Runge-Kutta-Nystrom Methods For Solving Oscillatory Problems
New Runge-Kutta-Nyström (RKN) methods are derived for solving system of second-order Ordinary Differential Equations (ODEs) in which the solutions are in the oscillatory form. The dispersion and dissipation relations are imposed to get methods with the highest possible order of dispersion and dis...
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| Main Author: | |
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| Format: | Thesis |
| Language: | English |
| Published: |
2010
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| Online Access: | http://psasir.upm.edu.my/id/eprint/12438/1/FS_2010_23A.pdf http://psasir.upm.edu.my/id/eprint/12438/ |
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| Summary: | New Runge-Kutta-Nyström (RKN) methods are derived for solving system of second-order
Ordinary Differential Equations (ODEs) in which the solutions are in the oscillatory form.
The dispersion and dissipation relations are imposed to get methods with the highest possible
order of dispersion and dissipation. The derivation of Embedded Explicit RKN (ERKN)
methods for variable step size codes are also given. The strategies in choosing the free
parameters are also discussed. We analyze the numerical behavior of the RKN and ERKN
methods both theoretically and experimentally and comparisons are made over the existing
methods.
In the second part of this thesis, a Block Embedded Explicit RKN (BERKN) method are
developed. The implementation of BERKN method is discussed. The numerical results are
compared with non block method. We find that the new code on Block Embedded Explicit
RKN (BERKN) method is more efficient for solving system of second-order ODEs directly.
Next, we discussed the derivation of Diagonally Implicit RKN (DIRKN) methods for solving
stiff second order ODEs in which the solutions are oscillating functions. The dispersion and dissipation relations are developed and again are imposed in the derivation of the methods.
For solving oscillatory problems with high frequency, method with P-stability property is
discussed. We also derive the Embedded Diagonally Implicit RKN (EDIRKN) methods for
variable step size codes. To see the preciseness and effectiveness of the methods, the
constant and variable step size codes are developed and numerical results are compared with
current methods given in the literature.
Finally, the Parallel Embedded Explicit RKN (PERKN) method is developed. The parallel
implementation of PERKN on the parallel machine is discussed. The performance of the
PERKN algorithm for solving large system of ODEs are presented. We observe that the
PERKN gives the better performance when solving large system of ODEs.
In conclusion, the new codes developed in this thesis are suitable for solving system of
second-order ODEs in which the solutions are in the oscillatory form. |
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