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The numerical solution of hyperbolic systems using bicharacteristics


Authors: R. L. Johnston and S. K. Pal
Journal: Math. Comp. 26 (1972), 377-392
MSC: Primary 65P05
DOI: https://doi.org/10.1090/S0025-5718-1972-0305628-6
MathSciNet review: 0305628
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Abstract: This paper is concerned with deriving methods for solving numerically a firstorder quasilinear hyperbolic system. The basic principle of the method is to integrate along one of the bicharacteristics of the system to obtain an equivalent integral system. The numerical methods are then obtained by making suitable approximations to this integral system. Stability and convergence properties are analyzed in some detail. The methods are relatively easy to implement and have been successfully applied to problems in one, two and three space dimensions in such areas as magnetohydrodynamics and dynamic elasticity.


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  • [1] S. D. Baxter, Numerical Methods for the Solution of Hyperbolic Differential Equations with the Aid of Electronic Computers, Ph.D. Thesis, University of Toronto, Toronto, Ont., 1958.
  • [2] D. S. Butler, ``The numerical solution of hyperbolic systems of partial differential equations in three independent variables,'' Proc. Roy. Soc. London Ser. A, v. 255, 1960, pp. 232-252. MR 22 #10193. MR 0119432 (22:10193)
  • [3] R. Courant, E. Isaacson & M. Rees, ``On the solution of non-linear hyperbolic differential equations by finite differences,'' Comm. Pure Appl. Math., v. 5, 1952, pp. 243-255. MR 0053336 (14:756e)
  • [4] G. F. D. Duff, Partial Differential Equations, Math. Expositions, no. 9, Univ. of Toronto Press, Toronto, 1956. MR 17, 1210. MR 0078550 (17:1210i)
  • [5] F. G. Friedlander, ``Sound pulses in a conducting medium,'' Proc. Cambridge Philos. Soc., v. 55, 1959, pp. 341-367. MR 22 #1289. MR 0110409 (22:1289)
  • [6] D. R. Hartree, Some Practical Methods of Using Characteristics in the Calculation of Non-Steady Compressible Flow, Report LA-HU-1, Harvard University, Cambridge, Mass., 1953.
  • [7] A. Jeffréy & T. Taniuti, Non-Linear Wave Propagation. With Applications to Physics and Magnetohydrodynamics, Academic Press, New York, 1964. MR 29 #4410. MR 0167137 (29:4410)
  • [8] H. O. Kreiss, ``On difference approximations of the dissipative type for hyperbolic differential equations,'' Comm. Pure Appl. Math., v. 17, 1964, pp. 335-353. MR 29 #4210. MR 0166937 (29:4210)
  • [9] P. D. Lax, ``Differential equations, difference equations and matrix theory,'' Comm. Pure Appl. Math., v. 11, 1958, pp. 175-194. MR 20 #4572. MR 0098110 (20:4572)
  • [10] P. D. Lax & B. Wendroff, ``Systems of conservation laws,'' Comm. Pure Appl. Math., v. 13, 1960, pp. 217-237. MR 22 #11523. MR 0120774 (22:11523)
  • [11] P. D. Lax & B. Wendroff, ``Difference schemes for hyperbolic equations with high order of accuracy,'' Comm. Pure Appl. Math., v. 17, 1964, pp. 381-398. MR 30 #722. MR 0170484 (30:722)
  • [12] S. K. Pal, Numerical Solution of First-Order Hyperbolic Systems of Partial Differential Equations, Ph.D. Thesis, University of Toronto, 1969; Technical Report #13, Department of Computer Science, University of Toronto, Toronto, Ontario.
  • [13] A. Ralston & H. S. Wilf (Editor), Mathematical Methods for Digital Computers. Vol. I, Wiley, New York, 1960. MR 22 #8680. MR 0117906 (22:8680)
  • [14] R. D. Richtmyer & K. W. Morton, Difference Methods for Initial-Value Problems, 2nd ed., Interscience Tracts in Pure and Appl. Math., no. 4, Interscience, New York, 1967. MR 36 #3515. MR 0220455 (36:3515)
  • [15] W. G. Strang, ``Accurate partial difference methods. II. Non-linear problems,'' Numer. Math., v. 6, 1964, pp. 37-46. MR 29 #4215. MR 0166942 (29:4215)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1972-0305628-6
Keywords: Bicharacteristics, finite difference, hyperbolic system, stability, convergence
Article copyright: © Copyright 1972 American Mathematical Society

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