Error estimates for finite difference approximations to hyperbolic equations for large time

Author:
William Layton

Journal:
Proc. Amer. Math. Soc. **92** (1984), 425-431

MSC:
Primary 65M10; Secondary 35L99

MathSciNet review:
759668

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Convergence results for bounded time intervals are well known for finite difference approximations to the Cauchy problem for hyperbolic equations. These results typically state that if the initial data is smooth and the approximation is stable in and accurate of order , then the error at time is bounded by , where is the initial data and .

This paper considers the error for long times. It is not possible for the error to be in uniformly in . However, it is shown here that if is a *bounded* domain the error in is bounded by , where is *independent of* . Thus, the global error will grow as more timesteps are taken but the local error will remain uniformly bounded.

**[1]**G. S. S. Ávila and D. G. Costa,*Asymptotic properties of general symmetric hyperbolic systems*, J. Funct. Anal.**35**(1980), no. 1, 49–63. MR**560217**, 10.1016/0022-1236(80)90080-4**[2]**Peter D. Lax and Ralph S. Phillips,*Scattering theory*, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR**0217440****[3]**William J. Layton,*Simplified 𝐿^{∞} estimates for difference schemes for partial differential equations*, Proc. Amer. Math. Soc.**86**(1982), no. 3, 491–495. MR**671222**, 10.1090/S0002-9939-1982-0671222-9**[4]**S. M. Nikol′skiĭ,*Approximation of functions of several variables and imbedding theorems*, Springer-Verlag, New York-Heidelberg., 1975. Translated from the Russian by John M. Danskin, Jr.; Die Grundlehren der Mathematischen Wissenschaften, Band 205. MR**0374877****[5]**Jaak Peetre and Vidar Thomée,*On the rate of convergence for discrete initial-value problems*, Math. Scand.**21**(1967), 159–176 (1969). MR**0255085****[6]**R. D. Richtmeyer and K. W. Morton,*Difference methods for initial value problems*, 2nd ed., Interscience, New York, 1967.**[7]**Gilbert Strang,*Accurate partial difference methods. I. Linear Cauchy problems*, Arch. Rational Mech. Anal.**12**(1963), 392–402. MR**0146970****[8]**Vidar Thomée,*Stability of difference schemes in the maximum-norm*, J. Differential Equations**1**(1965), 273–292. MR**0176240**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
65M10,
35L99

Retrieve articles in all journals with MSC: 65M10, 35L99

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1984-0759668-3

Keywords:
Finite difference method,
hyperbolic equation,
uniform error estimate

Article copyright:
© Copyright 1984
American Mathematical Society