Numerical Solution of Ordinary and Delay Differential Equations by Runge-Kutta Type Methods
Runge-Kutta methods for the solution of systems of ordinary differential equations (ODEs) are described. To overcome the difficulty in implementing fully implicit Runge-Kutta method and to avoid the limitations of explicit Runge-Kutta method, we resort to Singly Diagonally Implicit Runge-Kutta (S...
Saved in:
| Main Author: | |
|---|---|
| Format: | Thesis |
| Language: | English English |
| Published: |
1999
|
| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/8653/1/FSAS_1999_5_A.pdf |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | Runge-Kutta methods for the solution of systems of ordinary differential
equations (ODEs) are described. To overcome the difficulty in implementing fully
implicit Runge-Kutta method and to avoid the limitations of explicit Runge-Kutta
method, we resort to Singly Diagonally Implicit Runge-Kutta (SDIRK) method,
which is computationally efficient and stiffly stable. Consequently, embedded
SDIRK methods of fourth order five stages in fifth order six stages are constructed.
Their regions of stability are presented and numerical results of the methods are
compared with the existing methods.
Stiff systems of ODEs are solved using implicit formulae and require the use
of Newton-like iteration, which needs a lot of computational effort. If the systems can be partitioned dynamically into stiff and nonstiff subsystems then a more
effective code can be developed. Hence, partitioning strategies are discussed in
detail and numerical results based on two techniques to detect stiffness using
SDIRK methods are compared.
A brief introduction to delay differential equations (DDEs) is given. The
stability properties of SDIRK methods, when applied to DDEs, using Lagrange
interpolation to evaluate the delay term, are investigated.
Finally, partitioning strategies for ODEs are adapted to DDEs and numerical
results based on two partitioning techniques, interval wise partitioning and
componentwise partitioning are tabulated and compared. |
|---|