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4.1.6 Runge-Kutta Method
Figure 4.1:
a) EC formula (= RK of first order);
b) RK of second order
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(Also called half-step method, or Euler-Richardson algorithm.)
A much more powerful method that has found wide application is the RK
algorithm of order 4, as described in the table.
Advantages of RK:
- Self-starting (no preceding needed)
- Adjustable
But:
- Several evaluations of per step; may be too expensive
Stability of RK:
Half-step + relaxation equation:
.
EXERCISE:
Test various algorithms by applying them to an analytically solvable problem,
as the harmonic oscillator or the 2-body Kepler problem. Include in your
code tests that do not rely on the existence of an analytical solution (energy
conservation or such.) Finally, apply the code to more complex problems
such as the anharmonic oscillator or the many-body Kepler problem.
Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001