Richardson extrapolation

The Richardson extrapolation method was developed by Lewis Fry Richardson (1881-1953). It can be used when one has the approximations and for a problem due to two different discretizations (with the step sizes and ) in the numerical solution of a problem , and these approximations have been calculated with a -th order method. ${\displaystyle h_{u}}$ ${\displaystyle h_{g}}$ ${\displaystyle U_{u}}$ ${\displaystyle U_{g}}$ ${\displaystyle p}$

If these conditions are met, the extrapolation is

${\displaystyle U_{R}={\frac {U_{u}-U_{g}\left({\frac {h_{u}}{h_{g}}}\right)^{p}}{1-\left({\frac {h_{u}}{h_{g}}}\right)^{p}}}=U_{g}+{\frac {U_{u}-U_{g}}{1-\left({\frac {h_{u}}{h_{g}}}\right)^{p}}}}$

a better approximation for the result.

It is used, for example, in the Romberg integration . The method was used before Richardson by Takebe Katahiro in his calculation of pi (1723).

literature

• Hans-Görg Roos, Hubert Schwetlick: Numerical Mathematics. The basic knowledge for everyone. Vieweg + Teubner Verlag, Stuttgart et al. 1999, ISBN 3-519-00221-3 , p. 125 ( mathematics for engineers and natural scientists ).
• Martin Hermann: Numerical Mathematics. 2nd revised and expanded edition. Oldenbourg Wissenschaftsverlag, Munich et al. 2006, ISBN 3-486-57935-5 , p. 412.
• Guido Walz: The History of Extrapolation Methods in Numerical Analysis. University of Mannheim - Faculty of Mathematics and Computer Science, Mannheim 1991 ( Faculty of Mathematics and Computer Science at the University of Mannheim - Manuscripts 130, ZDB -ID 263563-x ), ( online version at the Mannheim University Library ).

Web links

• Richardson extrapolation ( PDF file; 49 kB)
• Richardson extrapolation on a website from Robert Israel (University of British Columbia)