Abstract
The concept of effective order allows for the possibility that the result computed in a Runge-Kutta step is an approximation to some quantity more general than the actual solution at a step point. This generalization is applied here to singly-implicit methods. The limitation that requires severe and inconvenient restrictions on the abscissae in the method is removed under this widening of the order requirement and all that is now needed is that the abscissae be distinct. Implementation questions, such as error estimation, stepsize change and dense output are also considered.
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Butcher, J., Chartier, P. The effective order of singly-implicit Runge-Kutta methods. Numerical Algorithms 20, 269–284 (1999). https://doi.org/10.1023/A:1019176422613
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DOI: https://doi.org/10.1023/A:1019176422613