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

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

MathSciNet review:
759668

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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.

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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