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes for Hyperbolic Conservation Laws. Journal of Scientific Computing, 82, 69.
https://doi.org/10.1007/s10915-020-01171-7

[11]   Fan, X.F. (2015) High-Order Difference Schemes for the First-Order Hyperbolic Equations with Variable Coefficients. Master’s Dissertation, Southeast University, Nanjing.

[12]   Chen, L. and Ma, H.P. (2013) Dissipative Spectral Element Method for First-Order Linear Hybperbolic Equation. Communication on Applied Mathematics and Computation, 27, 491-500.

 
 
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