Parallel implementation of fourth order block backward differentiation formulas for solving system of stiff ordinary differential equations

Abstract

Parallel implementation of fourth order Block Backward Differentiation Formulas (BBDF(4)) is presented for a numerical solution of first order stiff ordinary differential equations (ODEs). This method computes the numerical solution at two points concurrently in each block. The sequential and parallel codes are developed on Message Passing Interface (MPI) library and run on High Performance Computing (HPC). The performance of this new method is measured in terms of Speedup (Sp) and Efficiency (Ef). It is shown that the parallel implementation BBDF(4) produced a better speed up compared to the sequential timing.

Variable step 2-point block backward differentiation formula for index-1 differential algebraic equations

“…First, the block BDF method for ODEs 12 is presented and then the extension of the method for semi-explicit index-1 DAEs is explained.…”

“…In the derivation of the method for ODEs 12 , three back values, namely, y n−2 , y n−1 and y n , are used to compute two new values, y n+1 and y n+2 , simultaneously at each step. The step size between the back values and current values are qh and q, respectively, where q is the step size ratio.…”

“…The aim of this paper is to obtain two solution values of index-1 DAEs simultaneously using a variable step size block BDF method. A method for ODEs is used 12 . Unlike the conventional BDF methods, the proposed method has the advantage of computing more than one solution value per step, using three previously computed back values.…”

Stability Analysis of the 2-Point Diagonally Implicit Super Class of Block Backward Differentiation Formula with Off-Step Points

“…However, some numerical methods developed for solving eqn. (1) have been introduced in Ababneh et al [1], Abasi et al [2,3], Babangida et al, Dhalquist [4,5], Curtiss and Hirschfelder [6], Cash [7], Ibrahim et al [8][9][10][11], Musa et al [12][13][14][15][16][17][18][19], and Suleiman, et al [20] among others. According to researchers the stability problem appears to be the most serious limitation of block methods.…”

A New Numerical Method for Solving Stiff Initial Value Problems

“…This behaviour makes it difficult to develop suitable methods for solving stiff problems. However, efforts have been made by researchers, such as Abasi [4], Alt [5], Alvarez [6], Cash [7], Dahlquist [1], Ibrahim [8][9][10], Musa [11][12][13][14], Suleiman [2,3], Yatim [15] and Zawawi [16] among others, to develop methods for stiff ODEs. The need to obtain an efficient numerical approximation in terms of accuracy and computational time have attracted some researchers such as Alexander [17] with diagonally implicit Runge-Kutta for stiff ODEs, Ababneh [18] with design of new diagonally implicit Runge-Kutta for stiff problems, Ismail [19] with embedded pair of diagonally implicit Runge-Kutta for solving ODEs, Zawawi [20] with diagonally implicit block backward differentiation formulas for solving ODEs.…”