Geodesic
An algorithm of variable order and step based on stages of the Dormand-Prince method of the eighth order of accuracy
Numerical methods and programming, Tome 8 (2007) no. 4, pp. 317-325

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An inequality is obtained to control the stability of the 13-stage Dormand-Prince method of the eighth order of accuracy. A first-order method with an expanded stability domain is proposed on the basis of the first seven stages. An algorithm of variable order is formulated. Some numerical results for stiff systems are discussed; these results confirm an efficiency increase of the variable-order method in comparison with a fixed-order scheme.
Keywords: stiff systems, explicit methods, stability and accuracy control, variable order methods, ordinary differential equations, one-step difference methods. 
@article{VMP_2007_8_4_a1,
     author = {A. E. Novikov and E. A. Novikov},
     title = {An algorithm of variable order and step based on stages of the {Dormand-Prince} method of the eighth order of accuracy},
     journal = {Numerical methods and programming},
     pages = {317--325},
     publisher = {mathdoc},
     volume = {8},
     number = {4},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2007_8_4_a1/}
}
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A. E. Novikov; E. A. Novikov. An algorithm of variable order and step based on stages of the Dormand-Prince method of the eighth order of accuracy. Numerical methods and programming, Tome 8 (2007) no. 4, pp. 317-325. http://geodesic.mathdoc.fr/item/VMP_2007_8_4_a1/