Parallel Diagonally Implicit Runge-Kutta Methods For Solving Ordinary Differential Equations
This thesis focuses on the derivations of diagonally implicit Runge-Kutta (DIRK) methods with the capability to be implemented by parallel executions. A few new methods are proposed by having sparsity patterns which enable the parallelization of methods. In the first part of the thesis, a fifth o...
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| Format: | Thesis |
| Language: | English English |
| Published: |
2009
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/11983/1/FS_2009_46_A.pdf |
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| Summary: | This thesis focuses on the derivations of diagonally implicit Runge-Kutta (DIRK)
methods with the capability to be implemented by parallel executions. A few new
methods are proposed by having sparsity patterns which enable the parallelization of
methods. In the first part of the thesis, a fifth order DIRK suitable for two processors
parallel executions and DIRK methods of fourth and fifth orders suitable for three
processors are proposed. The executions of these methods are done by using fixed
stepsizes on a set of nonstiff problems. The regions of stability are presented and
numerical results of the methods are compared to the existing methods. Parallel
computations show significant time reduction when solving large systems of nonstiff
ordinary differential equations (ODEs).
The subsequent part of the thesis discusses on embedded DIRK methods suitable for
two processors implementations. Two 4(3) and also two 5(4) embedded DIRK
methods with adequate stability regions to solve stiff ODEs are proposed. Numerical experiments on stiff test problems are done based on variable stepsize strategy. An
existing code for solving stiff ODEs suitable for embedded DIRK with equal
diagonal elements is modified to accommodate the new methods with alternate
diagonal elements. Comparisons on numerical results to existing methods show a
competitive efficiency when solving small systems of stiff ODEs.
A parallel code is developed with the same capability of the modified sequential code
to handle stiff ODEs, linear and nonlinear problems. All algorithms are written in C
language and the parallel code is implemented on Sun Fire V1280 distributed
memory system. Three large scales of stiff ODEs are used to measure the parallel
performances of the new embedded methods. Results show that speedups increased
as the dimensions of the problems gets larger which is a significant contribution in
reducing the cost of computations. |
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