Shallow Water Hydraulics by Oscar Castro-Orgaz & Willi H. Hager

Author:Oscar Castro-Orgaz & Willi H. Hager
Language: eng
Format: epub
ISBN: 9783030130732
Publisher: Springer International Publishing

7.2 Basic Numerical Aspects

7.2.1 Remark on Basic Numerical Concepts

Some basic concepts linked to the numerical solution of hyperbolic conservation laws to be used in this chapter are (Hoffman 2001):

Accuracy: The truncation errors are an accuracy measure of the finite-difference (FD) scheme. These result from approximating the derivatives by FDs. In general, the accuracy of a numerical scheme is impaired by both truncation errors and (machine) round-off errors. However, the latter are usually negligible, given their smallness.

Convergence: Reference is made to the relation between the solution produced by the FD scheme and the solution of the original differential equation. Convergence is reached if the FD solution approaches the solution of the differential equation as the spatial and temporal steps, Δx and Δt, tend to zero.

Consistency: This refers to the relation between the algebraic equation produced by the FD scheme and the original differential equation. Consistency is reached if the FD equation approaches the original differential equation as Δx and Δt tend to zero. Consistency is needed to have a convergent solution, but it is not a sufficient condition.

Stability: A numerical scheme is stable if the solution remains bounded as it evolves in time. An unstable solution becomes unbounded due to the accumulation of errors and may result in a crashing of computations.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.