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Volume 17,     Number 3,     Fall 2009

 

A SYMBOLIC-NUMERIC APPROACH TO THE
SOLUTION OF THE BUTCHER EQUATIONS
SERGEY KHASHIN

Abstract. A new approach based on the use of new simplifying assumptions is introduced. This approach is based on the construction of a graduated finite-dimensional algebra for a given Butcher tableau, which allows us to discover some new families of Runge-Kutta (RK) methods of orders less than or equal to 8. Most of the methods constructed have new features different from those of previously known methods. A new order 9 method has been found having only 13 stages. For all of these families we have found representatives numerically and introduced a method to find their local dimensions. Using numerical information, we also derive analytical solutions in some cases.

 

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