Runge–Kutta–Fehlberg method
Encyclopedia
In mathematics
, the Runge–Kutta–Fehlberg method (or Fehlberg method) is an algorithm of numerical analysis
for the numerical solution of ordinary differential equations
. It was developed by the German
mathematician
Erwin Fehlberg and is based on the class of Runge–Kutta methods. The Runge–Kutta–Fehlberg method uses an O(h4) method together with an O(h5) method that uses all of the points of the O(h4) method, and hence is often referred to as an RKF45 method. Similar schemes with different orders have since been developed. By performing one extra calculation than would be required for an RK5 method, the error in the solution can be estimated and controlled and an appropriate step size can be determined automatically, making this method efficient for ordinary problems of automated numerical integration
of ordinary differential equations.
The Butcher tableau is:
The first row of coefficients gives the fourth-order accurate method, and the second row gives the fifth-order accurate method.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Runge–Kutta–Fehlberg method (or Fehlberg method) is an algorithm of numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
for the numerical solution of ordinary differential equations
Numerical ordinary differential equations
Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations...
. It was developed by the German
Germans
The Germans are a Germanic ethnic group native to Central Europe. The English term Germans has referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Erwin Fehlberg and is based on the class of Runge–Kutta methods. The Runge–Kutta–Fehlberg method uses an O(h4) method together with an O(h5) method that uses all of the points of the O(h4) method, and hence is often referred to as an RKF45 method. Similar schemes with different orders have since been developed. By performing one extra calculation than would be required for an RK5 method, the error in the solution can be estimated and controlled and an appropriate step size can be determined automatically, making this method efficient for ordinary problems of automated numerical integration
Numerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...
of ordinary differential equations.
The Butcher tableau is:
| 0 | ||||||
| | 1/4 | 1/4 | |||||
| | 3/8 | 3/32 | 9/32 | ||||
| | 12/13 | 1932/2197 | −7200/2197 | 7296/2197 | |||
| | 1 | 439/216 | −8 | 3680/513 | −845/4104 | ||
| | 1/2 | -8/27 | 2 | −3544/2565 | 1859/4104 | −11/40 | |
| | | 25/216 | 0 | 1408/2565 | 2197/4104 | −1/5 | 0 |
| | | 16/135 | 0 | 6656/12825 | 28561/56430 | −9/50 | 2/55 |
The first row of coefficients gives the fourth-order accurate method, and the second row gives the fifth-order accurate method.