Numerical long-term integration of geodesic equations of motion
Numerical long-term integration of geodesic equations of motion
Numerical long-term integration of geodesic equations of motion
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
<strong>Numerical</strong> properties <strong>of</strong> the <strong>equations</strong>⎛⎜⎝Y 1⎞.Y sScheme is implicit:⎟⎠ =⎛ ⎞ ⎛∑ sy nj=1 a ⎞1jf (Y j )⎜ ⎟ ⎜⎟⎝ . ⎠ + h ⎝∑. ⎠ (5)y sn j=1 a sjf (Y j )⇒ Use Fixed-Point iteration:⎛⎜⎝Y k+11.Y k+1s⎞⎟⎠ =◮ Converges towards unique solution if:[ ]h ∥ (Y) ∥ < 1,(Banach fixed-point theorem)⎛ ⎞ ⎛∑ sy nj=1 a 1jf (Y k ⎞j)⎜ ⎟ ⎜⎟⎝ . ⎠ + h ⎝∑. ⎠ (6)y sn j=1 a sjf (Y k j)∂f∂y8/26 | Jonathan Seyrich (<strong>Numerical</strong> <strong>long</strong>-<strong>term</strong> <strong>integration</strong>) Tübingen | January 21, 2013∀Y
Change language