×

Bounds on the truncation error by finite differences for the Goursat problem. (English) Zbl 0141.33003

PDF BibTeX XML Cite
Full Text: DOI

References:

[1] A. K. Aziz & J. B. Diaz, “On linear hyperbolic equations with initial conditions on higher derivatives.” (To appear.)
[2] R. Courant & D. Hilbert, Methods of Mathematical Physics, Vol. 2, Interscience, New York, 1962. · Zbl 0099.29504
[3] R. Courant, K. Friedrichs, and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100 (1928), no. 1, 32 – 74 (German). · JFM 54.0486.01
[4] 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. · Zbl 0047.11704
[5] Ralph T. Dames, Stability and convergence for a numerical solution of the Goursat problem, J. Math. and Phys. 38 (1959/1960), 42 – 67. · Zbl 0088.34004
[6] J. B. Diaz, On an analogue of the Euler-Cauchy polygon method for the numerical solution of \?_{\?\?}=\?(\?,\?,\?,\?\?,\?_{\?}), Arch. Rational Mech. Anal. 1 (1958), 357 – 390. · Zbl 0084.11501
[7] Kurt Friedrichs and Hans Lewy, Das Anfangswertproblem einer beliebigen nichtlinearen hyperbolischen Differentialgleichung beliebiger Ordnung in zwei Variablen. Existenz, Eindeutigkeit und Abhängigkeitsbereich der Lösung, Math. Ann. 99 (1928), no. 1, 200 – 221 (German). · JFM 54.0520.01
[8] George E. Forsythe and Wolfgang R. Wasow, Finite-difference methods for partial differential equations, Applied Mathematics Series, John Wiley & Sons, Inc., New York-London, 1960. · Zbl 0099.11103
[9] Herbert B. Keller, On the solution of semi-linear hyperbolic systems by unconditionally stable difference methods, Comm. Pure Appl. Math. 14 (1961), 447 – 456. · Zbl 0099.16701
[10] Peter D. Lax, On the stability of difference approximations to solutions of hyperbolic equations with variable coefficients, Comm. Pure Appl. Math. 14 (1961), 497 – 520. · Zbl 0102.11701
[11] Milton Lees, Energy inequalities for the solution of differential equations, Trans. Amer. Math. Soc. 94 (1960), 58 – 73. · Zbl 0104.34903
[12] Milton Lees, The solution of positive-symmetric hyperbolic systems by difference methods, Proc. Amer. Math. Soc. 12 (1961), 195 – 204. · Zbl 0096.29905
[13] Robert H. Moore, A Runge-Kutta procedure for the Goursat Problem in hyperbolic partial differential equations, Arch. Rational Mech. Anal. 7 (1961), 37 – 63. · Zbl 0097.12004
[14] Robert D. Richtmyer, Difference methods for initial-value problems, Interscience tracts in pure and applied mathematics. Iract 4, Interscience Publishers, Inc., New. York, 1957. · Zbl 0079.33702
[15] Hans J. Stetter, On the convergence of characteristic finite-difference methods of high accuracy for quasi-linear hyperbolic equations, Numer. Math. 3 (1961), 321 – 344. · Zbl 0104.10404
[16] Vidar Thomée, Difference methods for two-dimensional mixed problems for hyperbolic first order systems, Arch. Rational Mech. Anal. 8 (1961), 68 – 88. · Zbl 0104.32202
[17] Philipp Frank and Richard v. Mises, Die Differentialund Integralgleichungen der Mechanik und Physik, Mary S. Rosenberg, New York, 1943 (German). · Zbl 0061.16603
[18] H. F. Weinberger, Exact bounds for solutions of hyperbolic equations by finite difference methods, Symposium on the numerical treatment of partial differential equations with real characteristics: Proceedings of the Rome Symposium (28-29-30 January 1959) organized by the Provisional International Computation Centre, Libreria Eredi Virgilio Veschi, Rome, 1959, pp. 87 – 97.
[19] H. F. Weinberger, Error bounds in finite-difference approximation to solutions of symmetric hyperbolic systems, J. Soc. Indust. Appl. Math. 7 (1959), 49 – 75. · Zbl 0092.32704
[20] Burton Wendroff, On centered difference equations for hyperbolic systems, J. Soc. Indust. Appl. Math. 8 (1960), 549 – 555. · Zbl 0096.32502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.