Partitioning Techniques and Their Parallelization for Stiff System of Ordinary Differential Equations

A new code based on variable order and variable stepsize component wise partitioning is introduced to solve a system of equations dynamically. In previous partitioning technique researches, once an equation is identified as stiff, it will remain in stiff subsystem until the integration is completed. In this current technique, the system is treated as nonstiff and any equation that caused stiffness will be treated as stiff equation. However, should the characteristics showed the elements of nonstiffness, and then it will be treated again with Adam method. This process will continue switching from stiff to nonstiff vice versa whenever it is necessary until the interval of integration is completed.Next, a block method with R-points generate R new approximate solution values;is a strategy for solving a system and also for parallelizing ODEs. Partitioning this block method to solve stiff differential equations is a new strategy; it is more efficient and takes less computational time compared to the sequential methods. Two partitioning techniques are constructed, Intervalwise Block Partitioning (IBP) and Componentwise Block Partitioning (CBP). Numerical results are compared as validation of its effectiveness. Intervalwise block partitioning will initially treat the systems of equations as nonstiff and solve them using Adams method, by switching to the Backward Differentiation formula when there is a step failure and indication of stiffness. Componentwise block partitioning will place the necessary equations that cause instability and stiffness into the stiff subsystem and solve using Backward Differentiation Formula, while all other equations will still be treated as non-stiff and solved using Adams formula. Parallelizing the partitioning strategies using Message Passing Interface (MPI) is the most appropriate method to solve large system of equations. Parallelizing the right algorithm in the partitioning code will give a better perfonnance with shorter execution times. The graphs of its performance and execution time, visualize the advantages of parallelizing

Parallel block backward differentiation formulas for solving large systems of ordinary differential equations.

In this paper, parallelism in the solution of Ordinary Differential Equations (ODEs) to increase the computational speed is studied. The focus is the development of parallel algorithm of the two point Block Backward Differentiation Formulas (PBBDF) that can take advantage of the parallel architecture in computer technology. Parallelism is obtained by using Message Passing Interface (MPI).Numerical results are given to validate the efficiency of the PBBDF implementation as compared to the sequential implementation

Componentwise block partitioning: a new strategy to solve stiff ordinary differential equations

Componentwise Block Partitioning is a new strategy to solve stiff ODEs, based on Block Backward Differentiation Formulas (BBDFs), and block of Adam type formulas. In this partitioning technique, the ODEs system is initially solved by Adam formulas until the equation that cause instability and stiffness is identified. Then, the equations that caused instability are placed into stiff subsystem and solved using BBDF. Numerical comparisons with code in the literature such as ode15s show the efficiency of the proposed partitioning technique

Diagonally implicit block backward differentiation formula for solving linear second order ordinary differential equations

The three point block method for solving second order ordinary differential equations (ODEs) directly using constant step size is derived. The reliability of this new method is verified in the numerical results with the improved performance in terms of computation time while maintaining the accuracy. The comparison is presented between the new method and classical backward differentiation formulas (BDF) of order 3

Stability region of 3-point block backward differentiation formula

In this paper, we focus on the stability region of the variable step size of 3-point block backward differentiation formula (VSBBDF) method. The graphs are plotted using MAPLE software. To show the performance of the method, the accuracy is presented in the numerical results to solve first order stiff ordinary differential equations (ODEs)

Penyelesaian persamaan pembezaan biasa kaku menggunakan kaedah blok formulasi beza ke belakang.

Kertas kerja ini membincangkan kaedah Blok Formulasi Beza Ke Belakang (BFBB) bagi menyelesaikan persamaan pembezaan biasa (PPB) jenis kaku. Umumnya, persamaan jenis kaku diselesaikan dengan kaedah tersirat yang melibatkan lelaran Newton yang mengambil masa pengiraan yang panjang. Dengan yang demikian, suatu kaedah blok dua titik menggunakan saiz langkah berubah dibangunkan berdasarkan Formulasi Beza Ke Belakang (FBB). Sebelum ini pengiraan beza pembahagi yang berulangkali dan rumit dilakukan pada setiap langkah dalam pengiraan pekali pembezaan,tetapi dalam kod yang dibangunkan, pengiraan pekali pembezaan hanya dilakukan pada langkah awal sahaja. Dipaparkan perbandingan keputusan berangka antara BFBB dan FBB yang menunjukkan keberkesanan dalam mengurangkan masa pengiraan dan kejituan penyelesaian yang lebih baik apabila kaedah BFBB di gunakan dalam menyelesaikan masalah nilai awal PPB kaku

Fixed Coefficients Block Backward Differentiation Formulas for the Numerical Solution of Stiff Ordinary Differential Equations

This paper focuses on the derivation of implicit 2-point block method based on Backward Differentiation Formula (BDF) which will be called BBDF of variable step size for solving first order stiff initial value problems (IVPs) for Ordinary Differential Equations (ODEs). The method presented is similar to the form of standard BDF. This allows us to store the coefficients of the y values and thus avoiding calculating the differentiation coefficients at each step but robust enough to allow for step size variation.Plots of their regions of absolute stability for the method are also presented. The efficiency of the 2-point BBDF is compared with the conventional variable step variable order BDF(VSVOBDF) method. Numerical results indicate that the resulting 2-point BBDF method outperform the VSVOBDF method in both execution time and accuracy

3-point block backward differentiation formulas for solving fuzzy differential equations

In this paper, 3-point Block Backward Differentiation Formulas (3BBDF) is used for the numerical solution of Fuzzy Differential Equations (FDEs). Implementation of 3BBDF using Newton iteration is discussed. Numerical results obtained by the 3BBDF are presented and compared with the Modified Simpson method to illustrate the ability of the 3BBDF method for solving FDEs

Variable order block method for solving second order ordinary differential equations

This paper proposed 2-point block backward differentiation formulas (BBDF) of order 3, 4, and 5 for direct solution of second order ordinary differential equations. These methods were derived via backward difference interpolation polynomial with two solutions are produced simultaneously at each step. All the three different orders of 2-point BBDF is implemented in variable order scheme. The scheme utilizes the local truncation error, which is generated by the single order of 2-point BBDF method. Numerical results are presented to illustrate the validity of the proposed scheme

Variable order block method for solving second order ordinary differential equations

This paper proposed 2-point block backward differentiation formulas (BBDF) of order 3, 4, and 5 for direct solution of second order ordinary differential equations. These methods were derived via backward difference interpolation polynomial with two solutions are produced simultaneously at each step. All the three different orders of 2-point BBDF is implemented in variable order scheme. The scheme utilizes the local truncation error, which is generated by the single order of 2-point BBDF method. Numerical results are presented to illustrate the validity of the proposed scheme