P-Stable Methods of High Exponential Order

Based on the theory fully described in refs. 95 and 96 two P-stable methods of exponential and phase-lag order eight and ten are produced. Based on these methods a new variable-step algorithm is introduced. 

Based on the above family of methods Avdelas and Simos97 have given formulae for the coefficients of the P-stable method of exponential order 2m + 2 for arbitrary m. We note that the coefficients are determined in the computer automatically. The new approach is called generator of methods. This was the first generator of P-stable methods in the literature. We also note that the term generator is first introduced in this paper. 

Simos98 has derived another family of P-stable methods of exponential order 2m -l- 2 for m = 3(1)6. The methods here are of hybrid type and have the form  

P-stable method of exponential order 10 is produced. For the values 

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