Introduction to Scientific Computing - Tutorial 13: Recapitulation

Previous HW- Runge-Kutta-Fehlberg method
As k’s do not depend directly on u and t let us expand it in Taylor
serie:
k 1 = f
k 2 = f + c 2 hf t + ha 21 ff u + h2
2 [c2 2 f tt + 2c 2 a 21 ff tu + a21 2 f 2 f uu ]
k 3 = f + c 3 hf t + h(a 31 k 1 + a 32 k 2 )f u
+ h2
2 [c2 3 f tt + 2c 3 (a 31 k 1 + a 32 k 2 )f tu + (a 31 k 1 + a 32 k 2 ) 2 f uu ]
= f + h[c 3 hf t + (a 31 + a 32 )ff u ]
+ h2
2 [c2 3 f tt + 2c 3 (a 31 + a 32 )ff tu + (a 31 + a 32 ) 2 f 2 f uu ]
+h 2 a 32 (c 2 f t + a 21 ff u )f u .
1. Februar 2013 Bojana Rosic Introduction to Scientific Computing Seite 24