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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[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**255**(1960), 232–252. MR**0119432**, https://doi.org/10.1098/rspa.1960.0065**[3]**Richard Courant, Eugene Isaacson, and Mina Rees,*On the solution of nonlinear hyperbolic differential equations by finite differences*, Comm. Pure. Appl. Math.**5**(1952), 243–255. MR**0053336**, https://doi.org/10.1002/cpa.3160050303**[4]**G. F. D. Duff,*Partial differential equations*, Mathematical expositions no. 9, University of Toronto Press, Toronto, 1956. MR**0078550****[5]**F. G. Friedlander,*Sound pulses in a conducting medium*, Proc. Cambridge Philos. Soc.**55**(1959), 341–367. MR**0110409****[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. Jeffrey and T. Taniuti,*Non-linear wave propagation. With applications to physics and magnetohydrodynamics*, Academic Press, New York-London, 1964. MR**0167137****[8]**Heinz-Otto Kreiss,*On difference approximations of the dissipative type for hyperbolic differential equations*, Comm. Pure Appl. Math.**17**(1964), 335–353. MR**0166937**, https://doi.org/10.1002/cpa.3160170306**[9]**P. D. Lax,*Differential equations, difference equations and matrix theory*, Comm. Pure Appl. Math.**11**(1958), 175–194. MR**0098110**, https://doi.org/10.1002/cpa.3160110203**[10]**Peter Lax and Burton Wendroff,*Systems of conservation laws*, Comm. Pure Appl. Math.**13**(1960), 217–237. MR**0120774**, https://doi.org/10.1002/cpa.3160130205**[11]**Peter D. Lax and Burton Wendroff,*Difference schemes for hyperbolic equations with high order of accuracy*, Comm. Pure Appl. Math.**17**(1964), 381–398. MR**0170484**, https://doi.org/10.1002/cpa.3160170311**[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]***Mathematical methods for digital computers*, John Wiley & Sons, Inc., New York-London, 1960. MR**0117906****[14]**Robert D. Richtmyer and K. W. Morton,*Difference methods for initial-value problems*, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR**0220455****[15]**Gilbert Strang,*Accurate partial difference methods. II. Non-linear problems*, Numer. Math.**6**(1964), 37–46. MR**0166942**, https://doi.org/10.1007/BF01386051

Retrieve articles in *Mathematics of Computation*
with MSC:
65P05

Retrieve articles in all journals with MSC: 65P05

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