On Automatic Global Error Control in Multistep Methods with Polynomial Interpolation of Numerical Solution

2004 ◽  
pp. 345-354 ◽  
Author(s):  
Gennady Yu. Kulikov ◽  
Sergey K. Shindin
Keyword(s):  
Numerical Solution ◽  
Error Control ◽  
Polynomial Interpolation ◽  
Multistep Methods ◽  
Global Error

scholarly journals Trigonometrically-Fitted Methods: A Review

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1197
Author(s):  
Changbum Chun ◽  
Beny Neta
Keyword(s):  
Initial Value Problems ◽  
Polynomial Interpolation ◽  
Multistep Methods ◽  
Error Reduction ◽  
Trigonometric Polynomials ◽  
Linear Multistep Methods ◽  
Initial Value ◽  
First Order ◽  
The Subject ◽  
Trigonometrically Fitted Methods

Numerical methods for the solution of ordinary differential equations are based on polynomial interpolation. In 1952, Brock and Murray have suggested exponentials for the case that the solution is known to be of exponential type. In 1961, Gautschi came up with the idea of using information on the frequency of a solution to modify linear multistep methods by allowing the coefficients to depend on the frequency. Thus the methods integrate exactly appropriate trigonometric polynomials. This was done for both first order systems and second order initial value problems. Gautschi concluded that “the error reduction is not very substantial unless” the frequency estimate is close enough. As a result, no other work was done in this direction until 1984 when Neta and Ford showed that “Nyström’s and Milne-Simpson’s type methods for systems of first order initial value problems are not sensitive to changes in frequency”. This opened the flood gates and since then there have been many papers on the subject.

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